Mathologer

Exciting news everyone. We now have two simulators for the physical 4D 2x2x2x2 that the last Mathologer video was all about.

One I made myself:
www.qedcat.com/2x2x2x2%20hedgehog

And one super-professional version contributed by Ed Collen:
2x2x2x2.vercel.app/cube

(Chrome on my Macintosh has problems with these simulators, all other browsers are fine, Firefox, Safari, Opera, Edge.) My simulator just permutes the colors, whereas Ed’s is fully animated and even includes a button for solving the puzzle blindfolded!

My version also has a bunch of buttons for macro moves. Part of the fun of playing with these magnet-based puzzles is that you can explore them using different move sets:
1. Minimalistic, as in the video: just reorientations and twisting one face or two faces.
2. The first set plus the 180-degree turn and the axial twist. This is what Melinda & Co. have settled on, aiming for as many moves as possible that correspond to actual twists of the physical puzzle. You have to use this move set to solve Melinda’s puzzle if you want to get into the Hall of Fame.
3. All moves that correspond to twists of all sides and layers of the puzzle.
4. (My personal favorite) The second set plus twisting adjacent corners of a 2x2x2 side (count 0), swapping adjacent corners of a 2x2x2 side (count -1), essentially compressing the nuts-and-bolts corner-fixing moves from the Rubik’s cube into buttons. This set also includes the off-axial twist, which gives you a reasonably short and intuitive path to a monoflip.
5. Everything, including a monoflip button and a half-turn of one of the 2x2x2 sides.
6. Explode the puzzle and reassemble.

Also worth noting: in the classic 4D twisty puzzle app MagicCube4D you can program your own macros. Also, in the hedgehog and Melinda's puzzle, if you are careful you can avoid running into tricky monoflips altogether while solving these puzzles. See for example this video: https://www.youtube.com/watch?v=V4N2R.... Link to MagicCube4D and many other resources in the description of my video.

Anyway, have fun playing with these simulators and building some real 4D intuition until you get your hands on one of Melinda’s physical puzzles.

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1 month ago (edited) | [YT] | 320

Mathologer

Mathologer at IMO2025: This year, the International Mathematical Olympiad was held in Australia, and I was one of the four guest lecturers alongside Terry Tao, Cheryl Praeger, and Eddie Woo. Here is a link to the recording: youtube.com/live/tOiw8uEHaQE?si=7R5VDddWoupDEKWf&t… This presentation was an update of a Mathologer video on Steinbach’s fascinating higher-dimensional generalizations of the golden ratio. In particular, the update highlighted a striking occurrence of these golden ratios in nature: https://youtu.be/cCXRUHUgvLI Check it out!

I’m also planning to publish a new Mathologer video in about 12 hours, something else to look forward to :)

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1 month ago | [YT] | 351

Mathologer

Just a bit of fun! As some of you may know, quite a few math(s) YouTubers are into Pokémon Go—and yep, I’m one of them!

Well, I’ve been Best Friends with a very special trainer for ages, and the other day… it finally happened. Drumroll, please…

✨ WE’RE NOW LUCKY FRIENDS! ✨

The only clue I have about this mysterious trainer is that most of their gifts come from Kobe, Japan. So if you happen to know them, please tell them to get in touch! I’d love to find out whether there is an interesting mathematical story hiding behind their name :)

8 months ago | [YT] | 323

Mathologer

I made a critical mistake in the latest video, published two days ago. I’ve just reset it to private, so it’s no longer visible on the channel. I spent a lot of time on this one but ended up not thinking through a statement toward the end, which really messed things up. I just couldn’t live with that. Anyway, I’ll be fixing the video in the next couple of hours, and the plan is to republish it early tomorrow morning (7 Jan around 1 a.m. here in Melbourne, early morning in New York).

Could I ask all of you for a favour, please? I pulled a video once before, a couple of years ago, and the corrected version ended up performing poorly in terms of views and likes. This was mainly because the Mathologer diehards who were notified right away had already watched the messed-up version and never bothered to check out the corrected version.

So, even if you’ve already watched and liked the video, could you please click it and like it again when it’s re-released? It’s essential for a video to get enough of a boost on the first day it’s published to have a chance of doing reasonably well in the long run.

The corrected part is in the “Numbers of Nature” chapter, which starts around the 34-minute mark. Also, just in case you haven’t watched the video yet, there are a few extra special bits and pieces to look forward to. In particular, we start out in my office at Monash University (home to thousands of mathematical gadgets), Mathologer Junior makes an appearance, and this one is super visual. :)

All the best to all of you in 2025

9 months ago | [YT] | 4,790

Mathologer

In my last video, I asked everyone for off-the-beaten-track occurrences of the identities/equations 2×2=2+2 / x+y=xy and 1×2×3=1+2+3 / x+y+z=xyz. Here are three interesting examples submitted by viewers. Before I move on, I’d like to make one final appeal to all of you: if you’ve stumbled across any more gems related to this, please let me know!

