500 years of NOT teaching THE CUBIC FORMULA. What is it they think you cant handle? @Mathologer

Mathologer Why is it that, unlike with the quadratic formula, nobody teaches the cubic formula? After all, they do lots of polynomial torturing in schools and the discovery of the cubic formula is considered to be one of the milestones in the history of mathematics. It's all a bit of a mystery and our mission today is to break through this mathematical wall of silence! Lots of cubic (and at the very end quartic) surprises ahead.

A great starting point for further exploration of this topic is this wiki page:

en.wikipedia.org/wiki/Cubic_function

Closest to what I do in this video is this paper

A New Approach to Solving the Cubic: Cardan's Solution Revealed Author(s): R. W. D. Nickalls, The Mathematical Gazette, Vol. 77, No. 480 (Nov., 1993), pp. 354-359

Here is a writeup of the great feud Tartaglia v. Cardano (minus all the made up bits).

arxiv.org/abs/1308.2181

Tartaglia's poem

maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-poem

Here is a writeup of a way of solving the cube by completing the cube (not so easy to motivate as what I've got in the video):

http://mathforum.org/dr.math/faq/faq.cubic.equations2.html

Fun fact 1:
en.wikipedia.org/wiki/Cubic_function#Collinearities

Fun fact 2:
en.wikipedia.org/wiki/Cubic_function#Three_real_roots

Fun fact 3 (Marden's theorem)
maa.org/press/periodicals/loci/joma/the-most-marvelous-theorem-in-mathematics

Extra Superman commented: At 3:12, the cubic equation that you choose is in one of two infinite families.The first one: for odd n, x^3 - 3nx - (n^3+1). The second one: for odd n, x^3 + 3nx - (n^3-1).

Thank you very much to Marty for all his help with polishing the presentation and Andrea for his help with pronouncing all those Italian words.

Enjoy :)

P.S. For some places that sell the t-shirts that I am wearing today google "cube root t-shirt" and "square root t-shirt"
The music is Morning Mandolin by Chris Haugen youtu.be/i8fH6la-bJQ from the free YouTube audio library

Two ways to support Mathologer
Mathologer Patreon: patreon.com/mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)

14. Sep. 2021: Thank you very much Michael Didenko for your Russian subtitles.

updated 5 years ago

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you cant handle?

Mathologer 2019-08-24 | Why is it that, unlike with the quadratic formula, nobody teaches the cubic formula? After all, they do lots of polynomial torturing in schools and the discovery of the cubic formula is considered to be one of the milestones in the history of mathematics. It's all a bit of a mystery and our mission today is to break through this mathematical wall of silence! Lots of cubic (and at the very end quartic) surprises ahead.

A great starting point for further exploration of this topic is this wiki page:

en.wikipedia.org/wiki/Cubic_function

Closest to what I do in this video is this paper

A New Approach to Solving the Cubic: Cardan's Solution Revealed Author(s): R. W. D. Nickalls, The Mathematical Gazette, Vol. 77, No. 480 (Nov., 1993), pp. 354-359

Here is a writeup of the great feud Tartaglia v. Cardano (minus all the made up bits).

arxiv.org/abs/1308.2181

Tartaglia's poem

maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-poem

Here is a writeup of a way of solving the cube by completing the cube (not so easy to motivate as what I've got in the video):

http://mathforum.org/dr.math/faq/faq.cubic.equations2.html

Fun fact 1:
en.wikipedia.org/wiki/Cubic_function#Collinearities

Fun fact 2:
en.wikipedia.org/wiki/Cubic_function#Three_real_roots

Fun fact 3 (Marden's theorem)
maa.org/press/periodicals/loci/joma/the-most-marvelous-theorem-in-mathematics

Extra Superman commented: At 3:12, the cubic equation that you choose is in one of two infinite families.The first one: for odd n, x^3 - 3nx - (n^3+1). The second one: for odd n, x^3 + 3nx - (n^3-1).

Thank you very much to Marty for all his help with polishing the presentation and Andrea for his help with pronouncing all those Italian words.

Enjoy :)

P.S. For some places that sell the t-shirts that I am wearing today google "cube root t-shirt" and "square root t-shirt"
The music is Morning Mandolin by Chris Haugen youtu.be/i8fH6la-bJQ from the free YouTube audio library

Two ways to support Mathologer
Mathologer Patreon: patreon.com/mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)

14. Sep. 2021: Thank you very much Michael Didenko for your Russian subtitles.

Ptolemy’s Theorem and the Almagest: we just found the best visual proof in 2000 years

Mathologer 2024-09-07 | We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.

00:00 Introduction
04:27 Geometry 101
08:19 Applications
14:46 Ptolemy's inequality
18:34 LIES
25:35 Animated proofs
28:57 Thank you!
30:53 Degenerate Easter Egg

There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. jstor.org/stable/10.4169/math.mag.87.4.291
Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here youtu.be/p-0SOWbzUYI?si=GeGzZ0R_Dj1AsXqR&t=371

You can find a couple of full text versions of the Almagest here
wilbourhall.org/index.html#ptolemy
classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf

For more background info check out the very comprehensive wiki pages on:
Ptolemy’s theorem
en.wikipedia.org/wiki/Ptolemy%27s_theorem
Ptolemy’s inequality
en.wikipedia.org/wiki/Ptolemy%27s_inequality
Claudius Ptolemy
en.wikipedia.org/wiki/Ptolemy
The Almagest
sco.wikipedia.org/wiki/Almagest
Trigonometric identities
en.wikipedia.org/wiki/List_of_trigonometric_identities
Cyclic quadrilateral
en.wikipedia.org/wiki/Cyclic_quadrilateral
The optic equation
en.wikipedia.org/wiki/Optic_equation

There are very interesting higher-dimensional versions of Ptolemy's theorem just like there are higher-dimensional versions of Pythagoras theorem. I did not get around to talking them today. Google ...

Highly recommended:
T. Brendan, How Ptolemy constructed trigonometry tables, The Mathematics Teacher 58 (1965), pp. 141-149 jstor.org/stable/27967990

Tom M. Apostol, Ptolemy's Inequality and the Chordal Metric, Mathematics Magazine 40 (1967), pp. 233-235 jstor.org/stable/2688275

demonstrations.wolfram.com/PtolemysTableOfChords an interactive exploration of Ptolemy's table of chords

Ptolemy's theorem made a guest appearance in the the previous Mathologer video on the golden ratio: youtu.be/cCXRUHUgvLI

Here is a nice trick to make Ptolemy counterparts of Pythagorean triples. Take any two sets of Pythagorean triples:
5² = 3² + 4², 13² = 12² + 5², and combine them like this:
65² = 13² × 5²= 13²(4² + 3²) = 52² + 39²= 5²(12² + 5²) = 60² + 25².
Now combining the two right angled triangles 52-39-65 and 25-60-65 along the common diagonal in any of four different ways gives a convex quadrilateral with all sides integers. Note that you automatically get 5 integer lengths and then Ptolemy's theorem guarantees that the remaining side is a fraction. Scaling up everything by the denominator of that fraction gives one of the special integer-everywhere quadrilaterals. See also Brahmagupta quadrilaterals.

Here is a nice application of Ptolemy's theorem to a International Maths Olympiad problem youtube.com/watch?v=NHjtHOE1lks

In a cyclic quadrilateral the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points: geogebra.org/m/XQr5jJQg This extension of Ptolemy's theorem is part of the thumbnail for this video.

T-shirt: cowsine :)
Music: Floating branch by Muted and I promise by Ian Post.

Enjoy,

burkard

Way beyond the golden ratio: The power of AB=A+B (Mathologer masterclass)

Mathologer 2024-08-03 | Today's mission: saving another incredible discovery from falling into oblivion: Steinbach's amazing infinite family of counterparts of the golden ratio discovered around 1995. Lot's of my own little discoveries in this one :)

00:00 Intro
05:53 Ptolemy
09:18 Perfect cut
16:01 Golden rectangle
22:03 Fibonacci
33:07 A+B=AB
45:48 Images and music
47:27 Thank you!

The slide show for this video is made up of a new record of 750 slides!

Peter Steinbach's papers articles:
Sections beyond golden:
archive.bridgesmathart.org/2000/bridges2000-35.html
Golden fields: a case for the heptagon:
jstor.org/stable/2691048

A good online writeup with some extra insights (on a site dedicated to sacred geometry!:
tinyurl.com/4jhju7dw

A paper citing Peter Steinbach's paper
tinyurl.com/48vcmfdn by Scott Vorthmann, David Hall, and David Richter. Vorthmann also wrote the software vZome, which is an emulator of the Zometool construction system. The original Zometool is based on phi. However Vorthmann also added a special mode in vZome based on the heptagonal field. See also this Jupiter notebook tinyurl.com/bduzayr3

Alan H. Schoen's incredible infinite tiling site. For anybody who wants to explore some heptagonal Penrose rhombus tiling counterparts.

Also, check out the very good wiki pages dedicated to the golden ratio and the Fibonacci numbers.

The wiki page on Ptolemy's theorem features a great visual animated proof en.wikipedia.org/wiki/Ptolemy

Some relevant Mathologer videos:
The golden ratio spiral: visual infinite descent: youtu.be/ubHVK71F01M
Phi and the TRIBONACCI monster
youtu.be/e7SnRPubg-g
The fabulous Fibonacci flower formula
youtu.be/_GkxCIW46to
Infinite fractions and the most irrational number
youtu.be/CaasbfdJdJg

Golden ratio fact and fiction. Check out this paper by Georg Markowsky:
goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf

For more on this also check out the book: The golden ratio by Mario Livio

I am collecting a whole pile of other interesting bits and pieces that did not get mentioned in the video and/or popped up in the comments in my post pinned to the top of the comment section of this video.

Some questions for you to while your time away.
1. In the 3D golden spiral the left-over golden boxes converge to a point on one of the edges of the golden box we start with. In what ratio does this point divide the edge?
2. Which points in a golden rectangle can you reach by cutting off infinitely many squares as in the golden spiral construction? How about in 3d?
3. Nut out some details for the nonagon. What's Binet's formula in that case?
4. For which complex numbers n does Binet's formula spit out an integer/a real number?
5. Is it a coincidence that there is a 1/7 in my Binet's formula for the heptagon? (You can make the 1/5 th appear in Binet's formula itself by multiplying both denominator and numerator by phi +1/phi.)

T-shirt: Fibonightmare
teepublic.com/t-shirts?query=Fibonightmare

Music: Kashido - When you go out and Ardie Son - Spread your wings

Enjoy!

Burkard

PETRS MIRACLE: Why was it lost for 100 years? (Mathologer Masterclass)

Mathologer 2024-06-08 | Today’s topic is the Petr-Douglas-Neumann theorem. John Harnad told me about this amazing result a couple of weeks ago and I pretty much decided on the spot that this would be the next Mathologer video. I really had a lot of fun bringing this one to life, maybe too much fun :)

Very good Wiki page on the Petr–Douglas–Neumann theorem en.wikipedia.org/wiki/Petr-Douglas-Neumann_theorem

Napoleon's theorem and the PDN theorem at Cut-the-knot cut-the-knot.org/ctk/Napolegon.shtml

Petr's original paper https://dml.cz/handle/10338.dmlcz/120936

Worth a look for eigenpolygon decomposition and other application within geometry, etc.
https://forumgeom.fau.edu/FG2015volume15/FG201527.pdf
https://forumgeom.fau.edu/FG2009volume9/FG200920.pdf
researchgate.net/publication/265706727_A_harmonic_analysis_of_polygons

A fantastic app that does the eigenpolygon decomposition contributed by Steven De Keninck from the Computer Vision Group at the University of Amsterdam
enki.ws/ganja.js/examples/coffeeshop.html#AONlCLF1r&fullscreen

Check out Branko Grünbaum notes on "Modern Elementary Geometry" tinyurl.com/5av8jwnk and G.C Shephard's "Sequences of smoothed polygons" (paywalled) for how all this fits in the grand scheme of things.

John Harnad's YouTube video on a complete proof of the Petr-Douglas-Neuman theorem. youtube.com/watch?v=sZz7aAo6eAs
Here is John's (slightly updated) write-up that goes with his video http://www.qedcat.com/PDN_theorem_ext1.pdf
If you like this Mathologer video thank John, he lobbied for it :) John's website: http://www.crm.umontreal.ca/~harnad

I put a couple of Geogebra Geometry and Mathematica apps that made for this video in this folder http://www.qedcat.com/petr

Here is Geogebra Geometry online geogebra.org/geometry?lang=en-AU (you can also download a standalone version for free)

Nice geogebra classic animation by Ron Vanden Burg
geogebra.org/classic/sdftbe2y He chooses a different but equivalent approach to Petr, Douglas and Neumann. He attaches regular n-gons instead of the ears. Instead of connecting the tips of the ears, he connects the centers of the added regular n-gons. And instead of using ears with different angles each round, he adds the regular n-gons where each round has a different distribution of vertices that get to each side of the edge. It has a slider to switch from triangle (n=3) to decagons (n=10) and there is a play button to run through the different stages. Here is an updated version geogebra.org/classic/ztvrbdyy

Another really nice app by Christian Baune
programaths.com/petr

Andrew put together the following quick “howto” for tools in geogebra - a tool is a sequence of construction steps that you can reuse (can speed up constructions tremendously) . youtu.be/Vpj0SDYPd6A
The file he is working on in his video is here
geogebra.org/classic/zrzqed4s

A YouTube video animating some instances of PDN
youtube.com/watch?v=bpmNmAd_uL0

Eigenpolygon Decomposition of Polygons the Microsoft technical report by Pixar founder Alvy Ray Smith http://alvyray.com/Memos/CG/Microsoft/19_eigen.pdf
Alvy Ray's website http://alvyray.com

Nice applications:
In electrical engineering: en.m.wikipedia.org/wiki/Symmetrical_components
Check out the paragraph entitled "Intuition" for an explicit reference to Napoleons theorem.
Being able to find the center of mass of a polygon is another nice application in itself.

Nice remarks:
For a digon we are attaching 2-2=0 360/2-gons to arrive at ... the same digon ... which is automatically regular :)
Adding a 180-degree ear to a segment is the same as bisecting this segment. So the tip of the ear ends up in the middle of the segment.

Visualisation challenges: I sort of had it going in Mathematica just for 10-gons. An app that allows you to pick the vertices of a closed polygon on a canvas and then calculates the intermediate polygons. One problem with the intermediate polygons is that for acute angles the displacement is large and so that can quickly lead to the intermediate polygons growing too large for your canvas. Some rescaling is probably the way to go. Alternatively, since the end result is always the same no matter the order, it makes sense to apply angles in complementing pairs, jump out and in, and only show every second stage of the evolution. Maybe some app that allows to input a smooth curve and then allows to experiment with different polygon approximations to see whether we get some convergence. The decomposition into the special types is a great one to animate. If you’ve got Mathematica I’ve included what I got up to in the file directory I link to above.

Music: A tender heart by the David Roy Collective and Trickster by Ian Post (two slightly different versions)

T-shirt: Rock Paper Scissors Lizard Spock t-shirt (google it, lots of different versions)

Enjoy!

Burkard

Conways IRIS and the windscreen wiper theorem

Mathologer 2024-04-06 | Conway's whatever ... it's named after John Conway and so it must be good :)

Wiki page dedicated to John Conway
en.wikipedia.org/wiki/John_Horton_Conway

Wiki page Conway's circle
en.wikipedia.org/wiki/Conway_circle_theorem

Wiki page on his Game of Life
en.wikipedia.org/wiki/Conway%27s_Game_of_Life

Michael de Villiers (the connection with the windscreen wiper theorem, read this one first)
http://dynamicmathematicslearning.com/conway-circle-theorem-special-case-side-divider-theorem.pdf
http://dynamicmathematicslearning.com/conway-circle-as-special-side-divider-theorem.html
Also check out Michael's main page http://dynamicmathematicslearning.com/homepage4.html

Matt Baker's tribute blog to John Conway
https://mattbaker.blog/2020/04/15/some-mathematical-gems-from-john-conway/#more-4162

Tanya Khovanova's blog post on Conway's circle showing Conway's wearing a Conway circled Mathcamp t-shirt
blog.tanyakhovanova.com/2010/06/conways-circle

Colin Beveridge's proof without words
aperiodical.com/2020/05/the-big-lock-down-math-off-match-14

Colin Beveridge and Elizabeth A Wilson (his proof without words explained :)
aperiodical.com/2020/06/conways-circle-theorem-a-proof-this-time-with-words

Paul Farrel's beautiful animated proof
youtu.be/MvLBeOIPZ08

Me on Conway's circle and Conway's pinwheel tiling for the Gathering of Gardner
youtu.be/VrXnwmyxylg

Card Colm on Conway and his circle
cardcolm.org/JHC.html
cardcolm.org/Pics/RybaConwayCircle.pdf

Eric Braude
Conways Circle Theorem: A Short Proof Enabling Generalization to Polygons
arxiv.org/abs/2111.01835

Mathologer video on curves of constant width
New Reuleaux Triangle Magic
youtu.be/-eQaF6OmWKw

Mathologer videos on Conway's proof of Morley's miracle
Math is Illuminati confirmed
youtu.be/DfnBW6HvNwM
Part 2
youtu.be/gjhmh3yWiTI

Music: Rain by ANBR and Ethereal Ottom
T-shirt: ages old, don't remember where I got this one from

Enjoy!

Burkard

Simple yet 5000 years missed ?

Mathologer 2024-02-24 | Good news! You really can still discover new beautiful maths without being a PhD mathematician.

Stumbled across this one while working on the magic squares video. Another curious discovery by recreational mathematician Lee Sallows. A simple and beautiful and curious fact about triangles that, it appears, was first discovered only 10 years ago. Really quite amazing that this one got overlooked, considering the millennia old history of triangles.

