Conway's IRIS and the windscreen wiper theorem @Mathologer

Mathologer | Conway's IRIS and the windscreen wiper theorem @Mathologer | Uploaded 6 months ago | Updated 1 hour ago
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

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

Numberphile v. Math: the truth about 1+2+3+... 1/12

The ARCTIC CIRCLE THEOREM or Why do physicists play dominoes?

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

Infinity shapeshifter vs. Banach-Tarski paradox

Fermat’s HUGE little theorem, pseudoprimes and Futurama

$The fabulous Fibonacci flower formula You probably know that nature is crawling with the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, etc. But have you ever seen a simple explanation for this phenomenon? This video is the result of my own quest to distill a really accessible explanation from existing research. Enjoy :) In the last video on continued fractions I mentioned that part of the explanation involves the golden ratio and the fact that this number is the most irrational number. Ill talk about this in a follow-up video. If you cannot wait check out this website: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Also check out the following video produced as part of research by Douady and Couder about how simple displacement at the center of a plant gives rise to Fibonacci numbers of spirals https://youtu.be/U-at-y3MicE The paper itself can be found here https://www.math.ntnu.no/~jarlet/Douady96.pdf Another very interesting approach by Levitov involves a magnetic cactus, vortices in superconductors and the fabulous Farey numbers: http://www.ams.org/samplings/feature-column/fcarc-phyllotaxis#2$

Death by infinity puzzles and the Axiom of Choice

Liouvilles number, the easiest transcendental and its clones (corrected reupload)

$Way beyond the golden ratio: The power of AB=A+B (Mathologer masterclass) Todays mission: saving another incredible discovery from falling into oblivion: Steinbachs amazing infinite family of counterparts of the golden ratio discovered around 1995. Lots 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 Steinbachs papers articles: Sections beyond golden: https://archive.bridgesmathart.org/2000/bridges2000-35.html Golden fields: a case for the heptagon: https://www.jstor.org/stable/2691048 A good online writeup with some extra insights (on a site dedicated to sacred geometry!: https://tinyurl.com/4jhju7dw A paper citing Peter Steinbachs paper https://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 https://tinyurl.com/bduzayr3 Alan H. Schoens 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 Ptolemys theorem features a great visual animated proof https://en.wikipedia.org/wiki/Ptolemy Some relevant Mathologer videos: The golden ratio spiral: visual infinite descent: https://youtu.be/ubHVK71F01M Phi and the TRIBONACCI monster https://youtu.be/e7SnRPubg-g The fabulous Fibonacci flower formula https://youtu.be/_GkxCIW46to Infinite fractions and the most irrational number https://youtu.be/CaasbfdJdJg Golden ratio fact and fiction. Check out this paper by Georg Markowsky: https://www.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. Whats Binets formula in that case? 4. For which complex numbers n does Binets formula spit out an integer/a real number? 5. Is it a coincidence that there is a 1/7 in my Binets formula for the heptagon? (You can make the 1/5 th appear in Binets formula itself by multiplying both denominator and numerator by phi +1/phi.) T-shirt: Fibonightmare https://www.teepublic.com/t-shirts?query=Fibonightmare Music: Kashido - When you go out and Ardie Son - Spread your wings Enjoy! Burkard$