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The sierpinski triangle, henceforth abbreviated as s-tri, holds a place in my heart. It is the first fractal I ever came across, which lead to a probably life-long interest. I first encountered this fractal in the context of mod-2 pascals triangle, which we will cover in detail later. It blew my mind that such a shape was "hiding" in plain sight, only viewable when taking note of evens and odds. I immediately wanted to know more, why it appeared in pascals triangle, why does the iterations of the s-tri align perfectly to the rows with a power of 2. what about mods 3, 4, 5, ect, are there an infinite family of triangle based fractals corrisponding to mod N of pascals triangle?
But, we are getting ahead of ourselves. Let us first go over constructing the s-tri geometrically.
begin with a equallateral triangle.
/\
/ \
/ \
/ \
----------
Then, draw 3 lines connecting the midpoints to divide up the areas.
/\
/ \
/ \
/------\
/\ /\
/ \ / \
/ \ / \
/ \/ \
------------------
If done properly, the end result is 4 new equallateral triangles of equal size, each 1/4 as large as the first. imagine the middle triangle as being cut out and thrown away to form a 'hole'
This is the first iteration of the sierpinski triangle.
Now, we perform the next step of the process by subdividing each of the three new triangles yet again, Creating 12 new copies of the original.
/\
/ \
/ \
/------\
/\ /\
/ \ / \
/ \ / \
/ \/ \
------------------
/\ /\
/ \ / \
/ \ / \
/------\ /------\
/\ /\ /\ /\
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \/ \ / \/ \
-----------------\/-----------------
This is the second iteration, to get to the third repeat the process of dividing up the new triangles. Do this infinitely many times and you end up with the sierpinki triangle.
If you tried to measure the sieripinski triangle you would find it has infinite perimeter and zero area, wierd right? This is because our ideas of perimeter and area only apply to shapes which scale in 2 dimensions. The sierpinski triangle has a fractal dimension of about 1.585, which means you need to use a 1.585 dimensional analog of area and perimeter to measure it.
Click here for a video about fractal dimensions and scaling by 3B1B
(Note that dividing a triange like this is essentially the same thing as making new copies of the original triangle and sticking them together. Meaning that to build the sierpinski triangle you can either divide 1 triangle infinitely many times or add new copies of the original triangle infinitely many times. the end result is the same.)
The S-Triangle is essentially babies first fractal. It has an extremely simple construction method and is self-similar to a fault, being boring to zoom into or out of. Despite that, it shows up in MANY MANY MANY places you wouldnt expect it to, which is the interesting part.
a small list of places it appears:
Yep, thats alot of stuff to go through. I will create a section for each topic to explain how the S-Triangle appears in each setting exactly. I hope you like reading through each of them, but if some parts go over your head (like they do mine) dont be afraid to skip it.
To get the simple stuff out of the way first, The menger sponge is the square version of the sierpinski triangle. Instead of a triangle divided into 4 new slices, a square is divided into 9 new sections with the middle piece as the "hole".
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The 3D dimensional analog known as the sierpinski pyramid or tetrahedron is exactly what youd expect it to be. Instead of triangles its pyramids.
(sorry no ascii graphics, every image i tried to render turned out poorly. you can imagine it easy enough or look it up.)
Now the Cantor set analog is kind of wierd. Its not really apparent how the two are even related until you think about it for abit.
The cantor set is a set of real numbers along the numberline from 0 to 1.
To get the set, you first take the number line from 0 to 1 and remove the middle third of it from points .333R to .666R. you now have two number lines. At the next iteration, remove the middle from the new number lines again. Do this infinitely many times to get the cantor set.
0 .333R .666R 1 ################################################################################# ########################### ########################### ######### ######### ######### ######### ### ### ### ### ### ### ### ### # # # # # # # # # # # # # # # #
One would think that cutting out pieces an infinite number of times would result in no numbers left at the end, but the contrary is true. There are an infinite ammount of numbers in the cantor set and they are *nowhere* dense. for example, 0, .3333R, .6666R and 1 will never be removed. neither will .25
This process of removing middle sections over and over to create smaller copies of the whole is identical to the higher dimensional analogs.
