Scholar Sauce
PEMDAS is a Lie! (Or why it should really be PEMA)
updated
*The copyright of the original Animation vs. Geometry video is owned by Alan Becker. This video is made under fair use as it includes significant commentary and is transformative. All viewers are encouraged to watch the original Alan Becker video in its entirety at the link below. It's pretty awesome!*
*Video Used*
_Animation vs. Geometry_
Author: Alan Becker
YouTube Link: youtu.be/VEJWE6cpqw0?si=qQczavcYQKKdvTFI
Used under Fair Use with Commentary
*Music Used*
_Elevators Need Rock Too_
By Spence
_Fantasyland_
By Quincas Moreira
*Pictures Used*
_Fractal Antenna_
Source: commons.wikimedia.org/wiki/File:Fractal_antenna_patent.gif
License: Public Domain
_Sierpinski Carpet Fractal_
Author: Nadim-Daniel Ghaznavi
Source: commons.wikimedia.org/wiki/File:Sierpinski-carpet.gif
License: CC-BY-4.0 (creativecommons.org/licenses/by/4.0/deed.en)
Changes: No changes made to the file, but the file was added into this video.
_Mandelbrot Sequence New_
Author: Simpsons Contributor
Source: commons.wikimedia.org/wiki/File:Mandelbrot_sequence_new.gif
License: Public Domain
_Platonic Solids Transparent_
Author: Drummyfish
Source: commons.wikimedia.org/wiki/File:Platonic_Solids_Transparent.svg
License: CC0-1.0 Public Domain
_Cover of Flatland: A Romance of Many Dimensions / with Illustrations by the Author, A Square_
Source: Harvard University Houghton Library
Link: commons.wikimedia.org/wiki/File:Houghton_EC85_Ab264_884f_-_Flatland,_cover.jpg
License: Public Domain
_Animated 8-cell Tessaract_
Author: Jason Hise
Source: commons.wikimedia.org/wiki/File:8-cell-orig.gif
License: CC0-1.0 Public Domain
youtu.be/6G-ty4KVVMc
youtu.be/6G-ty4KVVMc
youtu.be/6G-ty4KVVMc
youtu.be/6G-ty4KVVMc
youtu.be/6G-ty4KVVMc
Dan McClellan's YouTube Channel:
youtube.com/@maklelan
Images Used:
Archimedes
Artist: Andre Thevet
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Archimedes1.jpg
Babylonian Tablet
Yale Babylonian Collection, YBC 07302
Source: https://cdli.mpiwg-berlin.mpg.de/artifacts/255051
Bible Closed
Artist: crisg
License: Public Domain
Source: http://www.publicdomainfiles.com/show_file.php?id=13933354812670
Music:
Top of the Morning by TrackTribe
Little Umbrellas by by TrackTribe
Survive the Montage by RKVC
Punk In Donuts by Hanu Dixit
youtu.be/6G-ty4KVVMc
youtu.be/Fs7uMI_7PEo
Videos Used:
Doctor Who
"42"
Copyright BBC Worldwide
Used under fair use with commentary.
Images Used:
Collatz Orbits Directed Graph
Author: Keenan Pepper
License: Public Domain
Music:
Shake It by Aakash Gandhi
Honky Tonk Jazzy Theme by Freedom Trail Studio
Controlled Distress by Biz Baz Studio
youtu.be/Fs7uMI_7PEo
youtu.be/Fs7uMI_7PEo
youtu.be/Fs7uMI_7PEo
youtu.be/Fs7uMI_7PEo
youtu.be/Fs7uMI_7PEo
youtu.be/Fs7uMI_7PEo
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
In his book, the Elements written in 300 BCE, Euclid described planar geometry using five postulates or axioms. One of these axioms was considerably more complicated than the others and seemed to some to be likely provable as a theorem from the other four more natural postulates. But no ancient Greek was ever able to prove Euclid's fifth postulate and this question persisted for over 2000 years until it was finally resolved in the 19th century CE. It took 2000 years to confirm that indeed you cannot prove Euclid's fifth postulate from the other four, but this proof resulted in us learning of another type of plane geometry that is completely consistent and satisfies the first four of Euclid's postulates but not the fifth. This type of geometry is called hyperbolic geometry and weird stuff happens there: triangles have less than 180 degrees, similar triangles are congruent, and rectangles are nowhere to be found. This discovery inspired further advancements in geometry including Riemannian and semi-Riemannian geometry which is the mathematical framework that Albert Einstein used to develop general relativity and advance our understanding of gravity and the universe.
