The Secret Music of Pascal's Triangle
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- Опубликовано: 29 авг 2024
- When exploring Pascal's Triangle with a student, I noticed an interesting pattern in the even and odd numbers. This pretty quickly turned into an obsession with the fractal patterns of Pascal's Triangle under different moduli, and the fascinating music that these patterns can yield.
To listen to the full videos of these patterns with no (auditory) commentary, check out the playlist here: • Hearing Pascal's Triangle
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I went about proving the zero rows on prime powers that you mentioned and realized that it was just Lucas' Theorem working in the background. The oversimplification of the phenomena is that if you write each row and column of your triangle in the base p, you'd see that the zeroes coincide with when the p-representation of the rows or columns contain a zero. For prime powers, the phenomena of the "nesting" comes from the behavior of prime power fields, I think an understanding of frobenius maps could clue it in.
There's a video on RUclips, unfortunately I forgot the name(I think it's a SoME 2/3 submission), where they explore similar patterns(I think the bounding shape was a triangle too) with different numbers.
They notice that there is a feature set of designs, generally linked to prime numbers and composite numbers usually exhibit the features of their factors. Some of them were quite interesting to watch.
They go really far, until a smooth gradient of values.
It’s from Kuvina Saydaki, and it’s “how I made my own fractal”. These were the bi versions.
@@kvanctok9234 called it
I’d never noticed these similarities between Pascal’s and Seprinski’s Triangles, this is actually really cool!
I have been looking into the superposition as you call it. It turns out there is an isomorphism between mod 6 numbers and pairs of numbers mod 2 and mod 3. You can add these pairs together by adding each coordinate individually, for instance (a,b)+(c,d)=(a+c,b+d). This seems to be what creates the sort of overlapping pattern because each combination of mod 2 and mod 3 numbers creates a unique mod 6 number. More generally there is an isomorphism between any composite number a tuple of each prime power in its prime factorization. I will comment a bunch of examples next.
The following example works because 12=2^2 * 3. (Splitting this into a tuple with two mod 2 can create an isomorphism but would need some more careful assignment that i don't fully understand, im not sure if addition could work the same too).
(mod 3, mod 4) ~ mod 12
(0,0) ~ 0
(1,1) ~ 1
(2,2) ~ 2
(0,3) ~ 3
(1,0) ~ 4
(2,1) ~ 5
(0,2) ~ 6
(1,3) ~ 7
(2,0) ~ 8
(0,1) ~ 9
(1,2) ~ 10
(2,3) ~ 11
addition working pairwise is relevant to the triangle because with 1 ~ (1,1) each entry of the pair is its own lower mod triangle number
Here is another example where we uniquely match each mod 30 number with each triplet of mod 2, mod 3, and mod 5 numbers.
(mod 2, mod 3, mod 5) ~ mod 30
(0,0,0) ~ 0
(1,1,1) ~ 1
(0,2,2) ~ 2
(1,0,3) ~ 3
(0,1,4) ~ 4
(1,2,0) ~ 5
(0,0,1) ~ 6
(1,1,2) ~ 7
(0,2,3) ~ 8
(1,0,4) ~ 9
(0,1,0) ~ 10
(1,2,1) ~ 11
(0,0,2) ~ 12
(1,1,3) ~ 13
(0,2,4) ~ 14
(1,0,0) ~ 15
(0,1,1) ~ 16
(1,2,2) ~ 17
(0,0,3) ~ 18
(1,1,4) ~ 19
(0,2,0) ~ 20
(1,0,1) ~ 21
(0,1,2) ~ 22
(1,2,3) ~ 23
(0,0,4) ~ 24
(1,1,0) ~ 25
(0,2,1) ~ 26
(1,0,2) ~ 27
(0,1,3) ~ 28
(1,2,4) ~ 29
The following is some of the mod 6 triangle but with (mod 2, mod 3) numbers instead.
(1,1)~1, the rest follows from pairwise addition
(1,1)
(1,1)_(1,1)
(1,1)_(0,2)_(1,1)
(1,1)_(1,0)_(1,0)_(1,1)
(1,1)_(0,1)_(0,0)_(0,1)_(1,1)
(1,1)_(1,2)_(0,1)_(0,1)_(1,2)_(1,1)
If you reference how the pairings are connected to mod 6 numbers you will see it is the same as the mod 6 triangle. Not only that but if you look at the first entries only you can see how naturally the mod 2 triangle is embedded, along with the second entries showing the mod 3 triangle. So each triangle directly contributes to the structure of the mod 6 triangle.
If anyone has more information about this that they would like to share, please do. Im interested in knowing more about the cases with prime factorization that have primes that are exponentiated.
@@adrienanderson7439I think you have to use the Chinese remainder theorem.🤔
Sounds like something Emerson, Lake & Palmer could have turned into a good number. Likewise, Mussorgsky before them.
Sounded more like Ozric Tentacles to me, just needs the synths and guitar solo on top of it.
@@Flatscores This sent this 62 year old off to hear what they sound like. They are definitely in the progressive rock tradition.
I can't remember the exact song (maybe bitches crystal) but my first thought when I heard the was elp
Editing cause I think it was from the begining I'm thinking of
I'm really intrigued by that piece at the end!!! ...Oh also, I think that super-position you observe around 4:50 can be understood as a consequence of a result called the "Chinese remainder theorem."
I saw one forum where someone labeled the Sierpinski just as SG (Sierpinski Gasket) 3, but that's boring. I propose we should just label these different levels of Sierpinski type fractals with the corresponding modulo. A regular triangle will be a unary Sierpinski, Sierpinski's triangle is a binary Sierpinski, and so on.