Wiki page dedicated to Lee Sallows
en.wikipedia.org/wiki/Lee_Sallows
His personal homepage
leesallows.com
The relevant subpage
http://tinyurl.com/y6tzsbjt

t-shirt: teepublic.com/t-shirt/300445-the-impossible-board-game
music: Campagna - Adventure of a Lifetime

Enjoy!

Burkard

Whats the curse of the Schwarz lantern?

Mathologer 2023-12-23 | Second coop with Andrew. This time it's about the Schwarz lantern a very famous counterexample to something that mathematicians believed to be obviously true. A 3D cousin of the famous pi = 4 paradox.

00:00 Intro
00:39 Troll math: the pi=4 meme
02:25 Archimedes chops off corners
05:51 Archimedes boxing of pi
07:40 Schwarz lantern
16:59 Area formula
17:12 Schwarz pi = 4 memes
20:17 Folding flat
21:12 Andrew's VR experiment
22:24 Soda can --- lantern
23:26 Thanks
25:00 Andrew's Christmas tree

The pi = 4 troll math meme on reddit:
reddit.com/r/math/comments/e6q4r/troll_math_pi_4_crosspost

Good discussion by GoldPlatedGoof
static.nsta.org/pdfs/QuantumV1N4.pdf
The Vi Hart video:
youtu.be/D2xYjiL8yyE

Closely related, the Staircase paradox:
en.wikipedia.org/wiki/Staircase_paradox

Good discussion of the history of this sort of paradox
http://tinyurl.com/3pe2eaav

Great wiki page on the Schwarz lantern:
en.wikipedia.org/wiki/Schwarz_lantern

Schwarz's original paper:
archive.org/stream/gesammeltemathem02schwuoft#page/310/mode/2up

Diamond-Cut Can
toyo-seikan.co.jp/e/technology/can/decorationshape/diamondcut

Lots of nice ideas (coffee cans, soda can creasing, collapsing cylinders, patent for effectively collapsing cans)
beachpackagingdesign.com/boxvox/pseudo-cylindrical-concave-polyhedral-packaging

The IKEA looking lantern features in the paper: Application of paper folding technique to three-dimensional space sound absorber with permeable membrane: Case studies of trial productions
http://tinyurl.com/223948tz

Great article in Quantum magazine (there is also an article about Gabriel's horn in the same issue):
static.nsta.org/pdfs/QuantumV1N4.pdf

Andrews Geogebra app in action:
Schwarz Lantern in Geogebra 3D & AR: youtu.be/YyKX00VukFw
Schwarz Lantern christmas tree: youtu.be/Ay1s4hizg1o
Geogebra 3D AR mode on an iPad how-to: youtu.be/yQAiBxTGbAg

Schwarz lantern origami with Deepali Karanjavkar:
youtu.be/ar7dMtsA-dY

Crushmetric water bottle (really impressive can crushing inspired toys):
crushmetric.com

Hydraulic press vs. metal pipe (square and circular)
youtube.com/watch?v=TM5dyY8zfxs

Mel Brooks: "May the Schwarz be with you"
youtube.com/watch?v=-6cho-gsaZM

Music today: Flat Rock by Bennett Sullivan
T-shirt: Google hohoho cubed t-shirts for lots of different versions ((ho)³ A case where (ab)³ ≠ a³b³ :)

Enjoy!

Burkard

Why are the formulas for the sphere so weird? (major upgrade of Archimedes greatest discoveries)

Mathologer 2023-11-25 | In today’s video we’ll make a little bit of mathematical history. I'll tell you about a major upgrade of one of Archimedes' greatest discoveries about the good old sphere that so far only a handful of mathematicians know about.

00:00 Intro to the baggage carousel
01:04 Archimedes baggage carousel
04:26 Inside-out animations
04:59 Inside-out discussion
10:38 Inside-out paraboloid
12:43 Ratio 3:2
13:28 Volume to area
18:40 Archimedes' claw
20:55 Unfolding the Earth
29:43 Lotus animation
30:38 Thanks!

Those fancy conveyor belts are called a crescent pallet conveyors, and sometimes "sushi conveyors" because they were originally designed for carrying sushi plates. en.wikipedia.org/wiki/Conveyor_belt_sushi Andrew also dug up an American patent dating back to 1925 patents.google.com/patent/US1757652A/en

Great wiki page on Archimedes
en.wikipedia.org/wiki/Archimedes

In "On the sphere and the cylinder"
en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder
Archimedes derives the volume and area formulas for the sphere. The proofs used in this work are quite complicated and conform to what was acceptable according to Greek mathematics at the time. His original original ingenious proof most likely involved calculus type arguments. Marty and I wrote about this here qedcat.com/archive_cleaned/99.html and here qedcat.com/archive_cleaned/100.html Also check out this page en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems

Why is the formula for the surface area the derivative of the volume formula? Easy:
V'(r) = dV/dr = A(r) dr / dr = A(r).
A nice discussion of the onion proof on this page I'd say check out the discussion of the onion proof on this page en.wikipedia.org/wiki/Area_of_a_circle
B.t.w. this works in all dimensions the derivative of the nD volume formula is the nD "area" formula. en.wikipedia.org/wiki/Volume_of_an_n-ball

Wiki page on Cavalieri's principle
en.wikipedia.org/wiki/Cavalieri%27s_principle
Includes both hemisphere = cylinder - cone and paraboloid = cylinder - paraboloid
Video on the volume of the paraboloid using Cavalieri by Mathemaniac
youtube.com/watch?v=U2_nvfWIAYE

Henry Segerman: en.wikipedia.org/wiki/Henry_Segerman
Henry's video about his 3d printed Archimedes claw: youtu.be/KD_hRn_97RI
Henry's 3d printing files: printables.com/model/651714

Andrew Kepert: https://www.newcastle.edu.au/profile/andrew-kepert
Andrew's playlist of spectacular video clips complementing this Mathologer video:
youtube.com/playlist?list=PL9JP5WCX_XJY9GmMO-kotRR5bRYOPOtn9
All of Andrew's animations featured in this video plus a few more (actual footage of a fancy baggage carousel in action, alternative proof that we are really dealing with a cylinder minus a cone, paraboloid inside-out action, inside-out circle to prove the relationship between the area and circumference of the circle, etc.)

There is one thing (among quite a few) that I decided to gloss over at the end of the video but which is worth noting here. At the end it’s not straight Cavalieri. Before you apply Cavalieri, you also need to put some extra thought into figuring out why the flat moon that runs along the semicircular meridian can be straightened out into something that has the same area (straighten meridian spine with interval fishbones at right angles). Here I was tempted to include a challenge for people to figure out why the red and blue surfaces in the attached screenshot have the same area: qedcat.com/ring.jpg

Funniest comment: Historians attempting to reconstruct the Claw of Archimedes have long debated how the weapon actually worked. The sources seem to have trouble describing exactly what it did, and now we know why. Turns out it was a giant disc that slid beneath the waters of a Roman ship, then raised countless eldritch crescents which inexplicably twisted into a sphere, entrapping the vessel before dragging it under the waves, all while NEVER LEAVING ANY GAPS in the entire process. No escape, no survivors, fucking terrifying. No wonder that Roman soldier killed Archimedes in the end, against the Consul's orders. Gods know what other WMDs this man would unleash on the battlefield if he were allowed to draw even one more circle in the sand. The Roman marines probably had enough PTSD from circles.

T-shirt: One of my own ones from a couple of years ago.
Music: Taiyo (Sun) by Ian Post

Enjoy!

Burkard

P.S.: Thanks you Sharyn, Cam, Tilly, and Tom for your last minute field-testing.

Is this a paradox? (the best way of resolving the painter paradox)

Mathologer 2023-10-21 | The painter's paradox, a.k.a. Gabriel's horn paradox a.k.a. Torricelli's horn paradox has been done to death on YouTube. So why do it again? Well, being all about some remarkable features of 1/x, this topic nicely complements the previous two videos that were also dedicated to 1/x. Now the first Mathologer trilogy is complete! Also, I thought of a couple of nice twists to make this treatment of the painter's paradox really stand out from the crowd (I hope :)

00:00 Intro
02:59 What's that horn?
07:09 A 700 year old trick
12:26 What's the volume exactly?
16:33 What paradox?
18:17 Fun fact.
20:12 Thanks

Must check out.
Wiki page on the horn: en.wikipedia.org/wiki/Gabriel%27s_horn
(also has a nice discussion of the converse of the paradox at the end)

Wiki page on Torricelli: en.wikipedia.org/wiki/Evangelista_Torricelli
(the anagram that featured in the video also gets a mention on this page)

Very nice write-up of the history of the horn by Paolo Mancosu and Ezio Vailati jstor.org/stable/233514 tinyurl.com/3rff7ubb

Wiki page on Oresme: en.wikipedia.org/wiki/Nicole_Oresme
(has a page from one of his books with some graphs. Somehow overlooked this one. Would have been nice to flash this in the video :(

Stanford Encyclopedia of Philosophy entry on Nicole Oresme (in particular, see the section on math) https://plato.stanford.edu/entries/nicole-oresme/#maths

Wiki page on the Method of shells: en.wikipedia.org/wiki/Shell_integration

Fun/intersting: The opposite of this is a vuvuzela, which is a horn with finite surface but infinite volume (as in loud)

One interesting observation is that both area and volume being finite or infinite is independent of what unit we use. Therefore it does make sense to compare finite/infinite volumes and surface areas of shapes. And one interesting observation in this respect is that solids of infinite volume and finite surface area don't exist. On the other hand, that also means that areas and volumes scaling differently is not part of the resolution of our paradox which only depends on the volume being finite and the surface area being infinite.

Music at the end: Young rich pixies - Year of life
T-shirt: TONEWAVE Music Tee Geometric Overtone Frequency Musician Sacred Geometry Tshirt etsy.com/ca/listing/212767891/unisex-tonewave-music-tee-geometric

Enjoy!

Burkard

The best A – A ≠ 0 paradox

Mathologer 2023-09-09 | This video is about a new stunning visual resolution of a very pretty and important paradox that I stumbled across while I was preparing the last video on logarithms.

00:00 Intro
00:56 Paradox
03:52 Visual sum = ln(2)
07:58 Pi
11:00 Gelfond's number
14:22 Pi exactly
17:35 Riemann's rearrangement theorem
22:40 Thanks!

Riemann rearrangement theorem.
en.wikipedia.org/wiki/Riemann_series_theorem This page features a different way to derive the sums of those nice m positive/n negative term arrangements of the alternating harmonic series by expressing H(n) the sum of the first n harmonic numbers by ln(n) and the Euler–Mascheroni constant. That could also be made into a very nice visual proof along the lines that I follow in this video youtu.be/vQE6-PLcGwU?si=iWTasKqo_JFn4etG&t=1321.

Gelfond's number
e^π being approximate equal to 20 + π may not be a complete coincidence after all:
@mathfromalphatoomega
There's actually a sort-of-explanation for why e^π is roughly π+20. If you take the sum of (8πk^2-2)e^(-πk^2), it ends up being exactly 1 (using some Jacobi theta function identities). The first term is by far the largest, so that gives (8π-2)e^(-π)≈1, or e^π≈8π-2. Then using the estimate π≈22/7, we get e^π≈π+(7π-2)≈π+20. I wouldn't be surprised if it was already published somewhere, but I haven't been able to find it anywhere. I was working on some problems involving modular forms and I tried differentiating the theta function identity θ(-1/τ)=√(τ/i)*θ(τ). That gave a similar identity for the power series Σk^2 e^(πik^2τ). It turned out that setting τ=i allowed one to find the exact value of that sum.
(@kasugaryuichi9767) I don't know if it's new, but it's certainly not well known. To quote the Wolfram MathWorld article "Almost Integer": "This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" e^π-π≈20 is true has yet been discovered."

Ratio of the number of positive and negative terms
It's interesting to look at the patterns of positive & negative terms when rearranging to Pi. We always only use one negative term before we switch. The first ten terms on the positive side are: 13, 35, 58, 81, 104, 127, 151, 174, 197, 220,... If you look at the differences between terms, you get: 22, 23, 23, 23, 23, 24, 23, 23, 23, 23, 23, 23, 23, 24,...
The reason for this is that Gelfond's number is approximately equal to 23. It turns out that if an arrangement of our series has the sum pi, then the ratio of the numbers of positive to negative terms in the finite partial sums of the series converges to Gelfond's number. This is just one step up from what I said about us being able to get arbitrarily close to pi by turning truncations of the decimal expansion of Gelfond's number into fractions. Similarly for other target numbers. For example, to predict what the repeating pattern for e is, you just have to calculate e^e :)

@penguincute3564 thus ln(0) = negative infinity (referring to +0/1-)

Bug report: At the 1:18 mark, I say minus one sixth when I should have just said one sixth.

Music: Silhouettes---only-piano by Muted
T-shirt: Pi Day Left Vs Right Brain Pie Math Geek T-Shirt tinyurl.com/3e3p5yeb

Enjoy!

Burkard

Why dont they teach simple visual logarithms (and hyperbolic trig)?

Mathologer 2023-08-05 | Simple visual logarithms. Is there such a thing? You bet :)

00:00 Intro
01:59 Rubik's cube and drill
03:26 What's the area?
05:15 Sum of 1+1/2+1/3+...
06:35 Mystery sum
11:32 What base?
17:25 What is Log_b(x)?
22:14 Is this a circle?
28:53 Proof that e^a = cosh(a) + sinh(a)
30:50 Thanks

Maths of spinning cubes by Lingguo Bu
tinyurl.com/yvp5udm6
Great history of hyperbolic functions by Janet Heine Barnett: maa.org/sites/default/files/321922717729.pdf.bannered.pdf
Very good wiki summary of hyperbolic function maths. Check out in particular the section "Comparison with circular functions" and "Relationship to the exponential function"
en.wikipedia.org/wiki/Hyperbolic_functions
Squeeze mapping and Lorentz transformation
en.wikipedia.org/wiki/Squeeze_mapping
Minute Physics video "Lorentz Transformations | Special Relativity Ch. 3"
youtu.be/Rh0pYtQG5wI
Stackexchange discussion of the different ways to pronounce the hyperbolic functions:
math.stackexchange.com/questions/176055/pronunciation-of-sinh-x-cosh-x-tanh-x-for-short

Bug alert: The 1 s here youtu.be/G0Fa5Zl-Z3c?t=1270 should be x s :(

Some neat comments: Call the hyperbolic sine and cosine "shine and coshine"
Hmm... the multiplication by halving and doubling looks awfully like Russian multiplication - never thought of a cross link to logarithms before (In the end quite different. Still neat observation.)
-An arc angle may pay you a visit if you sin too much :)

Music: Morning Mandolin by Chris Haugen
T-shirt. Google "Yes, I'm always right!" math t-shirt for many different versions of this t-shirt.

Enjoy!

Burkard

Ramanujans easiest hard infinity monster (Mathologer Masterclass)

Mathologer 2023-06-24 | In this masterclass video we'll dive into the mind of the mathematical genius Srinivasa Ramanujan. The focus will be on rediscovering one of his most beautiful identities.

00:00 Intro
02:48 How did his mind work?
09:12 What IS this?
15:11 Fantastic fraction
18:12 Impossible identity
23:38 Thanks!

This video was inspired by two 2020 blog posts by John Baez:
https://math.ucr.edu/home/baez/ramanujan/

Here are some links to selected Mathologer videos dealing with Ramanujan's mathematics:
Numberphile v. Math: the truth about 1+2+3+...=-1/12: youtu.be/YuIIjLr6vUA
How did Ramanujan solve the STRAND puzzle? youtu.be/V2BybLCmUzs
Ramanujan's infinite root and its crazy cousins: youtu.be/leFep9yt3JY

Check out the article "Inequalities related to the error function" by Omran Kouba for the nitty gritties about Ramanujan's infinite fraction: arxiv.org/abs/math/0607694v1
Further discussion of the error function: tinyurl.com/mu5vywsz
Another interesting stack exchange discussion: math.stackexchange.com/questions/1090857/bizarre-continued-fraction-of-ramanujan-but-wheres-the-proof
Survey of the problems that Ramanujan submitted to the Journal of the Indian Mathematical Society. See page 29 for a discussion of the identity that we talk about in this video. Also of interest in the problem discussed on page 30: https://faculty.math.illinois.edu/~berndt/jims.ps
This is the letter that Ramanujan sent to Hardy. Identity VII 6 is closely related to what we are talking about in this video qedcat.com/misc/ramanujans_letter.jpg
The answer to Ramanujan's challenge appeared in the February 1916 issue of the Indian Mathematical society (vol. VIII, no. 1, pp. 17–20) "Answer to Problem 541 by K.B. Madhava".

A couple of links and remarks about the "square root of the Wallis product":
Wiki page for the Wallis product: en.wikipedia.org/wiki/Wallis_product (among other things check out the discussion on the value of the derivative of the Riemann zeta function at 0 at the end of this page).
Mathologer video "Euler's infinite pi formula generator" has a proof for the Wallis product youtu.be/WL_Yzbo1ha4?t=465
Discussion on stackexchange of the asymptotic behaviour of the "square root" tinyurl.com/3yxyhjmp
Also check out the discussion in A. De Morgan, "On the summation of divergent series", The Assurance Magazine, and the Journal of the Institute of Actuaries, 12 (1865), pp. 245--252.
Here is a connection to the discussion of ways of associating meaningful values to certain divergent series in the Mathologer videos on 1+2+3+ "=" -1/12:
log (the product) "=" - log 1 + log 2 - log 3 + log 4 - log 5 + log 6 - ... = log 2 - log 3 + log 4 - log 5 + ....
and the last divergent series is known to have Cesaro sum log (pi/2)^(1/2). (essentially due to Euler, I think). See also exercise 207, page 515, in Knopp's book "Theorie and Anwendung der unendlichen Reihen", 2nd edition, Springer, 1924.

