Secret of row 10: a new visual key to ancient Pascalian puzzles

NEW (Christmas 2019). Two ways to support Mathologer
(see the Patreon page for details)

Today's video is about a recent chance discovery (2002) that provides a new beautiful visual key to some hidden self-similar patterns in Pascal's triangle and some naturally occurring patterns on snail shells. Featuring, Sierpinski's triangle, Pascal's triangle, some modular arithmetic and my giant pet snail shell.

Thank you very much to Marty for all his help with finetuning the script for this video and to Steve Humble and Erhard Behrends for making some photos available to me.

Enjoy :)

P.S.: The article I mentioned in this video is: Steve Humble, Erhard Behrends, ”Triangle Mysteries“, The Mathematical Intelligencer 35 (2), 2013, 10-15. There is also a followup article:
”Pyramid Mysteries“, The Mathematical Intelligencer 36 (3), 2014, 14 - 19.

Philip Smolen contributed this animation

Someone pointed out these links to some code wars problems:

Juan Mir Pieras pointed out these earlier references:

[Mathologer]

I felt that I had to take a break from all the heavy algebra that dominated recent videos. Today’s video is about a fairly recent chance discovery of some very beautiful mathematics by the mathematician Steve Morton. After I read about this discovery in an article by Steve Humble and his colleague Erhard Behrens in the Mathematical Intelligencer it occurred to me that this beautiful mathematics can be used as a very nifty visual key to some hidden self-similar patterns in Pascal’s triangle. Today is the first time that I tell anybody about this insight. Enjoy and please let me know how this video worked for you :)

[tamirerez]

One of Mathologer best videos.
The great graphic design,
The clear explanation,
The general idea, all are simply perfect!

[almightyhydra]

Mathologer: explains rules about two colours added -> the third one
me: ok, addition mod 3, easy
Mathologer: two same colours -> that colour
me: hmm
me: something something group theory???
Mathologer: -(a+b) mod 3
me: takes maths degree certificate off wall

[jfg]

This video is a great example of why Mathologer is the BEST Mathematics channel all over YouTube. Thank you Burkard, for making my day!!

[joeluquette]

Wow that blew my mind that Pascal's triangle mod 2 gives the Sierpinski triangle, and mod 3 makes a similar pattern. I wrote a script to play with this more and it seems that mod any prime number it gives a nice Sierpinski-like pattern, and mod composite numbers give more messy overlapping patterns that relate to the factors. It's always amazing when seemingly disparate areas of math connect together in unexpected ways!

[VibratorDefibrilator]

Such puzzles and mathematical "mysteries" are like a concentrated food for always hungry young mathematical intellects - they're teaching them to mathematical beauty, elegance and aesthetics, and hone their logic, heuristic ability, pattern recognition, intuition (I saw the shortcut too, yesss!), making seemingly impossible connections between semantically distant objects or facts.


For not such young minds it is pure pleasure and recreational activity to tackle such problems.
So, thank you, sir, for making my day.

[taibilimunduan]

I just won a lifetime subscription to Mathologer! So cool!

[froggoboom]

really love how much production quality goes into this (all the animation programming, etc)!

[joshuacoppersmith]

The sliding magenta triangles at 26:04 made my jaw drop. The concept, while amazing as are most automata, wasn't hyper-noteworthy, but the idea of these little protein-like slider triangles infixing colors upon corner matching was just a revolutionary conceptualization for me. Now I want to reconsider all sorts of automata and the Wolfram/Cook Turing completeness in light of little slider machines.

[Supremebubble]

"Are there any other special numbers?" Well obviously 2 works :p

[ericvilas]

1 - trivial
1 1 - still trivial
*1 2 1 - the middle ones have two as their greatest common factor*
*1 3 3 1 - GCF = 3*
1 4 6 4 1 - GCF = 2, NOT 4
*1 5 10 10 5 1 - GCF = 5*
1 6 15 20 15 6 1 - GCF = 1
*1 7 21 35 35 21 7 1 - GCF = 7*
1 8 28 56 70 56 28 8 1 - GCF = 2


So, when P is prime, doing Pascal mod P makes the triangles of size P^k + 1 special because the GCF of the middle numbers in the P'th row is P itself

[Artificial_Intellect]

Amazing video and I noticed a neat trick as an extension of this. If you have a number n, and want to know all mod(k), where k is all the numbers up to n, then take pascals triangle and extend out the top row by (n-1) zeros on the LHS and add an extra zero to the RHS after the 1 giving a row length of n+1 for the top row and that puts our value n, at the bottom position of a special triangle for mod(n) ... in fact you are at the bottom of many special triangles and can walk down each mod(k) by just cutting the top row off and our value in question will remain at the bottom of the new special triangle mod(k-1). Plus since the LHS is zero, the RHS is our value mod(k). Hope you find this trick neat and you really are inspiring and I hope to watch many more of your videos.

[jursamaj]

8:40 Up to left-right symmetry and color permutations, there are only 10 top rows to check:

1 way to fill with 1 color (aaaa)
5 ways to fill with 2 colors (abbb, babb, aabb, abab, abba)
4 ways to fill with 3 colors (aabc, abac, abca, baac)

[Ronenlahat]

This is one of the best mathologer videos so far, incredible. I inspires to play with math.

[Bubatu]

It's beautiful how complex behavior emerges from simple rules. Great video!

[MathAdam]

Saturday morning fun with Mathologer! What a pleasant surprise! 😎

[f-th]

2 minutes into the video amd here's what's on my mind: the rule is the same as the average modulo 3, meaning (x+y)/2. If you assign a number to each color, and use this average, the answer will be the number for the corresponding color. For example: (1+0)/2 =1/2 = 2 (inverse of 2 modulo 3 is 2). Average of x and x will be x.

This also reminds me of tricolorability of knots. (Or whatever it's called. I forget)

[stjernis]

What better way to begin a lazy Sunday morning than with some simple and beautiful mathematics presented so eloquently?

[Zolbat]

I recently solved that problem on codewars and came up with the 10 rule myself (looking at the results of many different triangles). Helped me solve it fast enough

[Jesus_.]

This is a gem of a video, congratulations. This is the definition of quality