Six Sierpiński Triangle Constructions (visual mathematics)
HTML-код
- Опубликовано: 17 июл 2024
- This video shows six different methods of creating the Sierpiński triangle including removing triangles, the chaos game, Pascal's triangle mod 2, the bitwise dominance order, a ternary branching tree, and the arrowhead construction.
0:00 : Six Sierpinski Triangle Constructions
0:09 : Introduction
0:50 : Removing Triangles
1:40 : The Chaos Game
2:53 : Halayudha/Pascal Triangle Modulo 2
4:26 : Ternary Branching Tree
5:00 : Bitwise dominance triangle
7:35 : Arrowhead construction
This is an amazing fractal object that is so fun to think about that I just had to animate these for myself. While I haven't seen the bitwise dominance discussed on RUclips, if you like this video, here are some other fantastic (and better) resources on RUclips for investigating the Sierpiński triangle:
1. @ThinkTwiceLtu's Chaos game : • Chaos Game | Fractals ...
2. @3blue1brown's arrowhead construction from Tower of Hanoi: • Binary, Hanoi and Sier... and • Binary, Hanoi, and Sie...
And I am sure there are others. You should also check out the wikipedia article for more resources:
en.wikipedia.org/wiki/Sierpi%...
The bitwise dominance part (and relatedly the Pascal mod 2 part) is based on the results in this paper by Tyler Ball, Tom Edgar, and Daniel Juda:
www.jstor.org/stable/10.4169/...
If you like this video, please consider liking and subscribing. Also, please comment and share the channel with your friends! :)
#math #manim #visualmath #fractal #sierpinski #pascal #pascaltriangle #arrowheadconstruction #ternarytrees #chaosgame #chaos #triforce #triangle #dynamicalsystems #selfsimilar #iteratedfunctionsystem #mathematics #iteachmath #mathvideo
To learn more about animating with manim, check out:
manim.community
_________________________________________
Music in this video:
Free Relaxing Chill Music | ARNOR by Alex-Productions | / @alexproductionsnocopy...
Music promoted by www.free-stock-music.com
Creative Commons Attribution 3.0 Unported License
creativecommons.org/licenses/...
![Introduction to Proof by Induction [Discrete Math Class]](http://i.ytimg.com/vi/OI-nSvpZTpE/mqdefault.jpg)
![Introduction to Proof by Induction [Discrete Math Class]](/img/tr.png)
Dude, this is pure beauty, simply amazing.
Glad you like it!
My favorite method is playing infinite Zelda games and keep adding triangles that way.
Other than that, nice video!
Hah! Thanks:)
@@MathVisualProofs SIERPINSKI HEXAGON
This one should undoubtedly win the some2 contest. Best one I've seen, bar none.
Hah! Thanks. I didn't submit this to #SoME2. And my submission didn't make the top 100 but I still enjoyed making it: ruclips.net/video/eHbtc50-qXo/видео.html
4:30 This is the closet to what I done with cubes (ruclips.net/user/shortsVzwvcMIDKjI?si=gLnWQjriNb_YZPyH). In fact, I call it a ternary cube tree.
Nice! BTW if you subdivide a cube into eight sub-cubes and repeat this process (octree) but each time removing the sub-cubes intersected by the main diagonal vector (1,1,1) the resulting structure contains a Sierpinski triangle (as can be seen when cutting through this 3d structure along a plane orthogonal to the main diagonal).
Very cool. Brings back memories for me, as the chaos game was one of my first (self taught) programming projects that I embarked on back in about 1984 or so (on one of the original IBM PCs).
Awesome! I first programmed it as part of a math class project but it ran and created a static image. Been enjoying watching manim create them in real time :)
@@MathVisualProofs My program actually showed the creation of the points and it was beautiful to watch the pattern develop (like this video)! Oh yeah, and my "initial condition" used random points for the vertices of the triangle, with some constraint to get a "reasonable" triangle, so each run was unique. It's crazy to think about how much programming has changed in 40 years.
@@AllThingsPhysicsRUclips so cool! I didn't have any idea about showing the creation of points back when I did this at Dickinson :) Do you remember what language you used?
@@MathVisualProofs I don't remember specifics, but it must have been BASIC. I also remember writing a (2D) graphing program a year or two later, inspired by one of my community college professors. I remember that this program involved some really intricate PEEKing and POKEing, which is why I'm pretty sure it was in BASIC. It took a lot of trial and error as I recall, but I ultimately got it to work and I remember being so stoked!
Idk much about the maths involved in this, but the triangle pattern thst it gets is rlly interesting
wonderful to find single pattern can help you to relocate connections between multiple theories
nature is beautiful
Man, your vids are awesome!! Great work!
Thanks!
2:47 I was going to ask what happens when you start from a point in the largest empty region, but then realized that wasn't what I wanted. What I wanted was to examine what happens when you pick a point such that the resulting mid-point to a vertex was in one of the empty regions. But, it seems that you can start from such a point, but the midpoints will eventually converge on denser regions.