Obviously, in the last part of the video, when we plug x=0 into the infinite fraction, we just go for it a la Nike: "Just do it", (or a la Ramanujan: 1+.2+3+...=-1/12). Having said that, us ending up with root pi over 2 which is exactly what we want, is really too weird a number to pop up by coincidence. As I said in the video, to really pin down why our manipulations give the right answers is tricky. For example, we need a justification for the way I arrive at the 1/1/2/3/4... fraction in the first place. Usually infinite fractions are evaluated by first turning them into a sequence of partial fractions. Then the value of the infinite fraction, if it exists, is the limit of this sequence. The partial fractions result by truncating the infinite fraction at the plus signs. For many infinite fractions you get a different sequence having the the same limit by truncating at the fractions bars instead.

A very good book on infinite fractions featuring, among many other things, the Wallis product and the error function: Sergey Khrushchev, Orthogonal polynomials and continued fractions, Cambridge University press, 2008 (p.198, has a high-level proof for our infinite fraction in x rep. of the error function.)

Bug report:
1. At some point I copied and pasted the warm-up infinite series instead of Ramanujan's infinite series.22:42
2. Almost invisible: An "(x)" is hiding in Ramanujan's hair :) at youtu.be/6iTdNmDHfV0?t=913

T-shirt: I bought today's t-shirt many years ago. When I just looked online I could not find it anymore. However, there are many similar designs available. Just google "Paranormal distribution".
Music: Down the Valley by Muted
The infinity sign turning into two question marks animation is based on an illustration entitled "Infinitely Many Questions" by Roberto Fernandez. See page 76 of my book Eye Twisters.

Enjoy!

Burkard

Powell’s Pi Paradox: the genius 14th century Indian solution

Mathologer 2023-05-06 | Around 1400 there lived an Indian astronomer and mathematician by the name of Madhava of Saṅgamagrāma. He was the greatest mathematician of his time and, among other mathematical feats, he and his followers managed to discover a lot of calculus 200 years before Newton and Leibniz did their thing. While preparing a video about this Indian calculus it occurred to me that some of Madhava's discoveries can be used to give a nice intuitive explanation of Powell's Pi Paradox, a very counterintuitive property of the famous Leibniz formula

π/4=1–1/3+1/5–1/7+1/9–...

that Martin Powell stumbled upon in 1983. In the end, giving an introduction to Madhava's discoveries and giving that intuitive explanation is what I ended up doing in this video. ("Leibniz formula" should really be "Madhava formula"!)

00:00 Intro
00:35 Powell's Piradox :)
04:08 Calculus made in India
15:18 Explanation of the paradox using Madhava's correction terms
19:37 Calculus: Neither Newton nor Leibniz
24:22 Palm leaf music sequence
24:56 Thanks!

Videos in which I prove the Madhava formula:
Euler's infinite pi formula generator: youtu.be/WL_Yzbo1ha4?t=558
Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+... :youtu.be/00w8gu2aL-w
Euler's real identity NOT e to the i pi = -1: youtu.be/yPl64xi_ZZA?t=956

The Wikipedia articles about Madhava, his school and his discoveries are excellent starting points if you are interested in more details:
en.wikipedia.org/wiki/Madhava_of_Sangamagrama
en.wikipedia.org/wiki/Madhava%27s_correction_term
en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics
en.wikipedia.org/wiki/Madhava_series

My explanation of how Madhava may have discovered his correction terms is based on this article by Hayashi, T., T. Kusuba, and M. Yano. "The Correction of the Madhava Series for the Circumference of a Circle." Centaurus 33 (1990): 149-174. This article is sitting behind a paywall. However, the wiki article linked to above is a good summary.

The original article by Powell in which he reports on his observation and asks for an explanation is here: jstor.org/stable/3616550
Five explanations were subsequently given in this article published in the same math journal: jstor.org/stable/3617175 (note on JSTOR this collection of articles is broken up into four parts. This link is only to the first part).

The most in-depth article about the Powell's Pi Paradox is this one here by the Borwein brothers and K. Dilcher on "Pi, Euler Numbers, and Asymptotic Expansions": maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_4.pdf
In this article they also analyse similar paradoxical behaviours of closely related series like ln(2)=1-1/2+.1/3-1/4+1/5-...

The photo of that palm leaf manuscript page shown at the end of the video was sourced from the slideshow of the 2022 International Congress of Mathematicians invited lecture by K. Ramasubramanian. youtube.com/watch?v=ctrROj3Tv-E . Also check out his website for LOTS of information about ancient Indian mathematics. kramasubramanian.com I have no idea what it says on this palm leave page, but I trust my colleague to have shown us the right thing here :)

The picture of Madhava in the thumbnail is what Google is pushing. However, this image is not a true likeness of the actual person: commons.wikimedia.org/wiki/File:Madhava_sangamagrama.jpg

A couple more links to check out:
The Discovery of the "Series Formula for π by Leibniz, Gregory and Nilakantha" by Ranjan Roy: jstor.org/stable/2690896 Goes into a lot of detail in terms of proofs.
https://www.pas.rochester.edu/~rajeev/papers/canisiustalks.pdf

Some bugs:
3:36 one of the digit difference towards the end not highlighted
14:39 In the 121 terms sums the correction terms features a minus in the place of a plus.
18:36 In the fourth correction term it should be ...N+9/(4N)

Music: Adventure of a lifetime by Campagna

Enjoy!

Burkard

The Korean kings magic square: a brilliant algorithm in a k-drama (plus geomagic squares)

Mathologer 2023-02-04 | A double feature on magic squares featuring Bachet's algorithm embedded in the Korean historical drama series Tree with deep roots and the Lee Sallow's geomagic squares.

00:00 Intro
02:52 Part 1: The king's magic squares
09:40 Proof
18:22 The order 5 and 7 magic squares
19:17 Part 2: Geometric magic square
30:59 Thanks!

The Korean historical drama Tree with deep roots is available here viki.com/tv/1585c-tree-with-deep-roots All the magic square action takes place in episodes 1 and 2.

Episode 1: the king's study with magic squares at 33:17 again at 42:00 (father "simplifies" magic squares)
Episode 2: lunchbox action starts around 46:00 then again at 58:39 (the AHA moment)

Lee Sallows's book: Geometric magic squares, Dover (2013)
His website: leesallows.com
His comprehensive online gallery of stunning geomagic squares: geomagicsquares.com

Nice write-up about the 33x33 magic square in Tree with deep roots: tinyurl.com/mszxrf2w

Wiki page on Claude Gaspar Bachet de Méziriac: en.wikipedia.org/wiki/Claude_Gaspar_Bachet_de_Méziriac

Eduard Lucas 3x3 magic square equation: en.wikipedia.org/wiki/Magic_square#Special_methods_of_construction

An app by Ilm Narayana that demonstrates the king's method for magic squares up to order 33 (thank you very much Ilm for accepting my coding challenge:) editor.p5js.org/ilmnarayana/full/KBhql96F9

Bachet's magic square algorithm write-up humanicus.medium.com/bachets-magic-square-c00c8ae4d56f (this article by Humanicus features the same proof that I present in this video)

Here is a magic square from an old Chinese manuscript en.wikipedia.org/wiki/Magic_square#/media/File:Suanfatongzong-790-790.jpg Among other things they are writing from top to bottom which would also have been done in Korea at the time. In fact, in the drama the tech support ladies can be seen writing from top to bottom. So that's a bit of a blooper when it comes to the writing on the tiles. A few other minor issues are discussed in the comments. What's also interesting here that they went for a 33x33 magic square and 33x33=1089. 1089 is a four-digit number and all the tiles are only labelled with three numerals. How did they write 1089 and still make sense? :) Why not use a 31x31 magic square? 31x31=961.

For the second method for building odd order magic squares check out this link: en.wikipedia.org/wiki/Magic_square#A_method_for_constructing_a_magic_square_of_odd_order

Some bugs:
- at 18:48, there is no green circle on the 2nd row 3rd column square.
- at 23:00, I should have said: take any 3 or more of the numbers that add to 15, then the corresponding pieces combine into the 4x4 with the bite (important because, for example, 7 and 8 don't work).
- at 28:20 one of the pentominoes is a hexomino :)

Today's music: Ardie Son - Counterparts
Today's t-shirt: 31415... Cannot remember where I found this t-shirt.

Enjoy!

Burkard

Whats hiding beneath? Animating a mathemagical gem

Mathologer 2022-12-17 | There is a lot more to the pretty equation 10² + 11² + 12² = 13² + 14² than meets the eye. Let me show you.

00:00 Intro
00:07 Animated visual proofs
03:35 Mathologer materializes
06:31 Three puzzles
07:45 Thanks!

Notes:
The beautiful visual proof for the squares pattern is based on a note by Michael Boardman in Mathematics Magazine: tinyurl.com/2d4y7wtf

As far as I can tell, I am the first one to notice that this beautiful argument also works for those consecutive integer sums (but I am probably wrong :)

I first read about the two patterns that this video is about in the 1966 book Excursions in Number Theory by Ogilvy and Anderson (pages 91 and 92).

The article "Consecutive integers having equal sums of squares" J.S. Vidger, Mathematics Magazine, Vol. 38, No. 1 (Jan., 1965), pp. 35-42. is dedicated to finding generalisations of the sort of equations that the squares pattern is all about. Here is a particularly, nice example derived at the very end of this article: 4² + ... + 38² = 39² + ... + 48². This article is on JSTOR jstor.org/stable/2688015.

I first encountered the Russian painting that puzzle 2 is about in an article by Ethan Siegel about 10² + 11² + 12² = 13² + 14² and Co. tinyurl.com/y7p5k4kw Nice find :)

365 is the smallest integer that can be expressed as a sum of consecutive square in more than one way 365 = 10² + 11² + 12² = 13² + 14² (and of course 365 also happens to be the number of days in a year :) Viewer Exception2001: Knowing the result, it's fun to think about making an efficient one-page calendar where the front is a 13x13 square and the back is a 14x14 square, with each square containing a date :D

Viewer k k notes that consecutive squares also take care of leap years :) 8² + 9² + 10² + 11² = 366

Christofer Hallberg did some computer experiments and found the following beautiful equation: 4³+...+28³=30³+31³+32³+33³+34³

There are some nice families of equations involving sums of alternating sums of consecutive squares. Check out Roger Nelsen's one glance proof tinyurl.com/2xauf83u
2² - 3² + 4² = -5² + 6²
4² - 5² + 6² - 7² + 8² = -9² + 10² -11² + 12²
...
Fun fact: the top part of the logo is the top part of the last image I show in the previous video youtu.be/94mV7Fmbx88?t=2547

Several viewers (Exception1, Nana Macapagal, B Smith, Shay) noticed that the projected cubes pattern differences are of the form n²(n+1)²/2 = 2(1 + 2 + 3 + ... + n)².
5³ + 6³ = 7³ - 2
16³ + 17³ + 18³ = 19³ + 20³ - 18
33³ + 34³ + 35³ + 36³ = 37³ + 38³ + 39³ - 72
56³ + 57³ + 58³ + 59³ + 60³ = 61³ + 62³ + 63³ + 64³ - 200
85³ + 86³ + 87³ + 88³ + 89³ + 90³ = 91³ + 92³ + 93³ + 94³ + 95³ - 450
And that actually means that the nice visual proofs in the video do extend to these modified cubes pattern because the six slices of the cube that I show in the video actually do form the shell of a smaller cube LESS two diametrically opposed corners.

For the the 4th powers differences the formula is 4³(1 + 2 + 3 + ... + n)³ = 8n³(n+1)³
7⁴ + 8⁴ = 9⁴ - 64
22⁴ + 23⁴ + 24⁴ = 25⁴ + 26⁴ - 1728
45⁴ + 46⁴ + 47⁴ + 48⁴ = 49⁴ + 50⁴ + 51⁴ - 13824
76⁴ + 77⁴ + 78⁴ + 79⁴ + 80⁴ = 81⁴ + 82⁴ + 83⁴ + 84⁴ - 64000
There is another nice piece of 4d hypercube geometry that goes with this observation.

The new emerging pattern then breaks again with 5th powers. Here the sequence of differences starts like this: 2002, 162066, 2592552, 20002600, 101258850, .... But who knows , ... :)

The triangular numbers tn=1+2+3+...+n that feature prominently in all this arrange themselves into a nice pattern like this
t1+t2+t3=t4
t5+t6+t7+t8=t9+t10
t11+t12+...+t15=t16+t17+t18
etc.

Solving x² + (x+1)² = (x+2)² has two integer solutions. The first is 3 corresponding to 3² + 4² = 5². The second is -1 corresponding to (-1)² + 0² = 1². You also get a second solution for every other equation in the square pattern.
(-1)² + 0² = 1²
(-2)² + (-1)² + 0² = 1² + 2²
etc.

Donald Sayers and qyrghyz point out that there is a nice discussion of minimal dissections of 6³ into eight pieces that can be reassembled into a 3³ a 4³ and a 5³ in Martin Gardner's book Knotted doughnuts and other mathematical entertainments, pages 198-200. A picture of a dissection like this is shown on this wiki page on Euler's conjecture tinyurl.com/27pkbj2c Another dissection here tinyurl.com/y6c6tbj4

If you are interested in more Mathologer animations of the type shown at the beginning of this video check out Mathologer 2 and the final sections of many/most regular Mathologer videos.

T-shirt: google "super pi t-shirt"
Music: Here to fight by Roman P. and Earth, the Pale Blue Dot by Ardie Son

Enjoy!

Burkard

Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

Mathologer 2022-12-03 | In 2007 a simple beautiful connection Pythagorean triples and the Fibonacci sequence was discovered. This video is about popularising this connection which previously went largely unnoticed.

00:00 Intro
07:07 Pythagorean triple tree
13:44 Pythagoras's other tree
16:02 Feuerbach miracle
24:28 Life lesson
26:10 The families of Plato, Fermat and Pythagoras
30:45 Euclid's Elements and some proofs
37:57 Fibonacci numbers are special
40:38 Eugen Jost's spiral
41:20 Thank you!!!
42:27 Solution to my pearl necklace puzzle

The two preprints by H. Lee Price and and Frank R. Bernhart and another related paper by the same authors:
arxiv.org/abs/0809.4324
arxiv.org/abs/math/0701554
tinyurl.com/y6k4eyx5

The wiki page on Pythagorean triples is very good and very comprehensive
en.wikipedia.org/wiki/Pythagorean_triple

Wiki page on Pythagorean triple trees
en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples

Mathematica code for the Pythagorean Christmas tree by chyanog tinyurl.com/2z66rfkb

Geoalgebra app for the Pythagoras tree fractal by Juan Carlos Ponce Campuzano
geogebra.org/m/VU4SUVUp

Connection to the Farey tree/Stern-Brocot tree in a paper by Shin-ichi Katayama
tinyurl.com/vmvcs729

David Pagni (on the extra special feature of the Fibonacci number)
jstor.org/stable/30215477

Eugen Jost's Fibonacci meets Pythagoras spiral (in German)
https://mathothek.de/katalog/fibonacci-meets-pythagoras-eine-begegnung-die-zur-spirale-wird/

Bug report:
06:06 - right circle doesn't touch line (I mucked up :(

Puzzle time codes:
11:41 Puzzle 1: a) Fibonacci box of 153, 104, 185 b) path from from 3, 4, 5, to this triple in the tree
16:02 Puzzle 2: Area of gen 5 Pythagorean tree
25:55 Puzzle 3: Necklace puzzle

Some interesting tidbits:
Jakob Lenke put together an app that finds the route from 3,4,5 to your primitive Pythagorean triple of choice inside the tree. Thanks Jacob pastebin.com/T71NP8Z9

theoriginalstoney and Michael Morad observed that at 39:28 (last section, extra special Fibonacci) the difference between the two righthand numbers (4 and 5, 12 and 13, 30 and 34, 80 and 89) are also squares of the Fibonacci numbers: F_(2n+3) - 2 F_(n+1) F_(n+2)=(F_n)^2

Éric Bischoff comments that the trick to get a right angle at 25:40 is popularized in French under the name "corde d'arpenteur". This term refers to a circular rope with 12 equally spaced nodes. If you pull 3, 4 and 5-node sides so the rope is tense, you get a right angle. See article "Corde à nœuds" on Wikipedia

Various viewers told me what F.J.M. stands for: Fredericus Johannes Maria Barning, Freek, b. Amsterdam 03.10.1924, master's degree in mathematics Amsterdam GU 1954|a|, employee Mathematical Center (1954-), deputy director Mathematical Center, later Center for Mathematics and Informatics (1972-1988) Deceased. Amstelveen 27.06.2012, begr. Amsterdam (RK Bpl. Buitenveldert) 04.07.2012.

John Klinger remarks that if the four numbers in the box are viewed as fractions, the two fractions are equal to the tangents of half of each of the two acute angles of the triangle.

Colin Pountney: Here is another piece in the jigsaw. The link to Pascals triangle. It only works for the Fermat series of triples (ie the set of "middle children"). Choose any row in Pascals triangle. Multiply the odd entries by 1, 2, 4, 8, ..... and add to get the top left entry in a Fibonacci box. Do the same with the even entries to get the top right entry. For example taking the 1 5 10 10 5 1 row, we have top left number = 1*1 + 2*10 + 4*5 = 21. Top right number = 1*5 + 2*10 + 4*1 =29. For example taking the 1 6 15 20 15 6 1 row we have top left = 1*1 + 2*15 + 4*15 + 8*1 = 99. Top right = 1*6 + 2*20 + 4*6 =70. Not obviously useful, but it seems to make things more complete.

