Mathologer

Parity of permutations, impossible puzzles and the magical determinant

48K views · 7mo ago · 38:56

This is a video I've been meaning to do for a long, long time. I've used the parity of permutations in quite a few videos and I've repeatedly promised to give a proper explanation in a separate video. This is it! Parity of permutations, the distinction between even and odd permutations or rearrangements, is one of the simplest nontrivial invariants in mathematics, yet it has far-reaching consequences across many areas.

Topics: Permutations, inversions and parity · Identity permutation and swaps flip parity · odd + odd = even · The 15-puzzle · My permutation visualiser app · The Rubik's cube (corners)

▶️ Mentioned videos: I Built an Original One-Glance Proof Dice · Video on youtube.com · Video on youtube.com · Video on youtube.com

I Built an Original One-Glance Proof from Dice

39K views · 1y ago · 14:38

Is there really something new to discover about our good old six-sided die? Well, I just did :) Nothing Earth-shattering and definitely a combo of things we've known for a while, still very neat I think. Let's see whether you agree :)

Topics: Magnet mechanism · App · Magnet cubes · Drilling cubes

▶️ Mentioned videos: Video on youtube.com

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org

How to build and solve a 4D Rubik's cubes in physical 3D (no simulator!)

37K views · 1y ago · 37:33

I’ve been meaning to make this video about building and solving physical 4D Rubik’s cubes ever since Melinda Green sent me one of her brilliant physical 2x2x2x2s back in 2017! Why did it take so long? Well, this one was especially tricky to get right, and I think it’s probably the video that took me the longest to put together. I really hope you like this one :)

Topics: Warm-up · Fancy moves · Solve · Melinda · Coding challenge · What's next?

▶️ Mentioned videos: Video on youtube.com · Video on youtube.com · Video on youtube.com · Video on youtube.com

📄 Mentioned papers: Rubik's Tesseract by Dan Velleman in Mathematics Magazine (1992), 65:1, 27-36

📰 Mentioned news: n-dimensional sequential move puzzle

How to Integrate with an AX? The Surprising Power of Planimeters – Visually Explained!

73K views · 1y ago · 35:13

Today’s mission is to reveal a surprisingly simple trick that lets you use any old ax(e) to calculate the area of plane shapes. More broadly, we’ll explore planimeters — mechanical instruments that compute area by tracing a shape’s edge. Real mathematical magic in action!

Topics: Cool machines (history and types of planimeters) · Green's theorem (the most common high-level calculus explanation) · Easy peasy (the easiest and at the same time pretty rigorous explanation) · Full sweep (the best but somewhat handwaivy explanation) · AX (making sense of the ax planimeter) · Cute stuff

📖 Mentioned book: Mathematica%C2%AE Action Problem Visualization Computation

📰 Mentioned news: Article on en.wikipedia.org

How Did Water Solve the 1800-Year-Old Talmudic Bankruptcy Problem?

115K views · 1y ago · 39:58

Today we are solving an ancient bankruptcy puzzle from the Talmud using the principle of communicating vessels. A very nice visual way of making sense of an otherwise tricky problem. "Game theoretic analysis of a bankruptcy problem from the Talmud" by Robert J Aumann and Michael Maschler

▶️ Mentioned videos: Video on youtube.com

📄 Mentioned papers: Paper on sciencedirect.com · Paper on sciencedirect.com

📰 Mentioned news: There is something called the Water filling algorithm in information theory

The Helicone Numberscope: Mathematical Superpowers Hidden in a Simple Toy

53K views · 1y ago · 48:21

This is a corrected version of a video that I uploaded two days ago. After a critical mistake was discovered in the "Nature's Numbers" part of the video, I decided to scrap the original video, fix the errors and republish the video (many more hours down the drain :( Anyway, had to be done. Just in case you are wondering my list of best approximations of pi did not fit in with my definition. This was first pointed out by @typha

Topics: Helicone lab! · Fibonacci Christmas tree · Number scope · Nuts and bolts · Fractional angles · 3, 22/7, ...

