This is one of my favorite videos I've ever created so far. I made all of the graphics in it myself. This episode contains 3 chapters: 0:00 - Part 1: The Sierpinski Triangle 4:36 - Part 2: Pascal's Triangle 8:22 - Part 3: How These Triangles Are Connected
I have done a large amount of study into Pascal's Triangle and could take you off the deep end if you would like. This includes going into higher triangles (simplexes). I can also do square extensions using properties of Pascal's triangle.
Tbh, while I love the premise of your channel and watch most of the episodes, they don't always hold my attention (not due to your presentation, just short attention span 😛) but this one has some of my favorite subjects. Awesome!😊
My intuitive understanding of fractal curves is that if one measures its length by "walking" a yardstick between two points at a time on it, then it becomes longer and longer the smaller the yardstick one uses. And the fraction of dimensions is the relationship (or slope) between the length of the yardstick and the length the curve when using that yardstick. Measured curve length = Yardstick length ^ Fractal dimensions (but with log, use a yardlog instead of a yardstick) The same is true for almost any curve, like y=x^2. It too gets longer when measuring it with a shorter yardstick. The difference is in the scaling behavior, and that's when I go for lunch.
Weird how you skipped over the part where Pascal rang a bell when he fed his triangle and found that later on, ringing a bell would make it salivate. Eventually, the triangle got hungry and fed up and got revenge by eating Pascal's math homework.
Very well done illustrations man. Excellent work. Very compelling. Also very heart felt hand written names of all your supporters. The clocks are also hilarious. Love what you do. Im always looking for the Marco Rodin relations in your mathematics. My brother and i studied Marco Rodins work together as teenagers and the visual illustrations that can be made from numbers by following sequences is entrancing. It makes you feel as tho youre touching god with your fingertips. Your work and videos revive that same feeling i used to have for Marco Rodins work
My! The control chaos in this video was all of the triangles! I must say, I was grinning ear to ear as you were unfolding the pattern! Your figures are journal worthy and it was cool to see both triangles appear to approach the same limit with different process! I’m curious if there’s a way to relate the series as equivalent? I might hop on MATLAB while I’m sick this weekend and generate the other mods!
12:05 I just realised that this approximation is _already_ happening with our monitors since we don't notice the square pixels that generate the approximation of the Sierpinski triangle! Mind blown.
As a bonus, if you take any of the first 32 rows of Pascal's Triangle mod 2, place the decimal (binary?) point anywhere that makes it an integer (i.e. not before the last 1), and read it as a binary number, a regular polygon with that number of sides can be constructed with straightedge and compass. If the number can't be made this way, a regular polygon with that number of sides can't be constructed with straightedge and compass.
@@landsgevaer You're right. I think it's likely that there aren't anymore though, and if there are, that just means there are more rows of Pascal's Triangle somewhere that also have this property.
Another method of generating serpinsky triangle from complete randomness is the chaos game: en.wikipedia.org/wiki/Chaos_game Great to see you evolving as a contant creator. The animations add a lot.
Awesome video. Another fractal shape that people could draw themselves is: A square, then right next to it attached, a square half the size. Then right next to it attached, a square half the size of the 2nd square. I don't know how to put this in formula with the keyboard my mobile can give. But the area of all those squares together are 2 if the big one you start with is1. And the area of the triangle goes to 0 when the first one starts at 1. Which clearly shows the difference to when something is build up by adding. Or broken down by subtracting.
As I'm sure you know, there are a lot of cool patterns in Pascal's Triangle. A couple of years ago, I decided to try to find a method of generating the sequences of numbers in the diagonals of Pascal's Triangle. I succeeded! In fact, I found 2 different methods for accomplishing that goal. And I'm not even a mathematician! I just enjoy playing with numbers.