Ricardo Guzman: Another cool property of Fibonacci numbers: Take any 3 consecutive Fibonacci numbers: 55,89,144. The difference of squares of the larger two, divided by the smallest, is the next Fibonacci. .... Thus, in interesting ways the Fibonacci numbers are intertwined with the squares.

CM63: This suggested the attached figure to me.
drive.google.com/file/d/1yjp6IZvwB5kPihXdFXRB_ZFd6FgSwpAN/view?usp=sharing
In reply I suggested to extend this picture a spiral using these identities: phi^2=phi+1, phi^3=phi^2+phi, phi^4=phi^3+phi^2 :)

According to this note on the relevant wiki page tinyurl.com/yv3fnac2 if you take overlaps of the Pythagorean tree into consideration the area of the tree is finite.

Today's music: Antionetta by Boreís and Dark tranquility by Anno Domini Beats
Today's t-shirt: google "Fibonacci cat t-shirt" for a couple of different versions. I just bought this t-shirt from somewhere but I think the cat is supposed to be superimposed onto this type of Fibonacci spiral tinyurl.com/2s3p7e3v

Enjoy!

Burkard

Pythagoras twisted squares: Why did they not teach you any of this in school?

Mathologer 2022-10-15 | A video on the iconic twisted squares diagram that many math(s) lovers have been familiar with since primary school. Surprisingly, there is a LOT more to this diagram than even expert mathematicians are aware of. And lots of this LOT is really really beautiful and important. A couple of things covered in this video include: Fermat's four squares theorem, Pythagoras for 60- and 120-degree triangles, the four bugs problem done using twisted squares and much more.

00:00 Intro
05:32 3 Squares: Fermat's four square theorem
12:51 Trithagoras
20:29 Hexagoras
22:06 Chop it up: More twisted square dissection proofs
23:42 Aha! Remarkable properties of right triangles with a twist
26:35 Mutants: Unusual applications of twisted squares
30:38 Op art: The four bugs problem
36:01 Final puzzle
36:32 Animation of Cauchy-Schwarz proof
37:16 Thanks!!

Here are a couple of links for you to explore:

My first Pythagoras video from four years ago:
youtu.be/p-0SOWbzUYI?t=732

A collection of over 100 proofs of Pythagoras theorem at Cut-the-knot cut-the-knot.org/pythagoras (quite a few with animations) I cover proofs 3, 4, 5 ( :), 9, 10, 76, 104. Other proofs closely related to what I am doing in this video are 55, 89, and 116.

A very good book that touches on a lot of the material in this video by Claudi Alsina and Roger B. Nelsen - Icons of Mathematics: An Exploration of Twenty Key Images (2011). Check out in particular chapters 1-3 and chapter 8.3.

Fermat's four square theorem:
Alf van der Poorten's super nice proof arxiv.org/abs/0712.3850
Fibonacci seems to be the discoverer of the connection between Pythagorean triples and arithmetic sequences of squares of length 3 en.wikipedia.org/wiki/Congruum

Trithagoras:
Wayne Robert's pages. Start here and then navigate to "The theory to applied to the geometry of triangles" tinyurl.com/3k6afad4
M. Moran Cabre, Mathematics without words. College Mathematics Journal 34 (2003), p. 172.
Claudi Alsina and Roger B. Nelsen, College Mathematics Journal 41 (2010), p. 370. (Trithagoras for 30 and 150-degree triangles)
Nice writeup about how to make Eisenstein triples from Eisenstein integers
http://ime.math.arizona.edu/2007-08/0221_cuoco_handout2.pdf
More people should know about Eisenstein:
mathshistory.st-andrews.ac.uk/Biographies/Eisenstein

Other twisted square dissection proofs:
There is an Easter Egg contained in the first proof. Five days after publishing the video only one person appears to have noticed it :) Here is an alternative version of the animation that only uses shifts that I put on Mathologer 2 youtube.com/shorts/vT9wUpu_Vco

The four bugs problem:
Actually I got something wrong here. Martin Gardner mentioned the four bugs for the first time in 1957 as a puzzle Martin Gardner actually mentioned the four bugs for the first time in 1957 as a puzzle (Gardner, M. November, 1957 Mathematical Games. Nine titillating puzzles, Sci. Am. 197, 140–146.) The 1965 article that is accompanied by the nice cover that I show in the video talks, among many other things, about the more general problem of placing bugs on the corners of a regular n-gon.

If you've got access to JSTOR, you can access all of Martin Gardner's articles through them.
jstor.org/journal/scieamer (all issues of the Scientific American)
jstor.org/stable/e24941962 (follow the Mathematical Games link)
jstor.org/stable/e24931930 (follow the Mathematical Games link)
jstor.org/journal/scieamer (all issues of the Scientific American)

Here are a couple of other online resources worth checking out.
tinyurl.com/4erz3zmf
tinyurl.com/ykvvj5sw

Explanation for distance 1: Because the bug that each bug is walking towards is always moving perpendicular to the first bug’s path, never getting closer or further away from the first bug’s motion. So it has to go exactly the same distance as it was at the beginning.

For the mathematics of various bits and pieces chasing each other check out Paul Nahin's book Chases and Escapes: The Mathematics of Pursuit and Evasion.

Solution for the puzzle at the end:
tinyurl.com/3eebxn2k

Today's music: A tender heart/The David Roy
T-shirt: google "Pythagoras and Einstein fighting over c squared t-shirt" for a couple of different versions.

Enjoy!

Burkard

Secrets of the lost number walls

Mathologer 2022-08-27 | This video is about number walls a very beautiful corner of mathematics that hardly anybody seems to be aware of. Time for a thorough Mathologerization :) Overall a very natural follow-on to the very popular video on difference tables from a couple of months ago ("Why don't they teach Newton's calculus of 'What comes next?'")

00:00 Intro
01:02 Chapter 1: What's in a wall
03:35 Chapter 2: Number wall oracle
14:31 Chapter 3: Walls have windows
16:34 Animations of Pagoda sequence
18:13 Chapter 4: Zero problems
25:31 Chapter 5: Determinants
32:49 Animation sequence with music
35:22 Thank you :)

References for number walls
The main reference for number walls is Fred Lunnon's article "The number-wall algorithm: an LFSR cookbook", Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.
cs.uwaterloo.ca/journals/JIS/VOL4/LUNNON/numbwall10.html

Also check out Fred's article "The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings", Electronic Proceedings in Theoretical Computer Science 1 (2009), 130–148. arxiv.org/abs/0906.3286. Among many other things this one features lots of pretty pictures :)

Conway and Guy's famous "The book of numbers" has a chapter dedicated to number walls. This is where I first learned about number walls. Sadly, Figure 3.24 on page 88 which describes the horse shoe rule is full of typos. Careful:
1. (formulae on right) Negate signs attached to w_l/w and e_l/e ;
2. (diagram on left) Leftward arrow missing from edge marked w_2 ;
3. The last row of arrows bears labels " s_3 " ... " s_2 " ... " s_1 " , which should instead read " s_1 " ... " s_2 " ... " s_3 " .

More articles/books to check out if you are really keen:
tinyurl.com/bdhyzscw
core.ac.uk/download/pdf/82737163.pdf
Jacek Gilewicz, Approximants de Padé, Springer Lecture Notes in Mathematics 667 (1978).

The Wiki page on linear recurrence with constant coefficients is a good resource for finding out about how the characteristic polynomial of a sequence translates into a "function rule"
en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients

Coding challenge
Create an online implementation of the number wall algorithm using determinants or, ideally, using the cross and horseshoe rules and do a couple of fun things with your program. Here are some possible ideas you could play with: 1. generate pictures of even number (or, more generally, mod p) windows of random integer sequences or of sequences grabbed from here oeis.org . 2. Explore the Pagoda sequence number wall, again mod various prime numbers. Here is the entry for this sequence in the on-line Encyclopaedia of integer sequences tinyurl.com/yc45cfvf 3. Be inspired by the examples in this article arxiv.org/abs/0906.3286 Send me a link to your app before the next Mathologer video comes out and I'll enter you in the draw for a copy of Marty and my book Putting two and two together :)

Research challenge
Prove the Pagoda sequence wall conjecture or find a counterexample.

Bug report
In the video I say that figuring out the factor rule is easy. This is only true for windows of 0s of even dimensions. Showing that the factor rule has a -1 on the right side for windows of odd dimensions is actually somewhat tricky. Details in the first article by Fred Lunnon listed above.

Today's music: Asturias by Isaac Albeniz performed by Guitar Classics and Taiyo (Sun) by Yuhi (Evening Sun)
Today's t-shirt: Yes, I am always right. If you are interested in getting one just google "Yes, I am always right math t-shirt" and pick the version you like best.

Enjoy!

Burkard

Why is calculus so ... EASY ?

Mathologer 2022-07-16 | Calculus made easy, the Mathologer way :)

00:00 Intro
00:49 Calculus made easy. Silvanus P. Thompson comes alive
03:12 Part 1: Car calculus
12:05 Part 2: Differential calculus, elementary functions
19:08 Part 3: Integral calculus
27:21 Part 4: Leibniz magic notation
30:02 Animations: product rule
31:43 quotient rule
32:18 powers of x
33:10 sum rule
33:52 chain rule
34:54 exponential functions
35:30 natural logarithm
35:56 sine
36:32 Leibniz notation in action
36:43 Creepy animations of Thompson and Leibniz
37:00 Thank you!

Online version of Silvanus P. Thompson's book "Calculus made easy" at Project Gutenberg:
gutenberg.org/ebooks/33283

There is also a version of this book annotated by the great Martin Gardner. That's the one to get if you after a hardcopy.
amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0333772431

Paranormal distribution maths t-shirt:
tinyurl.com/2p6x5jre for other versions of the same idea google "paranormal distribution math t-shirt"

Creepy animations: myheritage.com/deep-nostalgia

Music: Morning mandolin by Chris Haugen and Game changer by ikoliks.

Thank you very much to Eduardo Ochs for his subtitles in Brazilian Portuguese.

Burkard

Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

Mathologer 2022-05-14 | Today's video is about Heron's famous formula and Brahmagupta's and Bretschneider's extensions of this formula and what these formulas have to do with that curious identity 1+2+3=1x2x3.

00:00 Intro
01:01 1+2+3=1x2x3 in action
02:11 Equilateral triangle
02:30 Golden triangle
03:09 Chapter 1: Heron
06:18 Heron's formula
08:50 Brahmagupta's formula
10:20 Bretschneider's formula
11:52 Chapter 2: How? The proof
12:57 Heron via trig
20:09 Cut-the-knot
21:16 Albrecht Hess
21:46 Heron to Brahmagupta proof animation
25:10 Thank you!

Heron's formula on the Cut-the-knot site:
cut-the-knot.org/Curriculum/Geometry/HeronsFormula.shtml

Original article by Roger B. Nelsen "Heron's formula via proofs without words", featuring a version of the nice rectangle proof that I am presenting in this video: maa.org/sites/default/files/0746834212944.di020798.02p0691h.pdf

Simple derivation of Heron's formula just using Pythagoras's theorem:
https/www.mathpages.com/home/kmath196/kmath196

Job Bouwman's maths posts on Quora (you'll have to scroll a bit to get to Heron's formula)
http://shorturl.at/gzGOX
http://shorturl.at/dBX12
quora.com/profile/Job-Bouwman

A very comprehensive book about quadrilaterals:
Claudia Alsina, Roger B. Nelsen - A Cornucopia of Quadrilaterals (Dolciani Mathematical Expositions) (2020, American Mathematical Society)

Albrecht Hess's paper "A Highway from Heron to Brahmagupta"
https://forumgeom.fau.edu/FG2012volume12/FG201215.pdf

If you liked the rectangle proof of the sum = product identity you'll probably also like this proof of Pythagoras's theorem:
youtu.be/r4gOlttnJ_E
I also mentioned this one earlier in a video on this main channel
youtu.be/r4gOlttnJ_E

Two more interesting notes on the cut-the-knot page:
1. Let the angles of the triangle be 2α, 2β, 2γ so that α + β + γ = 90°. The identity RGP = r²(R + G + P) is equivalent to the following trigonometric formula: cotα + cotβ + cotγ = cotα cotβ cotγ, where "cot" denotes the standard cotangent function. More on this here tinyurl.com/yrsuhthk
2. A supercute way to derive Pythagoras from Heron with one line of calculus
cut-the-knot.org/pythagoras/HeronsDerivative.shtml

For a cyclic quadrilateral that also has an incircle we have a+b=c+d and it follows that the area is just square root of the product of all of the sides.

A 3d counterpart to Heron's formula:
en.wikipedia.org/wiki/Heron%27s_formula#Heron-type_formula_for_the_volume_of_a_tetrahedron
A different 3d connection (de Gua's theorem)
mathpages.com/home/kmath226/kmath226.htm

A couple of links to get you started on generalisations involving cyclic n-gons:
arxiv.org/pdf/1203.3438.pdf
arxiv.org/pdf/1910.08396.pdf
tinyurl.com/tyhzwpxj

Another interesting observation extending the fact that the 3-4-5 right-angled triangle has incircle radius 1: In general, the incircle radius of any right-angled triangle with integer sides is an integer.

Have a look at this for a related proof that arctan 1 + arctan 2 + arctan 3 = pi:
geogebra.org/m/A65eMkuN
math.stackexchange.com/questions/197393/why-does-tan-11-tan-12-tan-13-pi (2nd proof)

Not many integer solutions for x+y+z=xyz:
0+0+0=0x0x0
1+2+3=1x2x3
(-1)+(-2)+(-3)=(-1)x(-2)x(-3)

Other interesting little curiosities (some mentioned in the comments):
2+2=2x2=2^2 (of course)
3^3+4^3+5^3=6^3 = 6*6*6=216 illuminati confirmed
6+9+6*9 = 69
a+9+a*9 = 10a+9 (sub any digit)
en.wikipedia.org/wiki/Mathematical_coincidence
log(1+2+3)=log(1)+log(2)+log(3) follow from 1+2+3=1x2x3

Grégoire Locqueville 2:32 "Maybe one of you can check in the comments" is the new "left as an exercise to the reader" :)

Scaling the equations at this time code: youtu.be/IguNXoCjBEk?t=256:
length, area and "volume" start out the same with radius 1: length=area=volume.
When you scale by r, these values scale in this way Length = length * r, Area = area * r^2 and Volume = "volume" r^3. Therefore, Length = length * r = area *r and so (multiply through with r) Length* r = area *r ^2 = Area, etc.

Typo spotted: At the very end, in Brahmagupta's Formula the third bracket should be (A+C+D-B) not (A+B+D-B).

X minus Y maths t-shirt: Sadly the etsy shop I got this one from seems to have disappeared (Pacific trader). There is what appears to be a ripoff on zazzle by someone who does not know what they are doing :) tinyurl.com/24vrzpu9

Nice variation of the t-shirt joke by one of you: M - I - I = V :)

The Chrome extension I mentioned in this video is called CheerpJ Applet runner.

Music used in this video: Aftershocks by Ardie Son and Zoom out by Muted

Enjoy!

Burkard

Reinventing the magic log wheel: How was this missed for 400 years?

Mathologer 2022-04-02 | Today is about reinventing a really cool mathematical wheel and its many different slide rule incarnations, just using a rubber band.

00:00 Intro
04:40 Multiply!
06:02 Pi times e
07:15 Divide!
08:39 Sliding rules
10:53 Apollo
11:08 Star Trek
11:45 Rubber band proof
13:13 Logarithms
16:50 Dmitry's wheel
17:48 Thank you!

This video was inspired by Dmitry Sagalovskiy's Wheel of logarithms. Here is his original animation:
dsagal.github.io/circle-of-fifths/logwheel
The source code lives here (free for you to modify)
github.com/dsagal/circle-of-fifths/blob/master/src/logcircle.ts
Dmitry's original post on Reddit with an interesting discussion section lives here:
tinyurl.com/4dc9hb2r
Dmitry's company Grist a spreadsheet-database product lives here:
getgrist.com

Must see, the amazing Slide Rule Simulator Emulator Replica Collection: Aristo, Faber-Castell, Pickett, ..., they are all there.
sliderules.org

Sadly, all these slide rules are linear slide rules. There are some circular slide rules apps made for mobile devices. However, I don't like any of them, except for the German WWII submarine Angriffsscheibe (=attack disk) app Sub Buddy which contains a circular slide rule (not free :(

It would be great if one of you could make a nice circular slide rule online app. Optional features could include: 1. input fields for numbers that are multiplied or divided and then the automatic execution of the slide rule actions with the scales spinning as in this video; 2. infinite precision by making it possible to zoom in on the scales and have it refine automatically; 3. tick box for squaring, as we rotate the two inputs for multiplying are kept the same; 4. incorporation of other look-up scales or even log-log scales; 5. Change of base. :)

The Wiki page on slide rules is excellent
en.wikipedia.org/wiki/Slide_rule
Don't miss out on the bottom of the page, especially the part on "Contemporary use".

Nice contributions to the Mathologer coding challenge (infinite precision zoomable circular slide rule)
Cristian Merighi: https://js.pacem.it/2d/circular-slide-ruler
Mike Wessler: phoenixave.com/crule
Liam Applebe: tiusic.com/slide_rule.html
Juan Ignacio Almenara Ortiz: desmos.com/calculator/tul8psjh32 (demos)

Root of evil math t-shirt: A missed opportunity squaring the root of evil using the circular slide rule to find evil :( Will do in my next life. Some of you commented that the number shown on the t-shirt was just truncated at the 4th decimal and not rounded. Well, strictly speaking it's wrong no matter whether you round or just chop off as the designer of this t-shirt did :)

Nice concise summary of why the circumference of the wheel is ln(10): At any given moment, the numerical scale of the unwrapped band is proportional to 1/x, where x is the number entering the wheel. So this is a really nice way to see that the integral of 1/x is a logarithmic function.