Link to Part 1: youtu.be/pmGDea9ZQ4U
*Images*
_The Acropolis at Athens_
Artist: Leo von Klenze
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Akropolis_by_Leo_von_Klenze.jpg
_Papyrus Fragment of Euclid's Elements_
Artist: Euclid
License: Public Domain
Acquired: http://www.math.ubc.ca/~cass/Euclid/papyrus/tha.jpg
_Euclid of Megara_
Artist: Justus van Gent
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Euklid2.jpg
_Nikolai Lobachevsky_
Artist: Lev Kryukov
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Lobachevsky.jpg
_Bust of Janos Bolyai_
Artist: Szilas (photographer)
License: Creative Commons CC0 1.0 Universal Public Domain Dedication
Acquired: commons.wikimedia.org/wiki/File:Bust_of_J%C3%A1nos_Bolyai_at_the_Mikl%C3%B3s_Zr%C3%ADnyi_Military_Academy.jpg
_Johann Carl Friedrich Gauss_
Artist: Christian Albrecht Jensen
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg
_Eugenio Beltrami_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Beltrami.jpg
_Felix Klein_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Felix_Christian_Klein.jpg
_Bernhard Riemann_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Bernhard_Riemann_3.jpg
_Henri Poincare_
Artist: Unknown
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Henri_Poincar%C3%A9-2.jpg
_Hendrik Lorentz_
Artist: Museum Boerhaave
License: Pbulic Domain
Acquired: commons.wikimedia.org/wiki/File:Hendrik_Antoon_Lorentz.jpg
_Hermann Minkowski_
Artist: Raum und Zeit
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Hermann_Minkowski_Portrait.jpg
_Albert Einstein_
Artist: Ferdinand Schmutzer
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Einstein_1921_by_F_Schmutzer_-_restoration.jpg
*Videos*
All video clips are short and used under Fair Use as this work is transformative in each case.
_Star Trek IV: The Voyage Home_
Copyright Paramount Pictures Corporation
Acquired: yarn.co/yarn-clip/fd1c3db4-8966-4c1a-b2d0-12f2987f7851 and yarn.co/yarn-clip/242688c2-6dee-48e2-89fd-29485f7526b3
_The Flash (TV Series)_
"_The Darkness and the Light_"
Copyright Warner Bros. Entertainment, Inc.
Acquired: yarn.co/yarn-clip/3cf6cd52-945b-4934-8b8b-acf80bdfb2c7
_Justice League_
Copyright Warner Bros. Entertainment, Inc.
Acquired: yarn.co/yarn-clip/a70a67af-8dc5-46bc-bb17-c57ce2813636
_Avengers: Infinity War_
Copyright Marvel
Acquired: yarn.co/yarn-clip/e8dba66a-db48-49dd-a172-5d4507822d4a
_Despicable Me 2_
Copyright Universal Pictures
Acquired: yarn.co/yarn-clip/fc26c065-61cf-4844-a951-033164bffa6b
_Star Trek: The Next Generation_
"_Sarek_"
Copyright CBS Studio Inc.