As a side note: the structure of the cantor set resembles that of a binary bifurcating tree.
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So first off, what even *is* a pascals triangle?
Well its a kind of structure achieved through adding numbers together and putting the result below them on the next line. You start off with 1 and add the numbers next to it (blank spaces = 0)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
There are many many many different patterns and relationships in the pascals triangle. the one we are interested in for todays purposes is this: Circle every number divisible by 2.
Note: For more information on the other patterns in pascals triangle, i recommend watching
This video on it by Numberphile
1
1 1
1[2]1
1 3 3 1
1[4][6][4]1
1 5[10][10] 5 1
1[6]15 [20] 15[6]1
1 7 21 35 35 21 7 1
Its kind of hard to see with just ascii but if you look closely you will notice that the structure of numbers looks vaugely like the sierinski triangle and sure enough, yes it is.
I will do the triangle again, this time labeling numbers divisible by 2 "0" and those not divisible by 2 "1". This is what "mod 2" means really, instead of our base 10 numbers, its base 2.
1
1 1
1 0 1
1 1 1 1
1 0 0 0 1
1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 1 1 1 1 1 1
Can you see it?
1
1 1
1 0 1
1 1 1 1
-----------
1\0 0 0/ 1
1 1\0 0/ 1 1
1 0 1\0/ 1 0 1
1 1 1 1V 1 1 1 1
Note that the lines filled with 1111R are exactly on powers of 2.
As to why it shows up here, i dont know. I doubt anyone really does. Its just another beautiful 'coincidence' that mathematics seems so full of when you go looking.
The Chaos Games are really interesting. They are a sort of example of strange attractors in random dynamical systems. Instead of deterministic rules being used to create fractals, rules applied to random systems can also be used to generate fractals.
This one you can actually do in real life with some paper, a pen, and a 6 sided die, if you have some serious patience. Alternatively, its really easy for programming savvy people to whip up a program that does this, ive linked a webpage below for you to play with an interactive version.
Here is a video explaining the Chaos Game by Numberphile
Here is a video showcasing other examples of the Chaos Games with different shapes and rulesets
To start, draw a triangle, label the three verticies
A
/\
/ \
/ \
/ \
B----------C
Now, grab a regular 6 sided dice. The rules of the game are this:
1. Start by picking any random point inside the triangle
2. roll the die, if it lands on 1 or 2 draw a point MIDWAY between your randomly chosen point and vertex A. If it lands on 3 or 4 do the same thing but with B, and 5 or 6 is C.
3. Once your midway point is plotted, roll the die again and repeat step 2, drawing a midway point between the point you just plotted and a random vertice.
4. do this a couple hundred times until a shape starts to emerge from the seeming randomness
At first, it will seem completely random where the points are plotted, but eventually with enough points a shape starts to appear. Yup, you guessed it, that shape is the sierpinski triangle! The edges of the sieripnski triangle act as whats known as a "strange attractor" because they quite literally attract points to specific areas.
There is no easy way to show this in ascii, so i recommend looking at the videos linked above or trying it out yourself through the geogebra website. Again, you could do it in real life with a paper, pen and a die but it takes hundreds of points to get a clear enough image to really see it, which means hundreds of dice throws.
First off, what are the "Towers of Hanoi"? Well it is kind of puzzle.
imagine you have three poles horizontal to one another
| | |
| | |
| | |
A B C
Pole A has three disc of different size
Move 0:
[1] | |
[_2_] | |
[__3__] | |
A B C
The goal is to move all the disc one step at a time from pole A to pole B or C in the order of largest on bottom and smallest on top. The trick is that you CANT put a bigger disc on a smaller disc. I encourage you to do this yourself, find 3 different size books and stack them on eachother then see if you can solve the puzzle.