For real math gourmets: a slide rule for complex numbers :) demonstrations.wolfram.com/ComplexSlideRule

Sliderule nickname: Slipstick
Someone suggested another cute name: addalog computer (I like it :)
And another one: dial-log
Slide rule: only one child at a time (I like that one too :)
German: Rechenscheibe vs. Rechenschieber (calculating disk=circular slide rule vs. calculating slider=linear slide rule)

Check out what it took to win the international slide rule competition: sliderulemuseum.com/ISRC.htm

Different slide rule scales: http://www.quadibloc.com/math/sr02.htm (the whole site is really amazing http://www.quadibloc.com/math/slrint.htm)

A very detailed discussion of the use of circular slide rules in Star Trek: trekbbs.com/threads/props-re-used.81174/page-7

Circular slide rule in Dr Strangelove
tinyurl.com/35tkp6uj
https://www.fourmilab.ch/bombcalc/brico.html
youtube.com/watch?v=zZct-itCwPE

Slide rule in the movie Red October
tinyurl.com/4eehjp4p

Slide rule in the song Wonderful World by Sam Cooke
youtu.be/R4GLAKEjU4w?t=41

The largest slide rule (over 100 meters long!): http://mit-a.com/TexasMagnum.shtml

Something very interesting, a program for generating slide rule scales github.com/dylan-thinnes/slide-rules-generator

Connections to Vernier scales
aapt.scitation.org/doi/abs/10.1119/1.1991139?journalCode=ajp
patents.google.com/patent/US2424713A/en

Comment by TupperWallace: I’ll tell you why it was missed for hundreds of years: The rubber band wasn’t invented until St Patrick’s Day, 1845. The stretchy metaphor would not have been that understandable. :)

Real magic :) There are a few "easter eggs" hiding in this video which only the very observant will notice... e.g. youtu.be/ZIQQvxSXLhI?t=513

Music today: Trickster by Ian Post

Burkard

Tesla’s 3-6-9 and Vortex Math: Is this really the key to the universe?

Mathologer 2022-02-19 | Today, a long overdue foray into the realm of VORTEX MATHEMATICS :)

00:00 Intro
04:16 The vortex
08:10 The maths of remainders and digital roots
13:25 Demystifying the vortex
16:30 A matter of base. The 8 fingered Tesla.
19:21 Explanation why the digital root is the remainder on division by 9
24:01 Tristan's challenge
24:44 The magic of modular multiplication maths
25:19 Intuition for multiplier - 1 petals
28:23 Thank You!

Coding competition:
My wish list for the modular times table diagram app:
-Being able to color line segments according to length.
-Indication of the "direction" of multiplication. 1x2 = 2 and so there should really be a little arrow from 1 to 2 not just a simple connection :)
-different loops in different colors.
...
Here is the prize, a copy of my and Marty's new book.
bookstore.ams.org/mbk-141

That early Mathologer video featuring the modular times tables
Times Tables, Mandelbrot and the Heart of Mathematics
youtu.be/qhbuKbxJsk8

A really nice article about various ways to generate the cardioid by Dave Richeson
divisbyzero.com/2018/04/02/i-heart-cardioids

Nice debunking/demystifying article about vortex math by "Professor Puzzler"
theproblemsite.com/vortex

For a growing pile of implementation of modular times table diagrams see my comment pinned to the top of the comment section of this video.

Simon Plouffe's website
http://plouffe.fr/Simon%20Plouffe.htm
Articles by him relevant to this video can be found in this directory
http://plouffe.fr/Inverseofprimes/
See in particular the files
The shape of b^n mod p.pdf
La forme de bn mod p.pdf

What I am talking about in this video is really just the tip of a bizarre mathematical iceberg that most mathematically minded people are completely unaware of. Have a look at this presentation by Marko Rodin on vortex math (beware serious nutty and at the same time truely beautifully presented numerology ahead :) A LOT more than is usually reported on in popular YouTube videos.
sciencetosagemagazine.com/vbm-vortex-based-mathematics-with-marko-rodin
In turn this iceberg is just another tip of an even bigger iceberg of mainly wishful thinking. Have a look: sciencetosagemagazine.com/category/library

Today's music: Aftershocks by Ardie Son

Enjoy!

Burkard

How did Fibonacci beat the Solitaire army?

Mathologer 2022-01-22 | Fibonacci and a super pretty piece of life-and-death mathematics. What can go wrong?

00:00 Intro
02:20 Solitaire
03:12 Survivor challenge
05:32 Invasion
11:41 The triangles of death
20:22 Final animation
21:43 Thank You!

Here is an online version of Marty and my newspaper article about the possible positions of one remaining peg when playing peg solitaire on various boards
qedcat.com/archive_cleaned/212.html

Marty and my new book "Putting two and two together"
bookstore.ams.org/mbk-141

Martin Aigner's paper "Moving into the desert with Fibonacci". Bit of a pain to access it for free, possible though via this site.
jstor.org/stable/2691046
This paper contains the proof that I am focussing on in this video. It also has Conway's golden ratio based proof.

An implementation of the solitaire army game by Mark Bensilum. Use it to play solitaire army general. Note that this implementation starts with all of the bottom squares occupied by pegs. Please read carefully how you are supposed to play the game using this app :)
cleverlearning.co.uk/blogs/blogConwayInteractive.php

The paper "The minimum size required of a solitaire army" by George I. Bell, Daniel S. Hirschberg, Pablo Guerrero-Garcia considers all sorts of variations of the basic solitaire army game. The animation and challenge at the end of the video is based on some of the findings in this paper. Highly recommended.
arxiv.org/pdf/math/0612612.pdf

Reaching row 5 in Solitaire Army using infinitely many pegs (featuring a pretty spectacular animation at the bottom of the page) by Simon Tatham and Gareth Taylor
chiark.greenend.org.uk/~sgtatham/solarmy

A page of very interesting solitaire-army puzzles by Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso
isnphard.com/g/solitaire-army

Numberphile videos on "Conway's checkers" starring the mathematician Zvezdelina Stankova
youtu.be/FtNWzlfEQgY
youtu.be/Or0uWM9bT5w

Today's t-shirt: teeherivar.com/product/funny-math-fibonacchos

Today's music: I promise by Ian Post

Enjoy!

Burkard

The 3-4-7 miracle. Why is this one not super famous?

Mathologer 2021-12-30 | I got sidetracked again by a puzzling little mathematical miracle. And, as usual, I could not help myself and just had to figure it out. Here is the result of my efforts.

00:00 Intro
08:45 The Coin rotation paradox
16:00 Mystery number explanation
18:38 Challenges and the new book
19:53 One-minute animation on how to figure the sum of the angles in a star
21:04 Thank you :)

The winner of Marty and my book Putting Two and Two together is Alexander Svorre Jordan. Congratulations. :) Thank you again to everybody who submitted an implementation of the dance. Here are five particularly noteworthy submissions:
(Kieran Clancy) kieranclancy.github.io/star-animation (this was the very first submission submitted in record time :)
(Liam Applebe) tiusic.com/magic_star_anim.html (an early submission that automatically does the whole dance for any choice of parameters)
(Pierre Lancien) https://lab.toxicode.fr/spirograph/ (with geared circles)
(Christopher Gallegos) gallegosaudio.com/MathologerStars (very slick interface)
(Matthew Arcus) shadertoy.com/view/7tKXWy (implements the fact that BOTH types of rotating polygons are parts of circles rolling around DIFFERENT large circles)

Some fun and helpful links.

The animation in geogebra (by Juan Carlos Ponce Campuzano):
geogebra.org/m/tfqmub4g

The version of the animation I show in the video I stumbled across on Instagram, Twitter, etc.: tinyurl.com/36dy6nm3

The new book by Marty and me:
bookstore.ams.org/mbk-141

New short videos on Mathologer 2:
youtube.com/c/Mathologer2

New Mathologer instagram account:
instagram.com/the_real_mathologer

Hypotrochoids:
en.wikipedia.org/wiki/Hypotrochoid
geogebra.org/m/pTrc52nv

Spirograph:
en.wikipedia.org/wiki/Spirograph
(a nice app)
faishasj.github.io/spirograph

Coin rotation paradox
en.wikipedia.org/wiki/Coin_rotation_paradox

Tusi couple:
en.wikipedia.org/wiki/Tusi_couple

Funfair Twister ride:
youtube.com/watch?v=laQeA47NN0E

Today's music is Altitude by Muted.

Enjoy!

Burkard

Do you understand this viral very good math movie clip? (Nathan solves math problem X+Y)

Mathologer 2021-10-16 | Recently one of you requested that I explain the math(s) in this clip which recently went viral.

youtu.be/mYAahN1G8Y8

It's a clip taken from the movie X+Y aka A brilliant young mind. The math(s) problem that Nathan, the main character in this movie, is working on in this clip is a simplified version of the first part of a problem that was shortlisted for the 2009 International Mathematical Olympiad. Here is a link to the shortlist.

imo-official.org/problems/IMO2009SL.pdf

The problem in question is problem C1 (the number 50 in the problem has been replace by the number 2 in the clip). This problem was suggested to the makers of the movie by Lee Zhao one of the maths consultants of the movie. This Note that the file I've linked to ere also includes solutions.

This problem was invented by Michael Albert who is a mathematician and computer scientist working at the University of Otago in New Zealand.

en.wikipedia.org/wiki/Michael_H._Albert

The 4-colour puzzle that I am challenging you with at the end of this video was invented specifically for the movie X+Y by the U.K. mathematician Geoff Smith who was another math consultant for the movie.

en.wikipedia.org/wiki/Geoff_Smith_(mathematician)

Geoff Smith also served for many years as the leader of the United Kingdom team at the International Mathematical Olympiad and is the current president of the IMO board. He also appears in a cameo role in the movie. He is sitting next to Nathan's mother in the corridor outside the hall in which the math(s) olympiad is taking place at the end of the movie.

imgur.com/a/cjWLHKi

Here is scan of the fictional IMO questions that the students dealing with at the end of the movie.

imgur.com/a/C1Uxkbd

Actually they adapted the first problem a little bit to make it fit in better with the story

imgur.com/a/9lP4zKe

These questions and their solutions also appear in the book by Geoff Smith, A Mathematical Olympiad Companion UK Mathematics Trust

ukmt.org.uk/product/95

Here is my presentation of an animated solution of the 4-color problem on Mathologer 2.

youtu.be/4uQfY8NSM6w

Please also check the description of that video for more information and the announcement of the winner(s) of the t-shirt.

Thank you very much to my colleague Norman Do for digging out all this info about the origins of these problems.

Marty and my MMDB (Mathematical Movie Data Base) lives here

qedcat.com/moviemath

Here is the preview of Marty and my book Math goes to the movies on Google books (features a couple chapters)

tinyurl.com/f57wr72v

The music accompanying my Thank you at the end is I Promise by Ian Post (just like for the last video)

Enjoy!

Burkard

P.S.: Here is a great account featuring an ex IMO participant who contributed to the making of X+Y.

ncs6mathssociety.wordpress.com/2015/09/09/lee-zhou-zhao-visits-the-ncs

His name is Lee Zhou Zhao. He also made a cameo appearance in the movie

imgur.com/a/uFwS8Yv

Why dont they teach Newtons calculus of What comes next?

Mathologer 2021-10-02 | Another long one. Obviously not for the faint of heart :) Anyway, this one is about the beautiful discrete counterpart of calculus, the calculus of sequences or the calculus of differences. Pretty much like in Alice's Wonderland things are strangely familiar and yet very different in this alternate reality calculus.

Featuring the Newton-Gregory interpolation formula, a powerful what comes next oracle, and some very off-the-beaten track spottings of some all-time favourites such as the Fibonacci sequence, Pascal's triangle and Maclaurin series.

00:00 Intro
05:16 Derivative = difference
08:37 What's the difference
16:03 The Master formula
18:19 What's next is silly
22:05 Gregory Newton works for everything
28:15 Integral = Sum
32:52 Differential equation = Difference equation
36:06 Summary and real world application
39:22 Proof

Here is a very nice write-up by David Gleich with a particular focus on the use of falling powers. tinyurl.com/ymcyrapz

This is a nice lesson from a Coursera course on this topic coursera.org/lecture/discrete-calculus/differences-k4jBq

One volume of Schaum's outlines is dedicated to "The calculus of finite differences and difference equations" (by Murray R. Spiegel) Examples galore!

This is a really nice very old book Calculus Of Finite Differences by George Boole (published in 1860!) tinyurl.com/3bdjr932 Thanks to Ian Robertson for recommending this one.

There is a wiki page about our mystery sequence: tinyurl.com/uwc89yub It's got a proof for why the mystery sequence counts the maximal numbers of regions cut by those cutting lines. If you have access to the book "The book of Numbers" by John Conway and Richard Guy, it's got the best proof I am aware of.

Here is a sketch of how you solve the Fibonacci difference equation to find Binet's formula imgur.com/a/Btu5ZVk

Here are a couple more beautiful gems that I did not get around to mentioning:
1. When we evaluate the G-N formula for 2^n what we are really doing is adding the entries in the nth row of Pascal's triangle (which starts with a 0th row :) And, of course, adding these entries really gives 2^n.

2. Evaluating the G-F formula for 2^n at n= -1 gives 1-1+1-1+... which diverges but whose Cesaro sum is 2^(-1)=1/2!! Something similar happens for n=-2.

3. In the proof at the end we also show that the difference of n choose m is n choose m-1. This implies immediately that the difference of the mth falling power is m times the difference of the m-1st falling power.

Today's music is by "I promise" by Ian Post.

Enjoy!

Burkard

P.S.: Some typos and bloopers
youtu.be/4AuV93LOPcE?t=719 (396 should be 369)
youtu.be/4AuV93LOPcE?t=1709 (where did the 5 go?)
youtu.be/4AuV93LOPcE?t=2448 (a new kind of math(s) :)

The Iron Man hyperspace formula really works (hypercube visualising, Eulers n-D polyhedron formula)

Mathologer 2021-08-28 | On the menu today are some very nice mathematical miracles clustered around the notion of mathematical higher-dimensional spaces, all tied together by the powers of (x+2). Very mysterious :) Some things to look forward to: The counterparts of Euler's polyhedron formula in all dimensions, a great mathematical moment in the movie Iron man 2, making proper sense of hupercubes, higher-dimensional shadow play and a pile of pretty proofs.

00:00 Intro
01:17 Chapter 1: Iron man
06:05 Chapter 2: Towel man
11:16 Cauchy's proof of Euler's polyhedron formula
17:37 Chapter 3: Beard man
22:16 Tristans proof that (x+2)^n works
26:16 Chapter 4: No man
28:52 Shadows of spinning cubes animation
28:42 Thanks

Here is a link to a zip file with the Mathematica notebooks for creating the cube and hypercube shadows that I discuss at the end of the video in chapter 4.

qedcat.com/cube_hypercube.zip

If you don't have Mathematica, you can have a look at pdf versions of the programs that are also part of the zip archive or you can use the free CDF player to open the cdf versions of the notebooks.

Something I forgot to mention: There is also another purely algebraic incarnations of this process of growing the cubes. It comes in the form of a recursion formula that connects the different numbers of bits and pieces in consecutive dimensions. That recursion formula is also present at the bottom of the "iron man page". Have a close look :) Also, in the Marvel movies the cube that Tony Stark is holding in the thumbnail of this video is called the Tesseract. Probably worth pointing out that "tesseract" is another name for a 4-d cube. I also built an easter egg into the thumbnail that plays on this fact:
imgur.com/a/psIy28k

The formulae for n-d tetrahedra and octahedra can be found on this page;

https://people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html

Here is a link to my video on solving the 4d Hyper Rubik's Cube

youtu.be/yhPH1369OWc

Another proof of Moessner for cubes using cubical shells Anthony Harradine and Anita Ponsaing

qedcat.com/StrikeMeOut.pdf

Here is a really nice video on the 120-cell that I only mentioned in passing.

youtu.be/MFXRRW9goTs

Noteworthy from the comments:
Today's video was "triggered" by a comment made by Godfrey Pigott on the last video on Moessner's miracle in which he pointed out that (x+2)^n captures the vital statistics of the n-dimensional cube.

Z. Michael Gehlke There is an easy way to see this: (x^1 + 2*x^0) describes the parts of a line; all of the cubes are iterated products of lines: n-cube = (1-cube)^n. Therefore, all cubes are described by iterated powers of (x^1 + 2*x^0)^n. (Me: Nice insight. Of course needs some fleshing out to make this work on it's own, like in the comment by ...

HEHEHE I AM A SUPAHSTAR SAGA I came up with an even simpler visual proof. Take a cube of side length x+2. This cube has a volume (x+2)^3. Now, slice the cube six times. Each slicing plane is parallel to a face and 1 unit deeper than the face. Don't throw away any volume. What you're left with is an inner cube of side length x (volume x^3), 6 square pieces of volume x^2, 12 edge pieces of volume x, and 8 corner cubes with volume 1 each. Adding up these volumes gives you the original (x+2)^3 volume, so it's proven. This works in any dimension.

Here is a link to an animation of this idea that I put on Mathologer 2, as a reward to those of you who who are keen enough to actually read these descriptions. youtu.be/cAVvwmcKsFk

Typo: The numbers of vertices and faces of the dodecahedron got switched.

Today's music is Floating Branch by Muted.

Enjoy!

Burkard

The Moessner Miracle. Why wasnt this discovered for over 2000 years?

Mathologer 2021-07-17 | Today's video is about a mathematical gem that was discovered 70 years ago. Although it's been around for quite a while and it's super cool and it's super accessible, hardly anybody knows about it.

