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

From Mathologer. 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…

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

From Mathologer. 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…

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

From Mathologer. 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…

# Conway’s IRIS and the windscreen wiper theorem

From Mathologer. Conway’s whatever … it’s named after John Conway and so it must be good 🙂 Wiki page dedicated to John Conway https://en.wikipedia.org/wiki/John_Horton_Conway Wiki page Conway’s circle https://en.wikipedia.org/wiki/Conway_circle_theorem Wiki page on his Game of Life https://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…

# Simple yet 5000 years missed ?

From Mathologer. 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.…

# What’s the curse of the Schwarz lantern?

From Mathologer. 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…

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

From Mathologer. 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…

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

From Mathologer. 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!…

# The best A – A ≠ 0 paradox

From Mathologer. 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…

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

From Mathologer. 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…

# Ramanujan’s easiest hard infinity monster (Mathologer Masterclass)

From Mathologer. 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…

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

From Mathologer. 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…

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

From Mathologer. 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!…

# What’s hiding beneath? Animating a mathemagical gem

From Mathologer. 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…

# Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

From Mathologer. 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…

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

From Mathologer. 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…