Acquired: yarn.co/yarn-clip/02c82a09-7335-4f8d-92ce-4dd07f821acf
_The BEST FALLS on Runway by Fashion Channel_
Acquired: youtube.com/watch?v=kYLE_kp6coQ
*Music*
Various Tracks from the YouTube Audio Library
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
In his book, the Elements written in 300 BCE, Euclid described planar geometry using five postulates or axioms. One of these axioms was considerably more complicated than the others and seemed to some to be likely provable as a theorem from the other four more natural postulates. But no ancient Greek was ever able to prove Euclid's fifth postulate and this question persisted for over 2000 years until it was finally resolved in the 19th century CE. It took 2000 years to confirm that indeed you cannot prove Euclid's fifth postulate from the other four, but this proof resulted in us learning of another type of plane geometry that is completely consistent and satisfies the first four of Euclid's postulates but not the fifth. This type of geometry is called hyperbolic geometry and weird stuff happens there: triangles have less than 180 degrees, similar triangles are congruent, and rectangles are nowhere to be found. This discovery inspired further advancements in geometry including Riemannian and semi-Riemannian geometry which is the mathematical framework that Albert Einstein used to develop general relativity and advance our understanding of gravity and the universe.
*Books*
_Foundations of Geometry_
Author: Gerard Venema
Link: pearson.com/en-us/subject-catalog/p/foundations-of-geometry/P200000006404/9780136845188
*Images*
_The Acropolis at Athens_
Artist: Leo von Klenze
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Akropolis_by_Leo_von_Klenze.jpg
_Papyrus Fragment of Euclid's Elements_
Artist: Euclid
License: Public Domain
Acquired: http://www.math.ubc.ca/~cass/Euclid/papyrus/tha.jpg
_Euclid of Megara_
Artist: Justus van Gent
License: Public Domain
Acquired: commons.wikimedia.org/wiki/File:Euklid2.jpg
_Bronze Age Dwelling at Flag Fen_
Artist: Ian and Viv Hamilton
License: Public Domain
Acquired: en.m.wikipedia.org/wiki/File:Bronze_age_dwelling_at_Flag_Fen.JPG
_Modern Building_
Artist: Sharee Basinger
License: Public Domain
Acquired: http://www.publicdomainfiles.com/show_file.php?id=13969228218118
_1910 Ford-T_
Artist: Harry Shipler
License: Public Domain
Acquired: en.m.wikipedia.org/wiki/File:1910Ford-T.jpg
_Mercedes Benz Sports Car_
Artist: Alex Borland
License: Public Domain
Acquired: publicdomainpictures.net/en/view-image.php?image=256545&picture=mercedes-benz-sports-car
*Music*
Various Tracks from the YouTube Audio Library
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
youtu.be/Rov5IjhWUP0?si=6_rIKnz8P8XZNumS
*Images*
_Potrait of Zeno of Elea_ (1699)
Artist: Bernard Picart (1673-1733)
License: Public Domain
Acquired: lookandlearn.com/history-images/YR0044610/Portrait-of-Zeno-of-Elea?t=2&q=Portrait+of+Zeno+of+Elea&n=2
_Ice Cubes_
Artist: George Hodan
License: Public Domain
Acquired: publicdomainpictures.net/en/view-image.php?image=164892&picture=ice-cubes
*Videos*
_Fastest Rubik's cube 3x3 WR (3.13) sec WORLD RECORD_
Credit: Max Park
Link to Video: youtube.com/watch?v=gh8HX4itF_w
*Music*
_I Had a Feeling_ by TrackTribe
_March On_ by Ethan Meixsell
Not all the facts are explaining in a reasonable length video, but you might spot a fun Star Trek retcon explanation related to one of the facts and a couple of great puns instead. Below are the time stamps for each fact if you want to jump to just one of them.