00:00 Intro
04:58 Chapter 1: Making our own proof
09:55 Chapter 2: Some more amazing facts
13:11 Chapter 3: Post's proof
23:36 Supporters

If after watching this video you'd like to find out more about Moessner's result, the following PhD thesis features a very comprehensive bibliography: https://ebooks.au.dk/aul/catalog/book/213

The proof by Karel Post that I matholologerise in the second half of this video is contained in this paper:

Karel A. Post. Moessnerian theorems. How to prove them by simple graph theoretical inspection. Elemente der Mathematik, (2):46–51, 1990.

Post also proves a couple of generalisations of Moessner's theorem. Another good write-up of the same proof can be found in Ross Honsberger"s 1991 book More Mathematical Morsels. Honsberger says about Moessner that "he was internationally known in the field of recreational mathematics for many spectacular results in arithmetic". Have to have a closer look at some point at what else exactly he did :)

Post's article can be accessed here: https://www.e-periodica.ch (search for "Moessnerian theorems"). Sadly most other articles about Moessner's theorem are located behind paywalls.

Here is another very pretty proof of the basic cubes result by Anthony Harradine and Anita Ponsaing using actual 3d cubical shells qedcat.com/misc/StrikeMeOut.pdf

It's well worth exploring further than what I get around to reporting in this video. If you do, you'll discover interesting connections with super-factorials, higher-dimensional counterparts of Pascal's triangle, and so on.

Challenge for the programmers among you: write a program that turns a sequence of highlighted integers into the corresponding Moessner sequence.

Today's music is "Just Jump" by Ian Post. If you are interested in the t-shirt google "math whisperer t-shirt". If you don't understand the math whisperer bit, you did not watch the video to the end :)

Enjoy!

Burkard

14. Sep. 2021: I just added Russian subtitles prepared by Michael Didenko. Thank you very much Michael.

The Pigeon Hole Principle: 7 gorgeous proofs

Mathologer 2021-04-10 | Let's say there are more pigeons than pigeon holes. Then, if all the pigeons are in the holes, at least one of the holes must house at least two of the pigeons. Completely obvious. However, this unassuming pigeon hole principle strikes all over mathematics and yields some really surprising, deep and beautiful results. In this video I present my favourite seven applications of the pigeon hole principle.

Starting with a classic, the puzzle of hairy twins, we then have a problem with pigeons on a sphere, a pigeon powered explanation of recurring decimals, some party maths, a very twisty property of the Rubik’s cube, a puzzler from the 1972 International Mathematical Olympiad, and, finally, what some people consider to be the best mathematical card trick of all time.

00:00 Intro
01:49 Chapter 1: Hairy twins
06:46 Chapter 2: Five pigeons on a sphere
08:16 Chapter 3: Repeating decimals
13:14 Chapter 4: Partying pigeons
17:00 Chapter 5: Repeating Rubik
22:20 Chapter 6: Pigeons at the Olympiad
26:18 Chapter 7: The best mathematical card trick ever
31:24 Supporters

Here are some links for you to explore.

A scanned copy of Récréation mathématique: Composée de plusieurs problèmes plaisants et ... by Jean Leurechon on Google books. For the hair puzzle check out page 130) tinyurl.com/3b6amaxk

The Pigeonhole Principle, Two Centuries Before Dirichlet by Albrecht Heeffer and Benoit Rittaud A very nice article about the origins of the pigeon principle and the hairy twins problem. Also features an English translation of the relevant page in Récréation mathématique tinyurl.com/hpkcuepx

The 4/5 pigeons in a hemisphere puzzle was problem A2 of the 63rd Putnam competition in 2002 https://prase.cz/kalva/putnam/psoln/psol022.html

Why are repeated decimals fractions? Watch this video on why 9.999... =10 for a big hint youtu.be/SDtFBSjNmm0 Or just skip straight to the answer en.wikipedia.org/wiki/Decimal_representation#Conversion_to_fraction

If you don't own a Rubik's cube you can use this simulator to test what happens when you repeat some algorithms (move the faces using the keyboard) ruwix.com/online-puzzle-simulators

The website of the International Mathematical Olympiad. imo-official.org . The problem I am considering in this video is Problem 1 of the 1972 olympiad. You can download all the problems from here.
imo-official.org/problems.aspx

Check out this very nice article about the Fitch Cheney five-card trick by Colm Mulcahy tinyurl.com/wttkfdwe

Today's music is English Country Garden (and as usual Morning Mandolin at the end) from the free YouTube music library. Today's t-shirt I got from here theshirtlist.com/pizzibonacci-t-shirt

Enjoy!

Burkard

The ultimate tower of Hanoi algorithm

Mathologer 2021-03-06 | There must be millions of people who have heard of the Tower of Hanoi puzzle and the simple algorithm that generates the simplest solution. But what happens when you are playing the game not with three pegs, as in the original puzzle, but with 4, 5, 6 etc. pegs? Hardly anybody seems to know that there are also really really beautiful solutions which are believed to be optimal but whose optimality has only been proved for four pegs. Even less people know that you can boil down all these optimal solutions into simple no-brainer recipes that allow you to effortless execute these solutions from scratch. Clearly a job for the Mathologer. Get ready to dazzle your computer science friends :)

I also talk about 466/885, the Power of Hanoi constant and a number of other Hanoi facts off the beaten track. And the whole thing has a Dr Who hook which is also very cute.

00:00 Intro
01:58 Chapter 1: The doctor vs. the toymaker
14:27 Chapter 2: Hanoi constant
21:21 Chapter 3: The Reve's puzzle
28:04 A beautiful shortest solution for 10 discs and 4 pegs (discs and super-disks)
30:23 Chapter 4: Unprovable algorithm
35:43 A beautiful shortest solution for 10 discs and 5 pegs (discs, super-discs and super-super-discs)
37:17 Supporters

Here are some references for you to check out:

Andreas M. Hinz et al. - The Tower of Hanoi – Myths and Maths, 2nd edition (2018, Birkhäuser Basel) That's the book I mentioned in the video.

The Dr Who episode (well the part that's not been lost) dailymotion.com/video/x119oe7

The 3Blue1Brown video that I mention youtu.be/2SUvWfNJSsM

Thierry Bousch, La quatrième tour de Hanoi, http://tinyurl.com/4p3fudu7
That's the paper that pins down things for four pegs.

Andreas M. Hinz, Dudeney and Frame-Stewart Numbers,
A nice paper explaining the connection between Dudeneys's work and Frame-Stewart. Also worth reading for the historical details. http://tinyurl.com/t8xb2e5t

A. van de Liefvoort, An Iterative Algorithm for the Reve's Puzzle, http://tinyurl.com/h5cxfy5u
I found this one useful.

Paul K. Stockmeyer, http://www.cs.wm.edu/~pkstoc/toh.html A couple of very nice papers including a huge bibliography.

Ben Houston & Hassan Masum, Explorations in 4-peg Tower of Hanoi, tinyurl.com/mw95tnek This paper has some pictures of state graphs for the 4-peg puzzle.

http://towersofhanoi.info/Animate.aspx Very fancy animation of mulit-peg tower of Hanoi. Sadly, it just comes across as a mess of moves for more than three pegs. Programmers, you really should rise to my challenge to animate the 4-peg algorithm the way I present it in this video.

Here is the link to the wiki page for the Celestial toymaker Dr Who episode en.wikipedia.org/wiki/The_Celestial_Toymaker
Makes very interesting reading. Especially the fact that most of this episode has been lost I find pretty amazing. That's also why I only show a still image from the relevant part of the episode and play some audio snippet.

Music: Fresh Fallen Snow and Morning Mandolin both by Chris Haugen, Mumbai effect. All from the free YouTube audio library.

Enjoy!

Burkard

Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions)

Mathologer 2021-01-23 | Okay, as it says in the title of this video, today's mission is to figure out how many ways there are to make change for one googol, that is 10^100 dollars. The very strange patterns in the answer will surprise, as will the explanation for this phenomenon, promise.

0:00 Intro
1:15 Chapter 1: curious count
9:05 Chapter 2: googol
14:10 Chapter 3: infinite change
28:41 Acknowledgements

A very nice Mathematica file created by Andrew Neitsch in 2005 covers pretty much every aspect of change making mathematics. http://andrew.neitsch.ca/publications/m496pres1.nb
Here is a pdf version of this file: andrew.neitsch.ca/publications/m496pres1.nb.pdf
What I cover in the last part of this video is pretty much fleshing out and animating the section "Coin change revisited". All this is part of to Andrew's answer to a post on math.stackoverflow stackoverflow.com/questions/1106929/how-to-find-all-combinations-of-coins-when-given-some-dollar-value

The visual algebra approach to calculate the number of ways to make change at the very beginning of this video was inspired by this article G. Pólya, On Picture-Writing, Am Math Monthly 63 (1956), 689-697. jstor.org/stable/2309555

Concrete mathematics by Graham, Knuth and Patashnik, the book I mentioned at the end of the video does the whole analysis for the coin set 1, 5, 10, 25, 50 (so no dollar coins).

A complete list of all the different ways to make change for a dollar appears in this post maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-dollar

The book "Generatingfunctionology" by Herbert Wilf, is a great intro to generating functions :) https://www2.math.upenn.edu/~wilf/DownldGF.html

Ron Graham to who this video is dedicated did a couple of videos with Numberphile. So if you'd like to see him in action, check out those videos. A lot of other interesting articles about Ron Graham can be found on his wife's (also a math professor) website. http://www.math.ucsd.edu/~fan/ron/

As usual the music in the video is from the free YouTube audio library: No. 2 Remembering her by Ester Abrami, Morning Mandolin by Chris Haugen, First time experience and T'is the season by Nate Blaze

Today's t-shirts: google "only half evil t-shirt".

Enjoy!

Burkard

The ARCTIC CIRCLE THEOREM or Why do physicists play dominoes?

Mathologer 2020-12-24 | I only stumbled across the amazing arctic circle theorem a couple of months ago while preparing the video on Euler's pentagonal theorem. A perfect topic for a Christmas video.

Before I forget, the winner of the lucky draw announced in my last video is Zachary Kaplan. He wins a copy of my book Q.E.D. Beauty in mathematical proof.

00:00 Intro
00:35 Chapter 1: mutilated chessboards
07:23 Chapter 2: Monster formula
15:12 Chapter 3: Aztec gold
20:07 Chapter 4: Square dance
30:41 Chapter 5: Ice
34:35 Chapter 6: Hexagon
38:25 Credits
40:46 Mini masterclass

In response to my challenge here are some nice implementations of the dance:
Dmytro Fedoriaka: http://fedimser.github.io/adt/adt.html (special feature: also calculates pi based on random tilings. First program contributed.)
Viktor Chlumský youtu.be/CCL77BUymSY (no program but a VERY beautiful animation made with the software Shadron by Victor)
Cannot fit any more links in this description because of the character limit. For lots of other amazing implementations check out the list in my comment pinned to the top of the comment section of the video.

For the most accessible exposition of iterated shuffling that I am aware of have a look at the relevant chapter in the book "Integer partitions" by Andrews and Eriksson. They also have a nice set of exercises that walk you through proofs for the properties of iterated shuffling that I mention in this video.

I used Dan Romik's old Mac program "ASM Simulator" to produce the movie of the random tilings of growing Aztec diamond boards https://www.math.ucdavis.edu/~romik/software/ Sadly this program does not work on modern Macs.

The arctic heart at the end of the video is a "chistmasized" version of an image from the article "What is a Dimer" by Richard Kenyon and Andrei Okounkov ams.org/notices/200503/what-is.pdf Thank you for letting me use this image.

Around the same time that Kasteleyn published the paper I showed in the video, the physicists Temperley and Fisher published similar results, Dimer problem in statistical mechanics-an exact result, Philosophical Magazine, 6:68, (1961) 1061-1063. The way Kasteleyn as well as Temperley and Fisher calculated the numbers of tilings of boards with square tiles was a bit more complicated than the nice refinement that I show in the video which is due to Jerome K. Percus, One more technique for the dimer problem. J. Mathematical Phys., 10:1881–1888, 1969.

Some great articles and websites to check out:

A very accessible introduction to domino and other tilings by Federico Adila and Richard Stanley http://www.claymath.org/library/senior_scholars/stanley_ardila_tilings.pdf

An accessible article about tilings with rectangles by my colleague Norm Do at Monash Uni. In particular, it's got some more good stuff about the maths of fault lines in tilings that I only hinted at in the video:
http://users.monash.edu/~normd/documents/Mathellaneous-07.pdf

A nice article about Kasteleyn's method by James Propp. Includes a proof of the crazy formula arxiv.org/abs/1405.2615

A fantastic survey article about enumeration of tilings by James Propp. This one's got everything imaginable domino and otherwise. Also the bibliography at the end is very comprehensive http://faculty.uml.edu/jpropp/eot.pdf

An introduction to the dimer model by Richard Kenyon
arxiv.org/abs/math/0310326

Alternating sign matrices and domino tilings by Noam Elkies, Greg Kuperberg, Michael Larsen, James Propp arxiv.org/abs/math/9201305

Random Domino Tilings and the Arctic Circle Theorem by William Jockusch, James Propp, and Peter Shor arxiv.org/abs/math/9801068

A website by Alexei Borodin full of amazing 3d representations of domino tilings. A must-see http://math.mit.edu/~borodin/aztec.html

James Propp's name pops up a couple of times throughout this video and in this description. He's one of the mathematicians who discovered all the beautiful arctic mathematics that I am talking about in this video and helped me get my facts straight. Check out his blog http://mathenchant.org and in particular in this post he talks a little bit about the discovery of the arctic circle phenomenon mathenchant.wordpress.com/2016/1/16/how-to-be-wrong

As usual the music in the video is from the free YouTube audio library: Night Snow by Asher Fulero and Fresh fallen snow by Chris Haugen.

Today's t-shirts I got ages ago. Don't think they still sell those exact same ones. Having said that just google "HO cubed t-shirt" and "i squared keep it real t-shirt" ... :)

Jokes:
1. Aztec diamond = Crytek logo; 2. no. tilings of Arctic diamond: 2^(-1/12). 3. ℝeal mathematical magic, 4. (HO)³ : joke for mathematicians (HO)₃ : joke for chemists
Bug:
Here one of the tiles magically disappears (damn :( tinyurl.com/ya6mqmhh
Nice insight:
If all holes in a mutilated board can be tiled with dominoes the determinant will work. Why is that?

Merry Christmas,

burkard

700 years of secrets of the Sum of Sums (paradoxical harmonic series)

Mathologer 2020-11-21 | Today's video is about the harmonic series 1+1/2+1/3+... . Apart from all the usual bits (done right and animated :) I've included a lot of the amazing properties of this prototypical infinite series that hardly anybody knows about. Enjoy, and if you are teaching this stuff, I hope you'll find something interesting to add to your repertoire!

00:00 Intro
01:00 Chapter 1: Balanced warm-up
03:26 Chapter 2: The leaning tower of maths
12:03 Chapter 3: Finite or infinite
15:33 Chapter 4: Terrible aim
20:44 Chapter 5: It gets better and better
29:43 Chapter 6: Thinner and thinner
42:54 Kempner's proof animation
44:22 Credits

Here are some references to get you started if you'd like to dig deeper into any of the stuff that I covered in this video. Most of these articles you can read for free on JSTOR.

Chapter 2: Leaning tower of lire and crazy maximal overhang stacks

Leaning Tower of Lire. Paul B. Johnson American Journal of Physics 23 (1955), 240

Maximum overhang. Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, Uri Zwick arxiv.org/abs/0707.0093

Worm on a rubber band paradox: en.wikipedia.org/wiki/Ant_on_a_rubber_rope

Chapter 3: Proof of divergence

Here is a nice collection of different proofs for the divergence of the harmonic series http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf

Chapter 4: No integer partial sums

A harmonikus sorrol, J. KUERSCHAK, Matematikai es fizikai lapok 27 (1918), 299-300

Partial sums of series that cannot be an integer. Thomas J. Osler,
The Mathematical Gazette 96 (2012), 515-519

Representing positive rational numbers as finite sums of reciprocals of distinct positive integers http://www.math.ucsd.edu/~ronspubs/64_07_reciprocals.pdf

Chapter 5: Log formula for the partial sums and gamma

Partial Sums of the Harmonic Series. R. P. Boas, Jr. and J. W. Wrench, Jr.
The American Mathematical Monthly 78 (1971), 864-870

Chapter 6: Kempner's no 9s series:

Kempner in an online comic
smbc-comics.com/comic/math-translations

A very nice list of different sums contained in the harmonic series en.wikipedia.org/wiki/List_of_sums_of_reciprocals

Sums of Reciprocals of Integers Missing a Given Digit, Robert Baillie, The American Mathematical Monthly 86 (1979), 372-374

A Curious Convergent Series. A. J. Kempner, The American Mathematical Monthly 21 (1914), 48-50

Summing the curious series of Kempner and Irwin. Robert Baillie, arxiv.org/abs/0806.4410

If you still know how to read :) I recommend you read the very good book Gamma by Julian Havil.

Bug alert: Here youtu.be/vQE6-PLcGwU?t=4019 I say "at lest ten 9s series". That should be "at most ten 9s series"

Today's music (as usual from the free YouTube music library): Morning mandolin (Chris Haugen), Fresh fallen snow (Chris Haugen), Night snow (Asher Fulero), Believer (Silent Partner)
Today's t-shirt: rocketfactorytshirts.com/are-we-there-yet-mens-t-shirt

Enjoy!

Burkard

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Mathologer Patreon: patreon.com/mathologer
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(see the Patreon page for details)

The hardest What comes next? (Eulers pentagonal formula)

Mathologer 2020-10-17 | Looks like I just cannot do short videos anymore. Another long one :) In fact, a new record in terms of the slideshow: 547 slides!

This video is about one or my all-time favourite theorems in math(s): Euler's amazing pentagonal number theorem, it's unexpected connection to a prime number detector, the crazy infinite refinement of the Fibonacci growth rule into a growth rule for the partition numbers, etc. All math(s) mega star material, featuring guest appearances by Ramanujan, Hardy and Rademacher, and the "first substantial" American theorem by Fabian Franklin.