Awesome video. I was just messing with this, man!🌞 Since I'm not proficient as a coder in any language, I had Claude 3.5 Sonnet build a pretty cool Python program that color codes the values at each part of the triangle based on their mod values with a given, inputted integer n. It's VERY interesting, and it's amazing to see someone make a video about the exact same thing, at the EXACT same time. I shouldn't really be that surprised.😂
ah yeah, the Sierpinski and Pascals Triangles having a connection, i forgot they were connected, i cant remember where i first learned they were connected, i had the sneaking suspition i already knew about it, regardless, always remember: 10:45 Hexagons are the Bestagons!
I was familiar with the pascal triangle mod 2/xor, but hadn't seen the mod 3 version before - which makes it clear that Sierpinski is only one of a family of fractals. One thing you didn't show is the version of pascal's triangle mod 3 where remainder-1 and remainder-2 are coloured differently from each other - which a bit surprisingly causes it to lose the triangular symmetry. I haven't quite thought through why that's true.
Reminds me of a game I was playing, where I started with a triangle with n points and repeated the following steps: if the number of white regions is even, paint half of them black, starting from the smallest. If the number of white regions is odd, take the triangle, copy it 2 times and arrange them in a bigger simmilar triangle, creating a white region in the middle. Repeat. I always came across the zelda triforce, no matter with how many white regions I started. lol
I feel like I’ve seen an answer to this before, but what is above Pascal’s triangle if you extend it to more rows of numbers above it? Is there a single extension of it in that direction, or are there multiple sensible ways to do it?
There's a channel named "Dr Barker" with a video titled "What Lies Above Pascal's Triangle?". This may be what you're thinking of. ua-cam.com/video/q2daqMR3l24/v-deo.html
There is multiple ways, but only one is symmetrical in the previous row ... ½ -½ ½ -½ ½ ½ -½ ½ -½ ½ ... The next rows above that have more options though, even symmetrical ones ... 1½-x x-1 ½-x x ½-x x-1 1½-x ... The reason being that ... 1 -1 1 -1 1 -1 1 ... annihilates itself on the next row, so you can always add a multiple of that.
10:20 i think triangular grid could've worked better here. I understand that this is for the sake of showing pattern that emerges ignoring the details. But if we consider triangular grid version not as simply approximation, but as a legit part of this fractal, than it kind of reminds me of p-adic numbers. The first iterative view of the triangle is like approximation of the irregular number. But looking at it from the tip (altered pascal triangle) is like looking at the end of the p-adic numbers and zooming away to see more. Can you please explore p-adic numbers in a future and maybe show some geometrical or modular-arithmetical connections? Hm, considering that in our seemingly infinite universe exists plank length and plank time, could it be that its better to message and study this values in p-adic numbers? And is it true that you can represent 0 in this system as ...999.99...? What if you use negative or complex baze?. Does dual numbers have something in common or in contrary to the modular arithmetics?
If you labeled the centers of a triangular grid, an array like Pascal's triangle (where each cell has exactly 2 above it) wouldn't fit properly without any gaps. And if you labeled the vertices of a triangular grid, that would be the same structure as labeling the centers of a hexagonal grid, just harder to make look clear. Triangular grids and hexagonal grids are dual graphs of each other.
idea: Make a table of NxN where N is the number of colors. Top row = upper left color, side row = upper right coior, inner cells = what color the current cell should be given the upper colors. What patterns emerge with different lookup tables?
so the serpinski triangle is an infinitely large shape but you can look at it from the outside, seeing the whole shape all at once and you can look at it from the "atomic" level, seeing how you can build it pixel by pixel at the most zoomed in level so theres an "outer end" and an "inner end" to this fractal, with an infinite scale between them thats like, if the question "what is the last digit of pi" had a real, provable answer, with pi still being infinitely long
Funny how "fractional" dimensions come out of a very binary method like this. "Emergence" as the philosophers say when they don't understand why things are and where they come from. Philosophers might seem not very helpful. But one day maybe one of them slams his head to the desk so that something useful comes out of it.