The drums really add to the music and make the polyrhythmic graspable. Did you add them manually after feeling the vibe of each line, or are they algorithmic too?
Also algorithmic! Each number has a particular percussion sound associated with it, and when there's a block of two or more of a number in a row we hear it. I also added a little fill for the beginning of each new line.
Hello! This video was super cool, and there’s quite a bit of cool math that’s going on behind the scenes here. I’m gonna answer some of the questions from the video, rapid-fire, in this comment. Please feel free to ask for clarification!
First, as you indeed noticed, the nth row of the staircase modulo n is all zeros (except for the ones on the boundaries) exactly when n is a prime number. The fifth row is all multiples of fives, the seventh row is all multiples of seven, the eleventh row is all multiples of eleven, and so on. (Clarification: the uppermost row, which has only a single one in it, is called the “0th” row.)
The usual way to prove this is by using the explicit formula for the terms in the staircase, which involves factorials, but there’s also a neat combinatorial proof I found. I can explain either if someone asks.
As for your observation that sometimes the pattens seem to be overlaid on each other, you’ve actually run into a very deep fact from abstract algebra! The Chinese Remainder Theorem states that if n and m share no factors in common, then you can tell what a number is modulo nm from what it is modulo n and modulo m, and vice versa.
To remove a bit of the jargon, we can go into an example: 3 and 4 share no factors, so if a number is (for example) 1 mod 3 and 3 mod 4, then it must be 7 mod 12. There are no other options. You can try this yourself: pick a number from 0 to 2 and another from 0 to 3. There should be exactly one number between 0 and 11 which is equal to the first mod 3 and the second mod 4.
This is very powerful because, for example, it means that you can understand what the pattern is mod 60 just by looking at the patterns mod 3, 4, and 5 (these are the “prime power” factors of 60) and overlaying them.
One last point. You asked for the names of the shapes that were made by reducing modulo numbers other than two. While I’ve never seen them explicitly named, I’ve taken to calling them the “binary triangle” and “trinary triangle” and so on, naming the triangle mod n after the term for base-n numerals. (The mod-10 one, for example, would be the decimal triangle.)
This is because of another interesting way they show up: take the coordinate plane and color in every point (x,y) for which when you add x to y written in base-n, you don’t need to do any carrying. The resulting figure will look like the base-n triangle.
Oh, I just realized you’re also the person who made the Fibonacci music box video! You have a knack for these mathematical music videos, I must say. Keep it up!
I'm slightly colorblind too and the colors were great! Keep it up
Pascal's triangle with mod 2 is the same as rule 90 in an elementary cellular automaton
I think you should look into the Langlands program. The relationship between patterns created by modulo primes and their combinations is related to L-functions and Euler products.
Numberphile's latest video on that (with Frenkel) does a great job exposing how power series (ring a bell with pascal's triangle? related to polynomial expansions) relate to modular arithmetic, p-adics, and the relatively more grounded geometric interpretations of low rank equations. Sounds scary maybe but worth a watch!
@@oncedidactic There's a YT channel entirely dedicated to the subject: it's called "PeakMath".
@@lucassiccardi8764 Thanks I'll check it out! Trying to learn more about this stuff efficiently from a non-professional starting point ;D
I’m a little shocked this was recommended to me but I’m really happy it was, very cool video!
this is so nifty ooooh
i am v happy i found your channel :3
The patterns remind me of Wolframs cellular automatons
Yes, me too. I keep thinking about that comparison, especially with the composite numbers. The fact that the rows aren't all aligned make it a bit different, but maybe there's some broad definition of a cellular automaton that it fits under.
@@marcevansteinyou could align the rows so that it fits
@@marcevanstein You got to watch the New Kind of Science series Wolfram made (if you haven't already). It will change the way you think about this (Pascals Triangle)...but also practically everything.
It's a 16 part series, each part about 2-3 hours long, so it will take 32 - 48 hours to go through it all...but trust me, its the most valuable research you will ever need.
Cheers
1:03 - what is also XOR gate.
I wonder what patterns would other gates made?
AND it's multiplication modulo 2, though I don't think OR maps as cleanly to modular arithmetic as AND and XOR. That said, a OR b is the same as ~(~a AND ~b) by De Morgan's laws, and ~a can be treated as a + 1 ≡ a - 1, so we'd have a OR b = (a - 1)(b - 1) - 1 = ab - a - b + 1 - 1 = ab - (a + b) = (a AND b) XOR (a XOR b).
Then again, rolling things back reveals that the results really wouldn't be very interesting, as they'd either annihilate instantly or form a solid triangle of 1s. For things like NAND and NOR, it'd basically be the same, but inverted every row.
@@angeldude101 What about asymmetric logic operation like implication?
@@The0Stroy That has the potential for more interesting patterns, due to the asymmetry, but it's still fundamentally built from a single AND or OR with specific inputs or outputs inverted. Plus the asymmetry itself could make it look less appealing even if it wasn't as boring as AND or OR.
kuvina triangle
Is there a sheet of the piece at the end? I don't understand it with just the visuals.
1:17 -- Look! =D A Sierpiński triangle!
How many modules are there
It a very interresting video, unfortunately it is completely obscrured if watxh on a phone, the numberes are too small and the colors too dim.
That fractal is MY triangle and y’all can’t have it >:(
3:07 oh look, a prog song
The worst part about these songs is they never drop
They're just constantly pretending they're gonna drop and not doing it
not me thinking this was the ball bouncing erasing squares.
sierpiński's triangle!
It sounds like pillar man theme
SIERPINSKI TRIANGLE