I believe the one with Pascal's triangle is because of addition of even and uneven numbers.
Adding two even or two uneven numbers creates an even number, while adding an even and an uneven number creates an uneven number.
The triangle starts with a single 1, then two 1s side by side. The third layer however has an even number because there are two uneven numbers above it. Because it's now uneven-even-uneven, it generates a full row of unevens below it because there are no evens or unevens side by side. This then creates a row of evens with unevens at the side (keep in mind the outside is always uneven because it's always 1).
The rows of unevens at the sides grow while the row with evens shrink, because at the border between the evens and unevens, an uneven appears. This converges into a triangle until the row of evens shrinks completely. Meanwhile, at the sides, because the rows of unevens grow, there are new evens generated which then turn into unevens again because they border unevens. At some point, all of the (triangular) "holes" converge again to create a full row of unevens. This in turn creates a larger row of evens which converges to a larger triangle while at the sides new triangles are continuously created. This repeats simultaneously and infinitely, so it eventually turns into an approximation of Sierpinski's triangle.
Mathematics is beautiful.
edit: i really forgor the word for "odd" ☠️
:)
Thanks for this information
Welcome!
Brilliant video.. very well made..
Thanks!
What is the best software to make a menger sponge cube?
Muchas gracias. Tu trabajo es espectacular. Mi favorito fue el del triángulo de Pascal.
Gracias. Yo también :)
@@MathVisualProofs wow, do you actually speak Spanish? Well, sort of?
@@danielsantrikaphundo4517 I studied Spanish for a few years and did an immersion program in Spain for 6 weeks. But that was over 20 years ago, so a lot of it is gone :)
@@MathVisualProofs no me lo esperaba :0
@@danielsantrikaphundo4517 😀
My favorite construction is to initialize conway's game of life with a ray- pixels (0, i) are alive for i >= 0. This produces a noisy triangle full of all the typical gliders and oscillators, that slowly becomes more regular as you zoom out
Good one for sure!
Very cool!!
Thanks! Fun trying to figure out how to show all these. :)
Amazing!!!!!
Thank you :)
The Chaos Game is the one I have least understanding of.
Yes. The theorems involved are deeper and require a lot of mathematics so it’s a tough one to get to the bottom of :)
@@MathVisualProofs chaos is a ladder
@@missingtourist3746 one worth climbing?
@@MathVisualProofs Many who try to climb it fail, never to try again. The fall breaks them. And some given a chance to climb, they refuse. They cling to the realm, or the gods, or love, the illusions. Only the ladder is real. The climb is all there is.
@@missingtourist3746 whoa
wow.very nice. very impressive.
Thank you so much 😀
What about the Conway's square, in Conway's game of life if you have a square, it does nothing right, but if you move the square up one unit every frame(generation) it eventually makes the triangle
But if you choose the exact midpoint of the triangle as your first point, then no matter which point of the triangle you draw a line to, the midpoint of that line is not part of the Sierpinski triangle. Or does the initial point also have to be in the Sierpinski triangle?
Really it’s just the limiting shape that is the triangle.
The Sierpiński Triangle: The Sierpiński triangle is created through an iterative algorithm. Starting
with an equilateral triangle, the midpoints of each side are found and connected
to form an inverted smaller triangle which is then removed. The same process is
then applied to the remaining triangles at each stage.
The Sierpiński Triangle is made for Wacław Sierpiński.
You can create a Sierpiński Triangle with the Halayuda/Pascal Triangle.
Nice! The music made it a bit difficult to listen to. The auto-generated subs seem pretty good, though.
Thanks for the feedback. The sound editing is still a big hang up for me. I’ll keep on it :)
Not sure if one of your six ways to get to the final triangle is equivalent to one more I saw once on Wikipedia by the cellular automaton. One of the 256 possibilities gives the sipiersky triangle if I remember correctly…
If you take a square and divide it into four, delete a corner, then repeat for the small squares, if you do this a lot of times, a sierpinski triangle appears (Best method on checkered notebooks)
7:36 L-systems (Lindermeyer systems) are always interesting, as they are actually a set of rules for the evolution of an initial figure.
It is worth to mention that Lindermeyer first used this sort of process to try to describe the growth of some plants, as he was a botanic.
For sure!
Do you also provide the code that you used to make the animations, they would of great help of someone like me who is trying to make animation for example of a pascal's traingle.
Great video Btw
Triangle’s Majestic Divine.
Is there another geometric shape special like this? Its like a divine formula
The chaos game part: what if i place the first random dott in the center of the triangle?
Typically you have to throw away the first few dots if you want a perfect picture. But since they are dots, they actually won't be too noticeable... they only become noticeable when they aggregate together.
@@MathVisualProofs I also thought about that, but im really bad at math, so i wasn't sure
@@LorvinWolf bad at math? No way. You asked exactly the right question- that’s pretty good!
@@MathVisualProofs thanks.