▶️ Mentioned videos: The fabulous Fibonacci flower formula · Infinite fractions and the most irrational number · Video on youtube.com

📰 Mentioned news: The Moravian star has its own Wiki page. Worth checking out

What's the next freak identity? A new deep connection with Sophie Germain primes

125K views · 2y ago · 32:15

Guest: Sophie Germain

The third video in a trilogy of Mathologer videos dealing with sum-equals-product identities and equations. The other two are Way beyond the golden ratio The power of AB=A+B (Mathologer masterclass) Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

Topics: How many? · New from old · Sophie’s primes · Beyond Sophie

▶️ Mentioned videos: Video on youtube.com · Video on youtube.com · Combo class video on the same question

📄 Mentioned papers: Paper on tandfonline.com · arXiv 1311.3874

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

114K views · 2y ago · 32:54

We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality. Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here

Topics: Geometry 101 · Applications · Ptolemy's inequality · LIES · Animated proofs · Degenerate Easter Egg

▶️ Mentioned videos: Video on youtube.com · Video on youtube.com · Video on youtube.com · is a nice application of Ptolemy's theorem to a International Maths Olympiad problem

📄 Mentioned papers: Paper on jstor.org · Paper on jstor.org · Paper on jstor.org

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org · Article on en.wikipedia.org · Article on sco.wikipedia.org

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

167K views · 2y ago · 49:49

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 :)

Topics: Ptolemy · Perfect cut · Golden rectangle · Fibonacci · A+B=AB · Images and music

▶️ Mentioned videos: The golden ratio spiral: visual infinite descent · Video on youtube.com · Video on youtube.com · Video on youtube.com

📄 Mentioned papers: Paper on jstor.org

📰 Mentioned news: The wiki page on Ptolemy's theorem features a great visual animated proof

PETR'S MIRACLE: Why was it lost for 100 years? (Mathologer Masterclass)

174K views · 2y ago · 36:31

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 :)

▶️ Mentioned videos: John Harnad's YouTube video on a complete proof of the Petr-Douglas-Neuman theorem · howto · Video on youtube.com

📄 Mentioned papers: Paper on researchgate.net

📰 Mentioned news: Very good Wiki page on the Petr Douglas Neumann theorem · In electrical engineering

Conway's IRIS and the windscreen wiper theorem

90K views · 2y ago · 16:33

Conway's whatever ... it's named after John Conway and so it must be good :) Wiki page dedicated to John Conway Wiki page Conway's circle Wiki page on his Game of Life Michael de Villiers (the connection with the windscreen wiper theorem, read this one first)

▶️ Mentioned videos: Video on youtube.com · Video on youtube.com · Video on youtube.com · Video on youtube.com

📄 Mentioned papers: arXiv 2111.01835

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org · Article on en.wikipedia.org

Simple yet 5000 years missed ?

219K views · 2y ago · 13:48

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.

📰 Mentioned news: Article on en.wikipedia.org

What's the curse of the Schwarz lantern?

181K views · 2y ago · 25:10

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. The pi = 4 troll math meme on reddit

Topics: Troll math: the pi=4 meme · Archimedes chops off corners · Archimedes boxing of pi · Schwarz lantern · Area formula · Schwarz pi = 4 memes

▶️ Mentioned videos: Video on youtube.com · Schwarz Lantern in Geogebra 3D & AR · Schwarz Lantern christmas tree · Geogebra 3D AR mode on an iPad how-to

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org

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

252K views · 2y ago · 32:23

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.

Topics: Archimedes baggage carousel · Inside-out animations · Inside-out discussion · Inside-out paraboloid · Ratio 3:2 · Volume to area

▶️ Mentioned videos: Video on youtube.com · Henry's video about 3d printed Archimedes claw

📰 Mentioned news: sushi conveyors · Article on en.wikipedia.org · Article on en.wikipedia.org · Article on en.wikipedia.org

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

127K views · 3y ago · 21:31

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 :)

Topics: What's that horn? · A 700 year old trick · What's the volume exactly? · What paradox? · Fun fact.

📄 Mentioned papers: Very nice write-up of the history of the horn by Paolo Mancosu and Ezio Vailati

📰 Mentioned news: Wiki page on the horn · Wiki page on Torricelli · Wiki page on Oresme · Wiki page on the Method of shells

The best A – A ≠ 0 paradox

475K views · 3y ago · 24:48

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. Riemann rearrangement 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.

Topics: Paradox · Visual sum = ln(2) · Gelfond's number · Pi exactly · Riemann's rearrangement theorem

▶️ Mentioned videos: Video on youtube.com

📰 Mentioned news: Article on en.wikipedia.org

Why don't they teach simple visual logarithms (and hyperbolic trig)?