I am not clear on what your mean by "respect infinititude", but I have a problem with your claim that pascal's triangle, when completed, taking mod(2), IS a sierpinski triangle. I don't think it was fair of you to mention the limit of both shapes without a proof, because if pascal's triangle construction starts with a finite segment with non-zero area, its area diverges. I am not sure whether allowing this construction from a triangle with no area could make sense, but here is my argument for the two shapes differing: If you take an equilateral triangle with a positive non-zero area, and multiply it by three, ensuring that the two new triangles have their top vertices touching the bottom vertices of the first, with the new triangles having two vertices that intersect, and you repeat the process, applying it to the result, we still have a top triangle, and therefore, an area greater than 0. This process includes no subdivision, and there is therefore a smallest triangle with a defined area. You don't need to start with a sierpinski triangle for this process to give no area, though. When combining subdivision and multiplication, aiming for an area of 0, we can take an equilateral triangle, and then subdivide it, where we have a proportion 1:d between the exponents of multiplication and division such that (3)^n*(3/4)^dn converges to 0 as n->inf. If we divide 3 times for every multiplication, d=3, (3^n)*(3/4)^3n=3^n*(9/64)^n=(27/64)^n, which is 0, and therefore, in the limit as n->inf, is 0. The formula applies to similar fractals, like the sierpisnki carpet, etc... If we have a triangle that we want to subdivide, and it has no area, how can we subtract an interior or subdivide it? I feel like the sierpinski triangle is, in a sense, the inverse of pascal's triangle mod 2, if we consider size in both directions, but we can also create a fractal that merges the two processes. Pascal's triangle has a defined "top" triangle, but that sierpisnki's triangle doesn't, whereas sierpisnki's triangle has a bottom edge, and pascal's triangle does not. Of course, the shapes are identical visually when we consider a "completion" of pascal's triangle, but there is a shape that has neither a top triangle nor a bottom edge, which I think is the best triangle. The only problem is encoding the triangle of all trialgles into it. What would the string of binomial coefficients mod2 as the rows go to infinity look like? This is not necessarily a rhetorical question. I am wondering if we can use some pattern to determine where the middle will be a string of 1s or a string of 0s, and maybe we can average the strings? I'm assuming my question needs modification to yield any good answer.
WOW Stop fractalizing, my brain has gone somewhere in these paths of hidden patterns in "Pascal's triangle" 🌊✨♣🎶🩲📡⚛✡⏫⏏ It's interesting what fractal-fantasy worlds a simple contemplation of the hidden patterns in "Pascal's Triangle" can lead to... how unexpectedly the brain starts looking for some more variants of patterns in "Pascal's Triangle" and ways of visualizing them for better understanding... Unexpectedly, I wanted to build a 3-D pyramid with a fractal expansion from the center of the pyramid :)
This is one of my favorite videos I've ever created so far. I made all of the graphics in it myself.
This episode contains 3 chapters:
0:00 - Part 1: The Sierpinski Triangle
4:36 - Part 2: Pascal's Triangle
8:22 - Part 3: How These Triangles Are Connected
What an interesting topic. Thank you for it, Mr. Class. 😁
I have done a large amount of study into Pascal's Triangle and could take you off the deep end if you would like. This includes going into higher triangles (simplexes). I can also do square extensions using properties of Pascal's triangle.
I love Pascal’s triangle. Super excited for this one!
For real. It's gotta be the coolest integer triangle.
The graphics look awesome !! Well done
I love your enthusiasm that you have in every video. Math is exciting!
Serpinski triangle emerges so often you could do a whole series on it
That was actually extremely fucking cool. Holy shit.
“Two types of throdd” sounds like a description of a kooky person. Perhaps some guy teaching math in the outdoors with lots of clocks.
Domotro is n for which n+1 %3= 0 AND n+2 %3= 0
The configuration graph of the Tower of Hanoi puzzle also approximates the Sierpinski triangle!!
Tbh, while I love the premise of your channel and watch most of the episodes, they don't always hold my attention (not due to your presentation, just short attention span 😛) but this one has some of my favorite subjects. Awesome!😊
Isn’t this a fun surprise!