00:00 Intro
02:39 Chapter 1: Warmup
05:29 Chapter 2: Partition numbers can be deceiving
16:19 Chapter 3: Euler's twisted machine
20:19 Chapter 4: Triangular, square and pentagonal numbers
24:35 Chapter 5: The Ramanujan-Hardy-Rademacher formula
29:27 Chapter 6: Euler's pentagonal number theorem (proof part 1)
42:00 Chapter 7: Euler's machine (proof part 2)
50:00 Credits

Here are some links and other references if you interested in digging deeper.

This is the paper by Bjorn Poonen and Michael Rubenstein about the 1 2 4 8 16 30 sequence: http://www-math.mit.edu/~poonen/papers/ngon.pdf

The nicest introduction to integer partitions I know of is this book by George E. Andrews and Kimmo Eriksson - Integer Partitions (2004, Cambridge University Press) The generating function free visual proofs in the last two chapters of this vides were inspired by the chapter on the pentagonal number theorem in this book and the set of exercises following it.

Some very nice online write-ups featuring the usual generating function magic:
Dick Koch (uni Oregon) tinyurl.com/yxe3nch3

James Tanton (MAA) tinyurl.com/y5xj2dmb

A timeline of Euler's discovery of all the maths that I touch upon in this video:
imgur.com/a/Ko3mnDi

Check out the translation of one of Euler's papers (about the "modified" machine):
tinyurl.com/y5wlmtgb

Euler's paper talks about the "modified machine" as does Tanton in the last part of his write-up.

Another nice insight about the tweaked machine: a positive integer is called “perfect” if all its factors sum except for the largest factor sum to the number (6, 28, 496, ...). This means that we can also use the tweaked machine as a perfect number detector :)

Enjoy!

Burkard

Today's bug report:
I got the formula for the number of regions slightly wrong in the video. It needs to be adjusted by +n. In their paper Poonen and Rubenstein count the number of regions that a regular n-gon is divided into by their diagonals. So this formula misses out on the n regions that have a circle segment as one of their boundaries.

The two pieces of music that I've used in this video are 'Tis the season and First time experience by Nate Blaze, both from the free YouTube audio library.

As I said in the video, today's t-shirt is brand new. I put it in the t-shirt shop. Also happy for you to print your own if that works out cheaper for you: imgur.com/a/ry6dwJy

All the best,

burkard

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(see the Patreon page for details)

How did Ramanujan solve the STRAND puzzle?

Mathologer 2020-09-05 | Today's video is about making sense of an infinite fraction that pops up in an anecdote about the mathematical genius Srinivasa Ramanujan.

00:00 Intro
04:31 Chapter 1: Getting a feel for the puzzle
08:27 Chapter 2: Algebra autopilot
12:37 Chapter 3: Infinite fraction
17:51 Chapter 4: Root 2
21:19 Chapter 5: Euclidean algorithm
30:15 Chapter 6: The best of the best: 17/12
36:34 Chapter 7: Outramanujing Ramanujan

This was supposed to be a short video but in the end turned out to be quite a tricky to sort out. Anyway, as it sometimes happens, I got carried away and now the video really covers a lot of ground : Pell equations, visualising continued fractions by dissecting rectangles into squares, the relationship between continued fractions and the Euclidean algorithm, the irrationality of root 2. Overall quite a few things that you won't find anywhere else :)

The way I tell the anecdote in this video is based on the following account by Ramanujan's friend Prasanta Mahalanobis: Current Science, Vol. 9 (3), pp. 74-75.

"On another occasion, I went to his room to have lunch with him. The First World War had started some time ago. I had in my hand a copy of the monthly Strand Magazine which at that time used to publish a number of puzzles to be solved by the readers. Ramanujan was stirring something in a pan over the fire for our lunch. I was sitting near the table, turning over the pages of the Strand Magazine. I got interested in a problem involving a relation between two numbers. I have forgotten the details but I remember the type of the problem. Two British officers had been billeted in Paris in two different houses in a long street; the two numbers of these houses were related in a special way; the problem was to find out the two numbers. It was not at all difficult; I got the solution in a few minutes by trial and error. In a joking way, I told Ramanujan, 'Now here is a problem for you'. He said, 'What problem, tell me', and went on stirring the pan. I read out the question from the Strand Magazine. He promptly answered 'Please take down the solution' and dictated a continued fraction. The first term was the solution which I had obtained. Each successive term represented successive solutions for the same type of relation between two numbers, as the number of houses in the street would increase indefinitely. I was amazed and I asked him how he got the solution in a flash. He said, 'Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind. It was just as simple as this.' "

There is a complete digital archive of The Strand magazine. You can find the page with the puzzle here: tinyurl.com/y2lnb8xf (page 790)

If you read the puzzle in the Strand you'll find that the problem is actually phrased somewhat differently to what Mahalanobis remembers and Mahalanobis also does not spell out the infinite fraction that Ramanujan came up with. And if you do the math(s) some of the other things he says also don't quite sound right. What I am presenting in this video is my best guess for what really happened.

In particular, the continued fraction that I am talking about in video is probably the most natural candidate for Ramanujan's infinite fraction, but others have argued that it could have been a different continued fraction (which I don't buy :) You can find these other infinite fractions here: 'Ramanujan's Continued Fraction for a Puzzle" by Poo-Sung Park tinyurl.com/yyfdscgr and here 'On Ramanujan, continued fractions and an interesting street number' by John Butcher tinyurl.com/yy6nv2yg

Solution to the red cross puzzle from Dudeney's book "Amusements in Mathematics" p. 168 :) imgur.com/a/bBuLOZN

Another interesting way to systematically search for solutions to the Strand puzzle is this: The equation we want to solve is 2 x^2=y^2+y. You can rewrite this as x^2 = y(y+1)/2. The formula on the right is just the formula for 1+2+3+...+y. So just keep adding 1+2+3+... and at every step check whether the number you get is a square ... :)

Other short formulas: 1) Expanding (1+√ 2)^n gives a number a+b√2. Then a/b is the nth partial fraction. 2) Play with powers of the matrix {{2, 1}, {1, 0}}

Some number Easter eggs are hiding on this slide youtu.be/V2BybLCmUzs?t=1402 :)

Link to the unlisted Marching Squares video: youtu.be/f1yDExNAEMg

Here is a version of the t-shirt I am wearing: tinyurl.com/y5vgo7zb This one is about that other famous Ramanujan anecdote: tinyurl.com/y626c86x actually features prominently in another one of my videos.

The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits) and Nate Blaze 'Tis the season, all from the free YouTube music library

Enjoy!

Burkard

14.9.2021: Thank you very much Michael Didenko for your Russian subtitles.

What does this prove? Some of the most gorgeous visual shrink proofs ever invented

Mathologer 2020-07-25 | Bit of a mystery Mathologer today with the title of the video not giving away much. Anyway it all starts with the quest for equilateral triangles in square grids and by the end of it we find ourselves once more in the realms of irrationality. This video contains some extra gorgeous visual proofs that hardly anybody seems to know about.

0:00 Intro
0:47 First puzzle
2:24 Second puzzle
3:50 Edward Lucas
4:41 Equilateral triangles
13:15 3d & 3rd puzzle
19:52 30 45 60
29:31 Credits

Here are links to/references of some of the things I mention in the video:

Joel Hamkin's blog posts that inspired this video:
http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice
http://jdh.hamkins.org/no-regular-polygons-in-the-hexagonal-lattice
There is also a whole chapter about all this and much more related maths in his new book
amazon.com/Proof-Mathematics-Joel-David-Hamkins/dp/0262539799

Here is another really good article which includes a nice characterisation of the triangles that can be found in square grids plus a very good survey of relevant results:
Michael J. Beeson, Triangles with Vertices on Lattice Points, The American Mathematical Monthly 99 (1992), 243-252, jstor.org/stable/2325060?seq=1

Scherrer's and Hadwinger's articles:
Scherrer, Willy, Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik 1 (1946), 97-98.
tinyurl.com/y45p64t7
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[101]}
Hadwiger, Hugo Über die rationalen Hauptwinkel der Goniometrie, Elemente der Mathematik 1 (1946), 98-100.
tinyurl.com/yx98kkqt
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[102],%22view%22:%22info%22}

Another, nice paper on rational (and algebraic) cosines
arxiv.org/pdf/1006.2938.pdf

Here is a solution to the first puzzle (one way to find the general formula):
nrich.maths.org/657/solution

The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits)

A couple of remarks:
1. Probably the simplest way to deduce the sin and tan parts of the rational trig ratio theorem is to realise that they follow from the cos part via the trigonometric identities: sin(x)=cos(90-x) and tan^2(x) = (1-cos(2x))/(1+cos(2x)). Note that the second identity implies that if tan(x) is rational, then cos(2x) is rational (if tan(x)=c/d, then tan^2(x)=c^2/d^2=C/D and cos(2x)=(D-C)/(D+C)).

2. Bug report.
a) Here I redefine cos(120◦) = 1.
youtu.be/sDfzCIWpS7Q?t=1362
Remarkable :(
b) This transition to the good stuff I clearly did not think through properly.
youtu.be/sDfzCIWpS7Q?t=1018
It's possible to make this work for all regular n-gons. There is only one complication that occurs for n's that are of the form 2 * odd. For the corresponding regular n-gons, if you pick up the edges in the order that they appear around the n-gon and assemble them into a star, things close up into (n/2)-stars. For all other n, things work exactly as I showed in the video. Having said that you can also assemble the edges of one of the exceptions into stars. Have a look at this imgur.com/68A3fEe and you'll get the idea. Anyway lots more nice side puzzles to be explored here if you are interested :)

Enjoy!

Burkard

Two ways to support Mathologer
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(see the Patreon page for details)

14. Sep. 2021: Thank you very much Michael Didenko for your Russian subtitles.

What is the best way to lace your shoes? Dream proof.

Mathologer 2020-06-20 | A blast from the past. A video about my fun quest to pin down the best ways of lacing mathematical shoes from almost 20 years ago. Lots of pretty and accessible math. Includes a proof that came to me in a dream (and that actually worked)!

0:00 Intro
1:31 What's a mathematical lacing?
4:42 What does "best" mean?
5:15 What is the shortest lacing? Crisscross and bowtie lacings.
8:42 How to prove that the shortest are the shortest? Travelling salesman problem
12:36 What are the longest lacings? Devil and angel lacings.
13:48 What about real lacings?
15:16 What are the strongest lacings?
17:17 Can proofs hatched in dreams be true?

Some links:
Ian's shoelace site
fieggen.com/shoelace
and his explanations of what's wrong with the way a lot of people tie their shoelaces
fieggen.com/shoelace/grannyknot.htm

John Halton's proof that the crisscross lacing is always the shortest tight lacings
Halton, J.H. The shoelace problem. The Mathematical Intelligencer 17 (1995), 37–41
http://www.cs.unc.edu/techreports/92-032.pdf

My shoelace article in Nature
nature.com/articles/420476a.pdf

A preview of my shoelace book at Google books
https://books.google.com.au/books?id=-dAIAQAAQBAJ&printsec=frontcover&dq=the+shoelace+book&hl=en&sa=X&ved=2ahUKEwjE2bS254_qAhVgxDgGHVENDf8Q6AEwAHoECAUQAg#v=onepage&q=the%20shoelace%20book&f=false

Here is a page on the German travelling salesman problem that I mention in the video
http://www.math.uwaterloo.ca/tsp/d15sol/dhistory.html
I actually got the number of cities a bit wrong. It's 15,112 cites and not 18000.

My article on shoelaces was inspired by this fun article by Thomas Fink and Yong Mao about Designing tie knots by random walks (also in Nature)
tcm.phy.cam.ac.uk/~ym101/tie4/nature_tiepaper.pdf
The extended version
tcm.phy.cam.ac.uk/~tmf20/TIES/PAPERS/paper_physica_a.pdf
They also wrote a really nice book about tie knots
en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie

Enjoy!

Burkard

Eulers infinite pi formula generator

Mathologer 2020-05-02 | Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis's infinite product formula, Lord Brouncker's infinite fraction formula, Euler's Basel formula and it's infinitely many cousins. And we do this starting with one of Euler's crazy strokes of genius, his infinite product formula for the sine function.

This video was inspired by Paul Levrie's one-page article Euler's wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here link.springer.com/journal/283/34/4 Very pretty.

If you are a regular and some of what I talk about in this video looks familiar that's not surprising since we've visited this territory before in Euler's real identity NOT e to the i pi = -1: youtu.be/yPl64xi_ZZA

0:00 Intro
1:49 A sine of madness. Euler's ingenious derivation of the product formula for sin x
7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*.../1*3*3*5*5*...
9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+...
11:50 Brouncker's infinite fraction formula for pi: 4/pi = ...
18:31 Euler's solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+...
21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+... , etc.

Music (all from the free audio library that YouTube provides to creators):
youtube.com/audiolibrary/music?nv=1
Take me to the Depth (chapter transitions)
Fresh fallen snow
Morning mandolin
English country garden

Enjoy!

Burkard

Why did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS

Mathologer 2020-03-14 | The longest Mathologer video ever, just shy of an hour (eventually it's going to happen :) One video I've been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the whole thing took forever to make. Just to give you an idea of the work involved in producing a video like this, preparing the subtitles for this video took me almost 4 hours. Why do anything as crazy as this? Well, just like many other mathematicians I consider the law of quadratic reciprocity as one of the most beautiful and surprising facts about prime numbers. While other mathematicians were inspired to come up with ingenious proofs of this theorem, over 200 different proofs so far and counting, I thought I contribute to it's illustrious history by actually trying me very best of getting one of those crazily complicated proofs within reach of non-mathematicians, to make the unaccessible accessible :) Now let's see how many people are actually prepared to watch a (close to) one hour long math(s) video :)

0:00 Intro
4:00 Chapter 0: Mini rings. Motivating quadratic reciprocity
9:53 Chapter 1: Squares. When is a remainder a square?
16:35 Chapter 2: Quadratic reciprocity formula
24:18 Chapter 3: Intro to the card trick proof
29:22 Chapter 4: Picking up along rows and putting down by columns
29:21 Chapter 5: Picking up along columns and putting down along diagonals
45:16 Chapter 6: Zolotarev's lemma, the grand finale
55:47 Credits

This video was inspired by Matt Baker's ingenious recasting of of a 1830 proof of the LAW by the Russian mathematician Zolotarev in terms of dealing a deck of cards. Here is Matt's blog post that got me started (written for mathematicians):

https://mattbaker.blog/2013/07/03/quadratic-reciprocity-and-zolotarevs-lemma/

If you want to read up on the properties of the sign of a permutation that I am using in this video, Matt also has a nice write-up of this.

mattbakerblog.files.wordpress.com/2014/11/permutations.pdf

The relevant Wiki articles are these:
en.wikipedia.org/wiki/Zolotarev%27s_lemma
en.wikipedia.org/wiki/Quadratic_reciprocity

Zolotarev's original paper lives here:
http://archive.numdam.org/ARCHIVE/NAM/NAM_1872_2_11_/NAM_1872_2_11__354_0/NAM_1872_2_11__354_0.pdf

Here is a list of proofs of the law prepared by Franz Lemmermeyer

https://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html

Franz Lemmermeyer is also the author of the following excellent book on everything to do with quadratic reciprocity (written for mathematicians):

Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Springer Monographs in Mathematics, Berlin

The first teaching semester at the university where I teach is about to start and all my teaching and lots of other stuff will happen this semester. This means I won't have much time for any more crazily time-consuming projects like this. Galois theory will definitely has to wait until the second half of this year :( Still, quite a bit of beautiful doable stuff coming up. So stay tuned.

Thank you to Marty for all the relentless nitpicking of the script, his wordsmithing and throwing cards at me in the video. Thank you to Eddie, Tristan and Matt for all your help with proofreading and feedback on the script and exposition.

Enjoy!

Burkard

Why was this visual proof missed for 400 years? (Fermats two square theorem)

Mathologer 2020-01-25 | Today's video is about a new really wonderfully simple and visual proof of Fermat's famous two square theorem: An odd prime can be written as the sum of two integer squares iff it is of the form 4k+1. This proof is a visual incarnation of Zagier's (in)famous one-sentence proof.

0:00 Intro
2:20 Chapter 1: Discovering a theorem
7:05 Chapter 2: 400 years worth of proofs
9:59 Chapter 3: Zagier's one-sentence proof
15:40 Chapter 4: The windmill trick
22:12 Chapter 5: Windmill maths interlude
25:08 Chapter 6: Uniqueness !!
33:08 Credits

The first ten minutes of the video are an introduction to the theorem and its history. The presentation of the new proof runs from 10:00 to 21:00. Later on I also present a proof that there is only one way to write 4k+1 primes as the sum of two squares of positive integers.

I learned about the new visual proof from someone who goes by the YouTube name TheOneThreeSeven. What TheOneThreeSeven pointed out to me was a summary of the windmill proof by Moritz Firsching in this mathoverflow discussion: mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat
In turn Moritz Firsching mentions that he learned this proof from Günter Zieger and he links to a very nice survey of proofs of Fermat's theorem by Alexander Spivak that also contains the new proof (in Russian): Крылатые квадраты (Winged squares), Lecture notes for the mathematical circle at Moscow State University, 15th lecture 2007: http://mmmf.msu.ru/lect/spivak/summa_sq.pdf
Here is a link to JSTOR where you can read Zagier's paper for free:
jstor.org/stable/2323918
Here are the Numberphile videos on Zagier's proof that I mention in my video:
youtube.com/watch?v=SyJlRUBoVp0
youtube.com/watch?v=yGsIw8LHXM8

Finally here is a link to my summary of the different cases for the windmill pairing that need to be considered (don't read until you've given this a go yourself :)
http://www.qedcat.com/misc/windmill_summary.png

Today's t-shirt is one of my own: "To infinity and beyond"
Enjoy!

