Six Sierpiński Triangle Constructions (visual mathematics)

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 YouTube, if you like this video, here are some other fantastic (and better) resources on YouTube for investigating the Sierpiński triangle:

And I am sure there are others. You should also check out the wikipedia article for more resources:

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:

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:

_________________________________________
Music in this video:
Creative Commons Attribution 3.0 Unported License

[luciano.rezende]

Dude, this is pure beauty, simply amazing.

[LeoStaley]

This one should undoubtedly win the some2 contest. Best one I've seen, bar none.

[Tezhut]

My favorite method is playing infinite Zelda games and keep adding triangles that way.
Other than that, nice video!

[ahmedlutfi]

wonderful to find single pattern can help you to relocate connections between multiple theories
nature is beautiful

[AllThingsPhysicsYouTube]

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).

[ahmedh.]

The Pascal Triangle Modulo 2 looks like a variant of using the Rule 90 Elementary Cellular Automaton with a single cell on. That also uses parity. Thank you for a great video!

[ram_n_music]

Idk much about the maths involved in this, but the triangle pattern thst it gets is rlly interesting

[kyh]

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" ☠️

[jakobthomsen]

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).

[TesserId]

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.

[quadmasterXLII]

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

[BanMidouSan]

Muchas gracias. Tu trabajo es espectacular. Mi favorito fue el del triángulo de Pascal.

[beethovennine]

Man, your vids are awesome!! Great work!

[wendolinmendoza]

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.

[christopherop]

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)

[CesareVesdani]

What is the best software to make a menger sponge cube?

[didierleonard]

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…

[TimeTraveler-hkxo]

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?

[SuviTuuliAllan]

Nice! The music made it a bit difficult to listen to. The auto-generated subs seem pretty good, though.

[kijete]

you are the first person i've ever seen spelling sierpiński's surname correctly outside of poland