439K views · 3y ago · 32:32

Simple visual logarithms. Is there such a thing? You bet :) Maths of spinning cubes by Lingguo Bu Great history of hyperbolic functions by Janet Heine Barnett Squeeze mapping and Lorentz transformation Minute Physics video "Lorentz Transformations | Special Relativity Ch. 3"

Topics: Rubik's cube and drill · What's the area? · Sum of 1+1/2+1/3+... · Mystery sum · What base? · What is Logb(x)?

▶️ Mentioned videos: Video on youtube.com · Bug alert: The 1 s

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org

Ramanujan's easiest hard infinity monster (Mathologer Masterclass)

202K views · 3y ago · 26:45

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. This video was inspired by two 2020 blog posts by John Baez

Topics: How did his mind work? · What IS this? · Fantastic fraction · Impossible identity

📖 Mentioned book: Theorie and Anwendung der unendlichen Reihen

▶️ Mentioned videos: Numberphile v. Math: the truth about 1 2 3 ... -1 12 · How did Ramanujan solve the STRAND puzzle? · Ramanujan's infinite root and its crazy cousins · Euler

📄 Mentioned papers: Inequalities related to the error function

📰 Mentioned news: Wiki page for the Wallis product

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

567K views · 3y ago · 27:29

Guest: Powell’s Pi Paradox

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.

Topics: Powell's Piradox :) · Calculus made in India · Explanation of the paradox using Madhava's correction terms · Calculus: Neither Newton nor Leibniz · Palm leaf music sequence · one of the digit difference towards the end not highlighted

▶️ Mentioned videos: Euler's infinite pi formula generator · Fermat's Christmas theorem: Visualising the hidden circle in pi 4 1-1 3 1 5-1 7 · Euler's real identity NOT e to the i pi -1 · Video on youtube.com

📄 Mentioned papers: Paper on jstor.org · Paper on jstor.org · Series Formula for π by Leibniz, Gregory and Nilakantha

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org · Article on en.wikipedia.org · Article on en.wikipedia.org

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

139K views · 3y ago · 33:25

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. The Korean historical drama Tree with deep roots is available here All the magic square action takes place in episodes 1 and 2.

Topics: Part 1: The king's magic squares · Proof · The order 5 and 7 magic squares · Part 2: Geometric magic square

📰 Mentioned news: Wiki page on Claude Gaspar Bachet de Méziriac · Eduard Lucas 3x3 magic square equation · Bachet's magic square algorithm write-up · is a magic square an old Chinese manuscript

What's hiding beneath? Animating a mathemagical gem

154K views · 3y ago · 9:14

There is a lot more to the pretty equation 102 + 112 + 122 = 132 + 142 than meets the eye. Let me show you. Notes The beautiful visual proof for the squares pattern is based on a note by Michael Boardman in Mathematics Magazine

Topics: Animated visual proofs · Mathologer materializes · Three puzzles

▶️ Mentioned videos: Video on youtube.com

📄 Mentioned papers: Consecutive integers having equal sums of squares

Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

284K views · 3y ago · 42:35

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. The two preprints by H. Lee Price and and Frank R. Bernhart and another related paper by the same authors

Topics: Pythagorean triple tree · Pythagoras's other tree · Feuerbach miracle · Life lesson · The families of Plato, Fermat and Pythagoras · Euclid's Elements and some proofs

📄 Mentioned papers: arXiv 0809.4324 · Paper on arxiv.org · Paper on jstor.org

📰 Mentioned news: Article on en.wikipedia.org · Article on en.wikipedia.org

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

492K views · 4y ago · 38:33

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.

Topics: 3 Squares: Fermat's four square theorem · Trithagoras · Hexagoras · Chop it up: More twisted square dissection proofs · Aha! Remarkable properties of right triangles with a twist · Mutants: Unusual applications of twisted squares

▶️ Mentioned videos: Video on youtube.com

📄 Mentioned papers: Alf van der Poorten's super nice proof · all issues of the Scientific American) · follow the Mathematical Games ) · follow the Mathematical Games )

📰 Mentioned news: Article on en.wikipedia.org

Secrets of the lost number walls

185K views · 4y ago · 36:37

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?'")

Topics: Chapter 1: What's in a wall · Chapter 2: Number wall oracle · Chapter 3: Walls have windows · Animations of Pagoda sequence · Chapter 4: Zero problems · Chapter 5: Determinants

📄 Mentioned papers: s article

📰 Mentioned news: Article on en.wikipedia.org