P.S.: Added a couple of hours after the video went live:
One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far:
- Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video.
- Challenge 1 at the very end should (of course :) be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4.
-one of you actually ran some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case :)
- one of you actually found this !!! 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2
- a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares :)
- proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangle with short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty :)
- Mathologer videos covering the ticked beautiful proofs in the math beauty pageant:
e^i pi=-1 : youtu.be/-dhHrg-KbJ0 (there are actually a couple of videos in which I talk about this but this is the main one)
infinitely many primes: Mentioned a couple of times: This video has a really fun proof off the beaten track:youtu.be/LFwSIdLSosI
pi^2/6: Again mentioned a couple of times but this one here is the main video: youtu.be/yPl64xi_ZZA
root 2 is irrational: one of the videos in which I present a proof: youtu.be/f1yDExNAEMg
pi is transcendental: youtu.be/9gk_8mQuerg
And actually there is one more on the list, Brouwer's fixed-point theorem that is a corollary of of what I do in this video: youtu.be/7s-YM-kcKME
- When you start with the 11k windmill and then alternate swapping yz and the footprint construction, you'll start cycling through different windmill solutions and will eventually reach one of the solutions we are really interested in. Zagier et al talk about this in an article in the American Mathematical Monthly "New Looks at Old Number Theory" jstor.org/stable/10.4169/amer.math.monthly.120.03.243?seq=1

Fermats Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+...

Mathologer 2019-12-24 | NEW (Christmas 2019). Two ways to support Mathologer
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(see the Patreon page for details)

Leibniz's formula pi/4 = 1-1/3+1/5-1/7+... is one of the most iconic pi formulas. It is also one of the most surprising when you first encounter it. Why? Well, usually when we see pi we expect a circle close-by. And there is definitely no circle in sight anywhere here, just the odd numbers combining in a magical way into pi. However, if you look hard enough you can discover a huge circle at the core of this formula.

Here is a link to the relevant chapter in Hilbert and Cohn-Vossen's book Geometry and the Imagination (Google books). I am pretty sure that the idea and proof for the circle proof of the Leibniz formula that I mathologerise in this video first appeared in this book and is due to the authors: https://books.google.com.au/books?id=7WY5AAAAQBAJ&lpg=PA44&pg=PA37#v=onepage&q&f=false

Here is a link to a video in which 3blue1brown about the same hidden circle in Leibniz formula:
youtu.be/NaL_Cb42WyY
And another video by him about a hidden circle in the solution to the Basel problem:
youtu.be/d-o3eB9sfls

There is also a neat generalisation to what we talked about in this video to the solution of the Basel problem - in terms of the lattice points in a 4-dimensional sphere and the 4-square counterpart of the 4(good-bad) theorem. If you are interested in some details have a look at the last proof in this write-up by Robin Chapman: empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

Links to two Numberphile videos about the one-sentence proof by Don Zagier featuring Matthias Kreck: youtu.be/SyJlRUBoVp0 (intro), youtu.be/yGsIw8LHXM8 (the math)

Link to the original Jodocus Hondius engraving of Jodocus Hondius that Google tries to pass of as a portrait of the mathematician Albert Girard
swaen.com/zoomV5e.php?id=8653&referer=antique-map-of.php

Thank you very much to Marty for all his help with polishing the script of the video and Karl for his idea for the 2019 Easter egg.

Today's t-shirt: google "spreadshirt pi+tree+christmas+math"

Enjoy :)

Secret of row 10: a new visual key to ancient Pascalian puzzles

Mathologer 2019-11-30 | NEW (Christmas 2019). Two ways to support Mathologer
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Today's video is about a recent chance discovery (2002) that provides a new beautiful visual key to some hidden self-similar patterns in Pascal's triangle and some naturally occurring patterns on snail shells. Featuring, Sierpinski's triangle, Pascal's triangle, some modular arithmetic and my giant pet snail shell.

Thank you very much to Marty for all his help with finetuning the script for this video and to Steve Humble and Erhard Behrends for making some photos available to me.

Enjoy :)

P.S.: The article I mentioned in this video is: Steve Humble, Erhard Behrends, ”Triangle Mysteries“, The Mathematical Intelligencer 35 (2), 2013, 10-15. There is also a followup article:
”Pyramid Mysteries“, The Mathematical Intelligencer 36 (3), 2014, 14 - 19.
And there is a book by Erhard Behrends that has a couple of chapters dedicated to this topic: The Math Behind the Magic: Fascinating Card and Number Tricks and How They Work: bookstore.ams.org/mbk-122 :)

A Wolfram demonstration project that implements the 3-color game: http://demonstrations.wolfram.com/TriangleMysteries

Philip Smolen contributed this animation
trade-ideas.com/home/phil/Triangles/Circles.html

Someone pointed out these links to some code wars problems:
codewars.com/kata/coloured-triangles
codewars.com/kata/5a331ea7ee1aae8f24000175

Juan Mir Pieras pointed out these earlier references:
http://mathcentral.uregina.ca/mp/archives/previous2000 (Problem of the month June 2001)(http://mathcentral.uregina.ca/mp/archives/previous2000/june01sol.html) The webpage says the problem is from "Crux Mathematicorum 27:3 (April 2001) pages 204-205 - it is problem 3 from the Ninth Annual Konhauser Problemfest (Carleton College, prepared by David Savitt and Russell Mann."
https://www.macalester.edu/mscs/studentopportunities/competitions/konhauserproblemfest/kp2001/
cms.math.ca/crux/v27/n3/CRUXv27n3.pdf

Today's t-shirt: teepublic.com/en-au/posters-and-art-prints/5475843-funny-math-i-cant-even

Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)

Mathologer 2019-10-26 | The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :)

Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only real specialists dare to touch. I worked really hard on this one. Fingers crossed that after all this work the video now works for you :) Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite “proofs without words”, the mysterious Bernoulli numbers (the numbers to "rule them all" as far as power sums go), the (hopefully) most accessible introduction to the Euler-Maclaurin summation formula ever, and much more.

Here is a link to a couple of slides that show how to justify having all summations in sight run from 1 to n. This is the challenge that follows the discussion at youtu.be/fw1kRz83Fj0?t=2440 . Had this in the video originally and then decided to make this into a challenge: http://www.qedcat.com/misc/change_limits.pdf

As usual thank you very much to my friend Marty Ross for nitpicking this one to death (especially for not letting off until I finally inserted that "morph" shortcut in chapter 7 :)

Finally, check out the article "Gauss’s Day of Reckoning" by Brian Hayes which tells the story behind the famous story of Gauss adding 1+2+3+...+100 as a kid: tinyurl.com/y49buyak

Enjoy :)

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2000 years unsolved: Why is doubling cubes and squaring circles impossible?

Mathologer 2019-06-29 | Today's video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles?

00:00 Intro
05:19 Level 1: Euclid
08:57 Level 2: Descartes
16:44 Level 3: Wantzel
24:00 Level 4: More Wantzel
31:30 Level 5: Gauss
35:18 Level 6: Lindemann
40:22 Level 7: Galois

Towards the end of a pure maths degree students often have to survive a "boss" course on Galois theory and somewhere in this course they are presented with proofs that it is actually not possible to accomplish any of those four troublesome tasks. These proofs are easy consequences of the very general tools that are developed in Galois theory. However, taken in isolation, it is actually possible to present proofs that don't require much apart from a certain familiarity with simple proofs by contradiction of the type used to show that numbers like root 2 are irrational.

I've been meaning to publish a nice exposition of these "simple" proofs ever since my own Galois theory days (a long, long time ago.
Finally, today is the day :)

For some more background reading I recommend:
1. chapter 3 of the book "What is mathematics?" by Courant and Robbins (in general this is a great book and a must read for anybody interested in beautiful maths).
2. The textbook "Field theory and its classical problems" by Hadlock (everything I talk about and much more, but you need a fairly strong background in maths for this one).

Here is a great two-page summary by the mathematician Drew Armstrong of what is going on in this video
http://www.math.miami.edu/~armstrong/461sp11/ImpossibleConstructions.pdf

Here is a derivation of the cubic polynomial for the regular heptagon construction by (I think) the mathematician Reinhard Schultz http://math.ucr.edu/~res/math153/s10/history09a.pdf
(there is a little typo towards the bottom of the page. It should be 8 cos^3 theta + 4 cos^2 theta - (!) 4 cos theta -1 = 0. Replace cos theta by x and you get the cubic equation I mention in the video. )

Here is an interesting paper that explores why Wantzel's results did not get recognised during his lifetime sciencedirect.com/science/article/pii/S031508600900010X

Thank you to Marty and Karl for your help with creating this video. And thank you to Cleon Teunissen for pointing out that the picture of Pierre Wantzel that I use in this video is actually not showing Pierre Wantzel but rather Gustave Gaspard de Coriolis who was also a mathematician and lived around the same time as Pierre Wantzel. It appears that whenever there does not exist an actual picture of some person Google and other internet gods simply declare some more or less random picture to be the real thing. See also this page by the SciFi writer Greg Egan who made sure that no actual picture of himself is to be found on the internet: gregegan.net/images/GregEgan.htm

Enjoy :)

Burkard

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Why did we forget this simple visual solution? (Lills method)

Mathologer 2019-04-26 | Today's video is about Lill's method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horner's form, synthetic division and a newly discovered incarnation of Pascal's famous triangle.

00:00 Intro
04:14 Lill's method
07:31 Free meal
09:51 Square turtles
11:39 Origami turtles
14:16 Iterative turtles
17:32 QED
24:00 Pascal's turtle animation

Here is the page with an implementation of Lill's method for cubic polynomials that I show in the video.

http://www.qedcat.com/misc/lill_method

It's an adaptation of this webpage

http://heim.ifi.uio.no/magho/lill/

(I have not been able to find out who put this together originally).

The article that inspired this video is this:

Thomas C. Hull, Solving Cubics With Creases: The Work of Beloch and Lill, The American Mathematical Monthly , Vol. 118, No. 4 (April 2011), pp. 307-315. Here is a link to this article on Thomas Hull's webpage: http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf

Lill's original paper:
http://www.numdam.org/article/NAM_1867_2_6__359_0.pdf

Other good references include:
Polynomials as polygons by Serge Tabachnikov
https://www.math.psu.edu/tabachni/prints/Polynomials.pdf

Dan Kalman's book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lill's method)
https://books.google.com.au/books?id=JPq0pS3wrx4C&pg=PA7&source=gbs_toc_r&cad=3#v=onepage&q&f=false

Thank you very much to Marty, Karl and Danil for their help with this video.

One version of today's math t-shirt (Zombie addition): redbubble.com/people/manikx/works/8929883-zombie-math?p=t-shirt

The piece of music at the end is called "Fresh fallen snow" by Chris Haugen from the free YouTube music library.

Really neat 1-line Mathematica code for the generation of the Pascal turtle which appeared on Reddit after the video was posted there:
Graphics[Table[Line[ReIm[Accumulate[Table[2^(-n/2)Binomial[n,k]Exp[I(4+2k-n)Pi/4],{k,-1,n}]]]],{n,0,7}]]
and another nice implementation in Python (with a real turtle graphics turtle) by Alex Hall https://repl.it/repls/DeepskyblueFractalPoint

Enjoy :)

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The Secret of Parabolic Ghosts

Mathologer 2019-03-16 | NEW (Christmas 2019). Two ways to support Mathologer
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Today we'll perform some real mathematical magic---we'll conjure up some real-life ghosts. The main ingredient to this sorcery are some properties of x squared that they don't teach you in school. Featuring the mysterious whispering dishes, the Mirage hologram maker and some origami x squared.paper magic.

Here is a nice article about the Mirage hologram gadget. A complete ray-trace analysis of the Mirage toy by
Sriya Adhya, John W. Noé. The history part in particular makes for a fascinating read: spiedigitallibrary.org/conference-proceedings-of-spie/9665/966518/A-complete-ray-trace-analysis-of-the-Mirage-toy/10.1117/12.2207520.full?SSO=1

As usual, thank you very much to Marty and Danil for their help with this video.

Enjoy :)

New Reuleaux Triangle Magic

Mathologer 2019-02-16 | NEW (Christmas 2019). Two ways to support Mathologer
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Today's video is about plane shapes that, just like circles, have the same width in all possible directions. That non-circular shapes of constant width exist is very counterintuitive, and so are a lot of the gadgets and visual effects that are "powered" by these shapes: interested in going for a ride on non-circular wheels or drilling square holes anybody?

While the shapes themselves and some of the tricks they are capable of are quite well known to maths enthusiasts, the newly discovered constant width magic that today's video will culminates in will be new to pretty much everybody watching this video (even many of the experts :)

Here are a few links that you may want to check out:
http://www.etudes.ru/en/etudes/drilling-square-hole/
Drilling a square hole with rounded corners using a Reuleaux triangle (click on the video !!)
http://www.etudes.ru/en/etudes/reuleaux-triangle/ Same (wonderful) Russian site. An animated intro to shapes of constant width.
http://www.etudes.ru/en/etudes/wheel-inventing/ An animation of the cart with non-circular wheels that I talk about in the video.

http://www.qedcat.com/articles/waterwheel1.pdf Preprint of my write-up of all the stuff I talk about in this video. This was published in the Mathematical Intelligencer.

demonstrations.wolfram.com/DrillingASquareHole An interactive demo illustrating how a perfect square hole (NO rounded corners) can be drilled using a special shape of constant width.

Probably the most accessible intro to shapes of constant width is the chapter on these shapes in the book "The enjoyment of mathematics" by Rademacher and Toeplitz.

This article which I also mention in the video is behind a paywallMasferrer Leon, C. and Von Wuthenau Mayer, S. Reinventing the Wheel: Non-Circular Wheels, The Mathematical Intelligencer 27 (2005), 7–13.

Just found a Japanese toyshop the other day that sells wooden Nothing grinders global.rakuten.com/en/store/good-toy/item/c-006?s-id=rgm-top-en-browsehist and a wooden Reuleaux triangle that can be rotated inside a square global.rakuten.com/en/store/good-toy/item/c-018?s-id=rgm-top-en-browsehist

I didn't mention them in the video but there are also 3d shapes of constant width which are also very much worth checking out. All the touching stuff I talk about in this video generalises to these 3d shapes.

teepublic.com/t-shirt/626201-schrodingers-surprise
Today's t-shirt

The tune you can hear in the video is from the free audio library that YouTube provides to creators. youtube.com/audiolibrary/music . It's called Morning_Mandolin and it's by Chris Haugen.

As usual thank you very much to Danil for his Russian translation and to Marty for all his help with the script for this video.

Enjoy :)

The secret of the 7th row - visually explained

Mathologer 2019-01-26 | In 1995 I published an article in the Mathematical Intelligencer. This article was about giving the ultimate visual explanations for a number of stunning circle stacking phenomena. In today's video I've animated some of these explanations.

Here is a copy of a preprint of the Intelligencer article:
http://www.qedcat.com/misc/stacks.pdf

And here are links to a few beautiful interactive animations of the circle stacking marvels that I talk about in this video (on the Cut-the-knot site):

cut-the-knot.org/Curriculum/Geometry/NBallsAtBottom.shtml

cut-the-knot.org/Curriculum/Geometry/BallsInJar.shtml

(check for more links to related animations at the bottom of these pages)

As usual, many thanks go to Marty for all his help in getting this presentation just right and Danil for his Russian subtitles. Also, thank you very much dad for your help with building the stacking machine that features at the end of this video.

Enjoy :)

Irrational Roots

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For the final video for 2018 we return to obsessing about irrational numbers. Everybody knows that root 2 is irrational but how do you figure out whether or not a scary expression involving several nested roots is irrational or not? Meet two very simple yet incredibly powerful tools that they ALMOST told you about in school. Featuring the Integral and Rational Root Theorems, pi Santa, e(lf), and a really cringy mathematical Christmas carol.

As usual thank you very much to Marty and Danil for their help with this video.

Merry Christmas :)

Secrets of the NOTHING GRINDER

Mathologer 2018-12-07 | NEW (Christmas 2019). Two ways to support Mathologer
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This video is the result of me obsessing about pinning down the ultimate explanation for what is going on with the mysterious nothing grinder aka the do nothing machine aka the trammel of Archimedes. I think what I present in this video is it in this respect, but I let you be the judge. Featuring the Tusi couple (again), some really neat optical phenomenon based on the Tusi couple (I first encountered this here: youtu.be/pNe6fsaCVtI), the ellipsograph and lots of original twists to an ancient theme.

Here is a link to a .zip archive containing 3d printable .stl files of the models featuring the Mathologer logo that I showed in the video: http://www.qedcat.com/misc/grinder.zip

I usually trim the corners and excess material off the (slightly slanted) vertical edges of the three sliders to make them run without catching on anything. I also sharpen the points of the pins a bit before pushing them into the sliders. They lock in place automatically, you don't have to glue them in.

Other 3d printable incarnations featuring different numbers of sliders are floating around on the net—for example search nothing grinder/do nothing machine/Archimedes trammel on thingiverse.com

The wiki page on the nothing grinder is also worth visiting: en.wikipedia.org/wiki/Trammel_of_Archimedes

My current bout of nothing grinder obsession started with Naomi a year 10 student from Melbourne who did a week of mathematical work experience with me at Monash university a couple of weeks ago. As her project she chose to design a 3d printable version of the wooden model that you see in the video. Her Rhino3d files of the square and hexagon grinders served as the starting point for the models you can see in action in the video.

T-shirt: tinyurl.com/ybl55hez

As usual thank you very much to Marty and Danil for their help with this video.

Enjoy :)