My intuitive understanding of fractal curves is that if one measures its length by "walking" a yardstick between two points at a time on it, then it becomes longer and longer the smaller the yardstick one uses. And the fraction of dimensions is the relationship (or slope) between the length of the yardstick and the length the curve when using that yardstick.
Measured curve length = Yardstick length ^ Fractal dimensions
(but with log, use a yardlog instead of a yardstick)
The same is true for almost any curve, like y=x^2. It too gets longer when measuring it with a shorter yardstick. The difference is in the scaling behavior, and that's when I go for lunch.
Weird how you skipped over the part where Pascal rang a bell when he fed his triangle and found that later on, ringing a bell would make it salivate. Eventually, the triangle got hungry and fed up and got revenge by eating Pascal's math homework.
A commonly-skipped part of the history, unfortunately
Very well done illustrations man. Excellent work. Very compelling. Also very heart felt hand written names of all your supporters.
The clocks are also hilarious. Love what you do. Im always looking for the Marco Rodin relations in your mathematics. My brother and i studied Marco Rodins work together as teenagers and the visual illustrations that can be made from numbers by following sequences is entrancing. It makes you feel as tho youre touching god with your fingertips. Your work and videos revive that same feeling i used to have for Marco Rodins work
This guy is such a vibe
I loved the visualizations!
My! The control chaos in this video was all of the triangles!
I must say, I was grinning ear to ear as you were unfolding the pattern! Your figures are journal worthy and it was cool to see both triangles appear to approach the same limit with different process!
I’m curious if there’s a way to relate the series as equivalent? I might hop on MATLAB while I’m sick this weekend and generate the other mods!
12:05 I just realised that this approximation is _already_ happening with our monitors since we don't notice the square pixels that generate the approximation of the Sierpinski triangle! Mind blown.
As a bonus, if you take any of the first 32 rows of Pascal's Triangle mod 2, place the decimal (binary?) point anywhere that makes it an integer (i.e. not before the last 1), and read it as a binary number, a regular polygon with that number of sides can be constructed with straightedge and compass. If the number can't be made this way, a regular polygon with that number of sides can't be constructed with straightedge and compass.
Hmm, that latter sentence is not necessarily true. We know of five Fermat primes, but we do not know for sure there cannot be more, right?
@@landsgevaer You're right. I think it's likely that there aren't anymore though, and if there are, that just means there are more rows of Pascal's Triangle somewhere that also have this property.
Another method of generating serpinsky triangle from complete randomness is the chaos game:
en.wikipedia.org/wiki/Chaos_game
Great to see you evolving as a contant creator. The animations add a lot.
graphics were very helpful
Awesome video.
Another fractal shape that people could draw themselves is:
A square, then right next to it attached, a square half the size. Then right next to it attached, a square half the size of the 2nd square.
I don't know how to put this in formula with the keyboard my mobile can give.
But the area of all those squares together are 2 if the big one you start with is1.
And the area of the triangle goes to 0 when the first one starts at 1.
Which clearly shows the difference to when something is build up by adding. Or broken down by subtracting.
As I'm sure you know, there are a lot of cool patterns in Pascal's Triangle. A couple of years ago, I decided to try to find a method of generating the sequences of numbers in the diagonals of Pascal's Triangle. I succeeded! In fact, I found 2 different methods for accomplishing that goal.
And I'm not even a mathematician! I just enjoy playing with numbers.
Awesome video. I was just messing with this, man!🌞 Since I'm not proficient as a coder in any language, I had Claude 3.5 Sonnet build a pretty cool Python program that color codes the values at each part of the triangle based on their mod values with a given, inputted integer n. It's VERY interesting, and it's amazing to see someone make a video about the exact same thing, at the EXACT same time. I shouldn't really be that surprised.😂
The ultimate Triforce.
ah yeah, the Sierpinski and Pascals Triangles having a connection, i forgot they were connected, i cant remember where i first learned they were connected, i had the sneaking suspition i already knew about it, regardless, always remember:
10:45 Hexagons are the Bestagons!