VincentvanderN
I don't know if this is already down here somewhere in the comments, but 2 x 2 = 2 + 2 can be used to show that there are infinitely many prime numbers. We generate a sequence a_1, a_2, a_3 etc. by starting with a_1 = 3 and a_{n + 1} = a_n^2 - 2. Now we use our freak equation to show that if q is a prime divisor of some a_m, it cannot be a prime divisor of a_n for any n > m. (And then, as a result neither of an a_n with n < m, because otherwise we could repeat the proof with n in the role of m and get a contradiction.)

Here is the argument. Look at the sequence a_m, a_{m+1}, ... modulo q. We have a_m = 0 mod q, a_{m+1} = - 2 mod q, a_{m + 2} = 2 mod q, and then, by the freak equation 2 x 2 = 2 + 2 (in the form 2 x 2 - 2 = 2) we get that a_{m + k} = 2 for all k >= 2.

Neat, right? I believe I learned this from Proofs from the Book.

charlesstpierre9502
Use 2x2=2+2 to make a formula for generating Pythagorean triples. Start with two positive integers m > n. Then
(2×2)(mn)² = (2+2)(mn)²
(2×2)(mn)² = 2(mn)² + 2(mn)²
(2×2)(mn)² = 2(mn)² + 2(mn)² + (m⁴ - m⁴) + (n⁴ - n⁴)
(2×2)(mn)² = m⁴ + n⁴ + 2(mn)² - m⁴ - n⁴ + 2(mn)²
(2mn)² = (m² + n²)² - (m² - n²)²
Set:
a = m² - n²
b = 2mn
c = m² + n²
Then:
a² + b² = c²
This will not work for: aᵏ + bᵏ = cᵏ; k > 2


franknijhoff6009
Hi Burkard, the 3-variable equation plays a role in integrable systems. In fact, it appeared in Sklyanin's work on quadratic Poisson algebras (around 1982) providing solutions in terms of elliptic functions, and (with an extra constant) as the equation for the monodromy manifold of the Painleve II equation.

Sklyanin's paper is: Some algebraic structures associated with the Yang-Baxter equation. Functional.Anal.i Prilozhen 1982 vol 16, issue 4,pp 27-34 ; look at equation (27).
Furthermore, in L.O. Chekhov etc al., Painleve Monodromy Manifolds, Decorated Character Varieties and Cluster Algebras , IMRN vol 2017, pp 7639--7691 you can find in Table 1 a close variant of the 3variable equation with extra parameters.

10 months ago | [YT] | 330

Mathologer

DISCRETE FOURIER TRANSFORM and PETR'S MIRACLE FOLLOW-UP: Check out this fantastic app that does the "eigenpolygon" decomposition featured in the last video (contributed by Steven De Keninck from the Computer Vision Group at the University of Amsterdam) enki.ws/ganja.js/examples/coffeeshop.html#AONlCLF1…
This is getting better every time I look at it. Any other features you suggest Steven include in this?

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1 year ago | [YT] | 394

Mathologer

The Christmas video is out ... on the 24th (we give presents on the 24th in Germany :) Merry Christmas/Fröhliche Weihnachten everybody!

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1 year ago | [YT] | 1,051

Mathologer

Hello everyone, I just uploaded a new video. It delves into some new visually appealing enhancements to how we approach the volume and surface area formulas of spheres. While working on it, I played around with a few different designs for the thumbnail. Which of these designs do you find most "clickable" ?

1 year ago | [YT] | 409

Mathologer

I should know by now that once a nice topic has been covered on YouTube, no matter how meh, it's not a good idea to cover the same topic again to fix things. Because if you do, don't expect many people to watch your masterpiece :( In fact, I do know and yet I just made the same mistake again: A video on Gabriel's horn (done to death ... badly). Before that it was Conway's soldiers. Before that it was mathematical table turning, etc. Anyway, since you are reading this, I am assuming that you are one of the Mathologer regulars. And, as a regular, and if you are familiar with one of those Mathologer remakes, did it make enough of a difference to be worth the effort, at least as far as you were concerned? Should I keep giving in to the urge to fix things on YouTube?

1 year ago (edited) | [YT] | 1,835

Mathologer

YOUTUBE IS WEIRD. Quick question for you. I am assuming that since you are looking at this post you are also among my viewers who get alerts about new videos early on. When the last video on visual logarithms went up, relatively few of you actually watched it straightaway and this video remained fairly unpopular during the first 2-3 weeks (mostly regular viewers) of its life on YouTube. Only recently this video has started to attract more viewers. In fact, this is my only video the like/dislike ratio of which has been RISING since it was published. Anyway, just trying to figure out what is going on here. And, so, if you got notified early and you did not end up watching this video, why was that?

2 years ago | [YT] | 517