0:35 i^i is a real number
8:55 e^(i pi) + 1 = 0
10:08 Large numbers are incomprehensibly big, but never get any closer to infinity
13:08 Mobius band
14:34 Klein bottle
16:50 Fermat's Last Theorem and a Star Trek retcon
21:55 Solvable polynomials and Galois Theory (puns/wordplay on Abel and Galois)
24:52 It only takes 23 people in a room to have a more than 50% chance two of them have the same birthday
29:59 It is practically impossible that any deck of 52 cards have ever been shuffled the same way twice
*Andrew Wiles paper proving Fermat's Last Theorem:*
_Modular Elliptic Curves and Fermat's Last Theorem_
http://www.scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf
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*Music*
_End of the Rainbow_ by Quincas Moreira
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*Images*
_Pierre de Fermat_
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Pierre_de_Fermat.jpg
_Hands holding a bunch of playing cards_
License: Public Domain
Source: goodfreephotos.com/public-domain-images/hand-holding-a-bunch-of-playing-cards.jpg.php
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*Video*
Star Trek: The Next Generation
_The Royale_
Copyright CBS Studios Inc.
Short clip used under Fair Use as this work is transformative and contains significant commentary on the clip.
Star Trek: Deep Space Nine
_Facets_
Copyright CBS Studios Inc.
Short clip used under Fair Use as this work is transformative and contains significant commentary on the clip.
Images:
Neanderthal Flintworkers, Le Moustier Cavern, DorDogne, France
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Neanderthal_Flintworkers_(Knight,_1920).jpg
Galactic Year
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Milky_Way_Arms_ssc2008-10.svg
8-cell
License: Public Domain
Source: commons.wikimedia.org/wiki/File:8-cell.gif
2MASS LSS chart-New Nasa
License: Public Domain
Source: commons.wikimedia.org/wiki/File:2MASS_LSS_chart-NEW_Nasa.jpg
Revolving Earth
License: Public Domain
Source: NASA/Goddard Space Flight Center Scientific Visualization Studio
National Debt Screen Capture
Website captured: Peter G. Peterson Foundation
pgpf.org/national-debt-clock
Andrew Wiles in 2005
Copyright C. J. Mozzochi, Princeton N.J
Used in compliance with license granted by copyright holder.
Source: commons.wikimedia.org/wiki/File:Andrew_wiles1-3.jpg
Modular Elliptic Curves and Fermat's Last Theorem
Screen Grab of first page
Source: http://www.scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf
Cubic Formula
Source: artofproblemsolving.com/wiki/index.php/Cubic_Equation
Quartic Formula
Source: artofproblemsolving.com/wiki/index.php/Quartic_Equation
Hands holding a bunch of playing cards
License: Public Domain
Source: goodfreephotos.com/public-domain-images/hand-holding-a-bunch-of-playing-cards.jpg.php
Background of playing cards
License: Public Domain
Source: publicdomainpictures.net/en/view-image.php?image=18743&picture=background-of-playing-cards
Timeline of the Universe
License: Public Domain
Source: en.wikipedia.org/wiki/File:CMB_Timeline300_no_WMAP.jpg
Evariste Galois
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Evariste_galois.jpg
Neils Henrik Abel
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Niels_Henrik_Abel_(1802_-_1829).jpg
Paolo Ruffini
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Ruffini-paolo-colour.webp
Pierre de Fermat
License: Public Domain
Source: commons.wikimedia.org/wiki/File:Pierre_de_Fermat.jpg
Music:
Strollin' by TrackTribe
Shake It by Aakash Gandhi
I Had a Feeling by TrackTribe
Rolling Hills by Sir Cubworth
Confident Kurt by TrackTribe
Honky Tonk Jazzy Theme by Freedom Trail Studio
Waterpark Radio Ad by Freedom Trail Studio
Music
Good Days by Yung Logos
Rainbow Forest by Quincas Moreira
Smokey's Lounge by TrackTribe
Play-Doh Meets Dora by Carmen Maria and Edu Espinal
Modern Attempt by TrackTribe
This video was submitted to the #some3 competition sponsored by @3blue1brown and @LeiosLabs.
---------------
*Music*
_Elevators Need Rock Too_ by Spence
*Images*
_Blue Marble 2002_
Source: NASA Visible Earth: Blue Marble: Land Surface, Shallow Water, and Shaded Topography
Link: en.wikipedia.org/wiki/File:Blue_Marble_2002.png
License: Public Domain
*Video Game Captures (Fair Use)*
_Final Fantasy VI Pixel Remaster_
Square Enix
_Final Fantasy VII_
Square Enix
_Final Fantasy VIII Remastered_
Square Enix
_Wild Arms_
Media Vision
If you like this video and math and science edutainment, please subscribe!