I was familiar with the pascal triangle mod 2/xor, but hadn't seen the mod 3 version before - which makes it clear that Sierpinski is only one of a family of fractals. One thing you didn't show is the version of pascal's triangle mod 3 where remainder-1 and remainder-2 are coloured differently from each other - which a bit surprisingly causes it to lose the triangular symmetry. I haven't quite thought through why that's true.
this guy went from underrated to overrated and back to underrated
Bro is the youtube sine wave (hey wait he was never overrated)
@@bigfgreatsword his bonus channel suddenly exploded for absolutely no reason
Well, he was always good. Just, his videos,as any creative work, require an inspiration. Which isn't always reliable thing.
Thank you!!
14:14 - its the Domotro's Radioactive Triangular fractal :)
Reminds me of a game I was playing, where I started with a triangle with n points and repeated the following steps: if the number of white regions is even, paint half of them black, starting from the smallest. If the number of white regions is odd, take the triangle, copy it 2 times and arrange them in a bigger simmilar triangle, creating a white region in the middle. Repeat. I always came across the zelda triforce, no matter with how many white regions I started. lol
12:59 math truly is the language of god
6:22 Yup. It returns densitiy😎
I feel like I’ve seen an answer to this before, but what is above Pascal’s triangle if you extend it to more rows of numbers above it? Is there a single extension of it in that direction, or are there multiple sensible ways to do it?
There's a channel named "Dr Barker" with a video titled "What Lies Above Pascal's Triangle?". This may be what you're thinking of.
ua-cam.com/video/q2daqMR3l24/v-deo.html
There is multiple ways, but only one is symmetrical in the previous row
... ½ -½ ½ -½ ½ ½ -½ ½ -½ ½ ...
The next rows above that have more options though, even symmetrical ones
... 1½-x x-1 ½-x x ½-x x-1 1½-x ...
The reason being that ... 1 -1 1 -1 1 -1 1 ... annihilates itself on the next row, so you can always add a multiple of that.
Alignment: chaotic good.
13:58 black filled in cells 🤭
I also heard this
This is eerily similar to Cantor Dust
The Cantor Set has dimension ln2/ln3. The Sierpiński Triangle (aka Gasket) has dimension ln3/ln2, which is its reciprocal.
10:20 i think triangular grid could've worked better here.
I understand that this is for the sake of showing pattern that emerges ignoring the details. But if we consider triangular grid version not as simply approximation, but as a legit part of this fractal, than it kind of reminds me of p-adic numbers. The first iterative view of the triangle is like approximation of the irregular number. But looking at it from the tip (altered pascal triangle) is like looking at the end of the p-adic numbers and zooming away to see more.
Can you please explore p-adic numbers in a future and maybe show some geometrical or modular-arithmetical connections? Hm, considering that in our seemingly infinite universe exists plank length and plank time, could it be that its better to message and study this values in p-adic numbers? And is it true that you can represent 0 in this system as ...999.99...? What if you use negative or complex baze?. Does dual numbers have something in common or in contrary to the modular arithmetics?
If you labeled the centers of a triangular grid, an array like Pascal's triangle (where each cell has exactly 2 above it) wouldn't fit properly without any gaps. And if you labeled the vertices of a triangular grid, that would be the same structure as labeling the centers of a hexagonal grid, just harder to make look clear. Triangular grids and hexagonal grids are dual graphs of each other.
@ComboClass Okay, my spatial perception was wrong at the moment. Thank you for clarifying
1:00 ¿What's the name of that fractal vegetable pictured?
It's called romanesco broccoli (or romanesque cauliflower), since it's related to normal broccoli/cauliflower. It tasted good too!