Wonderful video and presentation.. Tried the Pascal triangle method in my pc.. It went haywire after row 60..
Cool! Numbers too large?
@@MathVisualProofs Yes.. wonder how you pulled it off..
@@SridharGajendran they key is to reduce to “mod 2”. So reduce binomial coefficients to 0 or 1 in a given row and then use pascal recurrence to get next row. Then keep doing this. You never get numbers larger than 2 :)
@@MathVisualProofs wow.. thank you very much.. Can't wait to try it out tomorrow..
you are the first person i've ever seen spelling sierpiński's surname correctly outside of poland
I try my best with those types of things.
"You take the blue pill - the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill - you stay in Wonderland and I show you how deep the rabbit hole goes."
Arrowhead construction is making another fractal simular the the Sierpin'ski triangle
Can I use this for my RUclips space documentary? Please
The bitwise dominance thing is, I think, basically the same as the pascal triangle one, in the following way: the pascal triangle gives the binomial coefficients. If one takes a prime number p (in this example, pick p=2) and expresses n and k in base p, then the binomial coefficient (n choose k), will be equivalent mod p, to the product of the binomial coefficients of the respective base-p digits.
And, for p=2, this product is 1 if all the terms in the product are 1, and is 0 otherwise.
And, (0 choose 0), (1 choose 0, and (1 choose 1) are all 1, with only (0 choose 1) being 0,
and so the “binary digit dominance” thing ends up being whether the corresponding binomial coefficient is even or odd,
So that’s why it gives the same thing as previous process.
yes. they are equivalent via Lucas' theorem (as you note :) ). But they are different in general because if you perform the similar task for different bases, you don't always get Pascal's triangle mod b (you do if you mod out by primes, but not composites).
What hapens if i putt the first random dott in the middle ?
Is a good question. You still get this shape with just a few extra points. The points don’t aggregate so you won’t really see them.
You will eventually end up with the same pattern ruined by one or two stray dots.
4:22
Level 8
?
@@MathVisualProofs Look closer, this is a level-8 Sierpinski.
@@revinhatol Ah! I see what you meant. Thanks!
How On earth do people come up with this kind of idea, I get Mixed feelings of getting amazed and noob as Not able to think like this
what is the use of sierpinski triangle ?
It’s just a fascinating object with interesting properties.
@@MathVisualProofs we made sierpinski triangle for our college exhibition, they asked that what are it's properties and uses.
if u could tell me some properties, it will be great help to me😇
Yes, two more ways
1. Conway's game of life
We are in an infinite square grid and we can decide a square is alive or dead. A cell only has eight possible neighbours, its alive if it has two or three alive neighbours and dies if it only has one alive neighbour or more than three alive neighbours. We make a straight line that has the number of squares from the power of 2 (4097 is fine). When we simulate it, it makes a chaotic Sierpinski.
You can search it if you dont understand it much and its a simulation called Cellular Automaton
2. Wolfram Cellular Automata
We are on an infinite white square grid we always start with one black square. We need to add rules to simulate if its three neighbouring squares on the bottom should be black or white by setting a table in binary descending like this
111 110 101 100 011 010 001
=0 =0 =0 =1 =0 =0 =1
000 < Input
=0 < Output
This is called Rule 18. It gets its name from the outputs
00010010 which is 18 in binary.
Since we have our rule it grows like this
Rule 18:
1
101
10001
1010101
100000001
10100000101
1000100010001
101010101010101
You get the idea. Also, The ones represent the black squares and the zeros represent the white squares. The blank spaces are zeros too.
There are many rules too that generate the Sierpinski like Rule 90, Rule 129 and etc
Edit: wait so are you going to do now Visual Proofs?
Woah, you sound like Code Parade!
I don’t know code parade. I’ll check it out. Is it a good thing ? :)
@@MathVisualProofs
Not a bad thing! He also does some code/math videos. I had just misheard the voice at first
Do check out his "Extraordinary Conics" video
@@NonTwinBrothers Excellent channel! Thanks for pointing me to it :)
When you’re putting the dots down, you are just shading in the odd numbers in the Pascal Triangle.
See time stamp 2:55 :)
3:44 fact: 2^n row numbers all are odd number
Definitely true. The digital dominance argument actually can be modified to prove this.
so we take a line and make it squigglier and squigglier and look it’s a sierpinski triangle
bihari viewers know what i’m talking about
*I CAME HERE TO SEE THE HEXAGON MADE OF SIERPINSKI TRIANGLES!!! WHERE THE [BEEP] IS IT???*
Hah! Sorry.
Zelda has reached the multiverse
👍😃
Hold me ... these things scare me
. O
OO
O O
OOOO
O O
OO OO
O O O O
OOOOOOOO
O O
OO OO
O O O O
OOOO OOOO
O O O O
OO OO OO OO
O O O O O O O O
OOOOOOOOOOOOOOOO seirpinski triangle
you can’t escape it lol
😀
This video has a criminally low amount of views
Thanks!
math is bad