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*Music*
_Frailich 2_ by E's Jammy Jams
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*Music*
_Survive the Montage_ by RKVC
---------------
*Images*
_Blue Marble 2002_
Source: NASA Visible Earth: Blue Marble: Land Surface, Shallow Water, and Shaded Topography
Link: en.wikipedia.org/wiki/File:Blue_Marble_2002.png
License: Public Domain
*Music*
_Elevators Need Rock Too_
Artist: Spence
Source: YouTube Audio Library
If you haven't yet (and you took the time to read this), please go ahead and click that subscribe button. It helps us be able to create more content like this!
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*Music*
_End of the Rainbow_
Quincas Moreira
*Images*
_The Runners_
Published in Brockton, MA in 1900
Link: loc.gov/item/2018696402
*Movie Clips (Fair Use)*
*Obtained from yarn.co*
_The Wizard of Oz_
Metro-Goldwyn-Mayer, 1939
2 clips
_Rookie of the Year_
20th Century Fox, 1993
_The Crown_
Left Bank Pictures and Sony Pictures Television, 2016
Since 2016, Dr. Palais and Dr. Alan Parry have held several debates on this topic as I think that we should not abandon π so easily. In today's video, Dr. Parry takes on the Godfather of τ in a debate that was filmed on Pi Day 2023 and is being released on Scholar Sauce on Tau Day 2023. We hope that you enjoy it. Let us know in the comments which circle constant you prefer and why. And don't forget to eat two pies today!
--------------------------------------------
Images
Most images from Dr. Palais' slides have links to the images within the slide in the video.
Google Earth animation created by Alan Parry using Google Earth.
Other animations courtesy of Dr. Bob Palais.
Remaining images mainly created using either MS Powerpoint or GeoGebra.com.
---------------------------------------------
Video Clips
Tau Replaces Pi
Numberphile
Clip from 3:09-3:40
Link: youtu.be/83ofi_L6eAo
Pi Is (still) Wrong.
Vihart
Clip from 4:43-5:02
Link: youtu.be/jG7vhMMXagQ
Please subscribe to both Numberphile and Vihart, they're pretty awesome channels!
Quite often people think that π is an amazing number because it has an infinite decimal representation that doesn't repeat indefinitely. But these facts are true of the vast majority of real numbers and so really aren't what makes π special. In this video, we talk about why π is not special for these popular properties that it has, and highlight why it really is special.
Music
Smokey's Lounge by Track Tribe
Subscribe to the channel for more fun math and science content!
Check out other amazing math and science facts on Scholar Sauce. Be one of the first 500 subscribers!
Music:
Ditch Diggin' by Jingle Punks
This is part of our series of Meme Review Videos where Dr. Alan Parry, Dr. Zach Hurdle, and Dr. Wiktor Mogilski take turns reviewing popular math memes and comics.
Pictures/Memes:
Varepsilon
Large vs. Two Pizzas
Matrix Transform
Divisibility by 7
The Natural Numbers
Source: reddit.com/r/mathmemes
Fields Arranged by Purity
Mathematically Annoying Advertisements
Source: xkcd.com
Math Procedures and Historical Context
Counting Episodes
Source: scrolldroll.com/funny-maths-memes
Transforming Calculus 2 Integral
Source: tumblr.com/math-memes
Music:
March On by Ethan Meixsell
Thor's Hammer by Ethan Meixsell
Wicked Things by Quincas Moreira
Coast by Anno Domini Beats
Island Summer Breeze by Coyote Hearing
Check out other amazing math and science facts on Scholar Sauce. Be one of the first 500 subscribers!
Images:
Earth
Source: Google Earth
M81 Galaxy
Source: images.nasa.gov/details-PIA09337
Music:
Stars Align by The 126ers