I love how @ComboClass ate it too xD I did a bunch of research on fractal morphology in college. Always loved fractals.
idea: Make a table of NxN where N is the number of colors. Top row = upper left color, side row = upper right coior, inner cells = what color the current cell should be given the upper colors. What patterns emerge with different lookup tables?
so the serpinski triangle is an infinitely large shape
but you can look at it from the outside, seeing the whole shape all at once
and you can look at it from the "atomic" level, seeing how you can build it pixel by pixel at the most zoomed in level
so theres an "outer end" and an "inner end" to this fractal, with an infinite scale between them
thats like, if the question "what is the last digit of pi" had a real, provable answer, with pi still being infinitely long
Elementary 1D cellular automata with one neighbor come to mind....
10:45 bro actually said "black filled incels" without skipping a beat
Funny how "fractional" dimensions come out of a very binary method like this. "Emergence" as the philosophers say when they don't understand why things are and where they come from. Philosophers might seem not very helpful. But one day maybe one of them slams his head to the desk so that something useful comes out of it.
Thank you for brocoli example - tasty examples are most welcome :)
Kudos to Wacław Sierpiński ;) and that he was opposed to socialist's government.
I am not clear on what your mean by "respect infinititude", but I have a problem with your claim that pascal's triangle, when completed, taking mod(2), IS a sierpinski triangle. I don't think it was fair of you to mention the limit of both shapes without a proof, because if pascal's triangle construction starts with a finite segment with non-zero area, its area diverges.
I am not sure whether allowing this construction from a triangle with no area could make sense, but here is my argument for the two shapes differing:
If you take an equilateral triangle with a positive non-zero area, and multiply it by three, ensuring that the two new triangles have their top vertices touching the bottom vertices of the first, with the new triangles having two vertices that intersect, and you repeat the process, applying it to the result, we still have a top triangle, and therefore, an area greater than 0. This process includes no subdivision, and there is therefore a smallest triangle with a defined area. You don't need to start with a sierpinski triangle for this process to give no area, though. When combining subdivision and multiplication, aiming for an area of 0, we can take an equilateral triangle, and then subdivide it, where we have a proportion 1:d between the exponents of multiplication and division such that (3)^n*(3/4)^dn converges to 0 as n->inf. If we divide 3 times for every multiplication, d=3, (3^n)*(3/4)^3n=3^n*(9/64)^n=(27/64)^n, which is 0, and therefore, in the limit as n->inf, is 0. The formula applies to similar fractals, like the sierpisnki carpet, etc...
If we have a triangle that we want to subdivide, and it has no area, how can we subtract an interior or subdivide it?
I feel like the sierpinski triangle is, in a sense, the inverse of pascal's triangle mod 2, if we consider size in both directions, but we can also create a fractal that merges the two processes. Pascal's triangle has a defined "top" triangle, but that sierpisnki's triangle doesn't, whereas sierpisnki's triangle has a bottom edge, and pascal's triangle does not. Of course, the shapes are identical visually when we consider a "completion" of pascal's triangle, but there is a shape that has neither a top triangle nor a bottom edge, which I think is the best triangle. The only problem is encoding the triangle of all trialgles into it.
What would the string of binomial coefficients mod2 as the rows go to infinity look like? This is not necessarily a rhetorical question. I am wondering if we can use some pattern to determine where the middle will be a string of 1s or a string of 0s, and maybe we can average the strings? I'm assuming my question needs modification to yield any good answer.
But what is the fractical use for this. 🤔
yoooooo
10:44 I heard "blackpilled incels" haha I'm too far-gone. Great video though, love Pascal's triangle.
The triforce is infinite 😀
third?
Ekip
WOW
Stop fractalizing, my brain has gone somewhere in these paths of hidden patterns in "Pascal's triangle" 🌊✨♣🎶🩲📡⚛✡⏫⏏
It's interesting what fractal-fantasy worlds a simple contemplation of the hidden patterns in "Pascal's Triangle" can lead to... how unexpectedly the brain starts looking for some more variants of patterns in "Pascal's Triangle" and ways of visualizing them for better understanding...
Unexpectedly, I wanted to build a 3-D pyramid with a fractal expansion from the center of the pyramid :)