Move 1:
| | |
[_2_] | |
[__3__] [1] |
A B C
Move 2:
| | |
| | |
[__3__] [1] [_2_]
A B C
move 3:
| | |
[1] | |
[__3__] | [_2_]
A B C
move 4:
| | |
| | [1]
[__3__] | [_2_]
A B C
And so on.
At this point i bet youre wondering how we can pull the sierpinski triangle out of our ass this time. Fear not, i shall explain. But from this point i dont have much in the way of visuals. I recommend watching 3B1B's two part video series "Binary, Hanoi, and Sierpinski part 1 & 2". He explains it far better than i probably can, with fancy animations to boot.
Part 2 (The part with the visualizations were interested in, timestamp 7:41 for relevant animations)
Alright. So.
It appears when we consider the supergraph of all possible positions, as well as how those positions are connected.
Move 1, move 2 and move 3 are all examples of different positions you can arrange the disc. But how many different positions are there exactly? The answer depends on the amount of disc you have to play with, but in the case of 3 disc its 27. 27 possible arrangements of disc positions.
Is there a way that we can graph these different arangements so that we can see the relationship of how one position goes to the next? Yes!
To start off, draw three bubbles in a triangle-ish arangement. inside these bubbles will be each particular arrangement of disc.
In the bottom left bubble, draw move 0, all disc on A. Next, consider the two next legal moves we could take given the rules. At the start, we had the option of moving disc 1 to either B or C.
So, draw the arrangement with disc 1 on B in the upper bubble and on C in the lower right.
Draw a line connecting the bubbles if you can go across positions in a single valid move.
After that, its a matter of drawing all 27 bubbles and connecting them in this arrangement:
all disc on B --> []
/ \
[]----[]
/ \
[] []
/ \ / \
[]--[]----[]----[]
/ \
move 4 -> [] []
/ \ / \
move 2 -> []--[] <- move 3 []----[]
/ \ / \
move 1 -> [] [] [] []
/ \ / \ / \ / \
All disc on A -> []---[]---[]---[]----[]--[]---[]---[] <-- All disc on C
(move 0)
Hello sierpinski my old friend!
As we increase the number of disc, the graph becomes higher iterations of the sierpinski triangle.
As it turns out, theres also a path that walks through every position once and only once. this path is the sierpinski arrowhead curve.
_
/ \
\ /
_/ \_
/ \
\_ _/
_ \ / _
/ \_/ \_/ \
\ /
_/ \_
/ _ _ \
\_/ \ / \_/
_ / \ _
/ \ \_ _/ / \
\ / _ \ / _ \ /
_/ \_/ \_/ \_/ \_/ \_
/ \
\_ _/
_ \ / _
/ \_/ \_/ \
\ _ _ /
_/ / \ / \ \_
/ _ \ / \ / _ \
\_/ \_/ \_ _/ \_/ \_/
_ \ / _
/ \ _/ \_ / \
\ / / _ _ \ \ /
_/ \_ \_/ \ / \_/ _/ \_
/ \ _ / \ _ / \
\_ _/ / \ \_ _/ / \ \_ _/
_ \ / _ \ / _ \ / _ \ / _ \ / _
/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \
Cellular Automata as defined by wolfram alpha:
"A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells."
If you are familiar with Conways Game of life, Cellular automata is like a more advanced version where you can change the ruleset. It seems to operate similar to pascals triangle where the top row influences the next.
Ill be honest with you, i dont really know that much about cellular automata. Its abit too mathy for me. So do your own research if you want to know more. What i do know, however, is that a couple of these rule sets generate sierpinskis triangle. these rulesets are
Speaking of Conways Game of Life, you can also arrange the cells in a way to generate the S-Triangle.
Theres not too much to talk about with this one. A couple of physicist used a quantum simulation to arrange specific atoms in such a way to arrange electrons in the formation of the sierpinski triangle. Their findings are very interesting, indicating that the electrons are actually confined to the fractal dimension 1.585, and even they dont know what that really means for the electrons. There are two ways of bonding the atoms which either let the electrons flow easily or flow more difficultly.
The research paper is locked behind a paywall, so here is
ABS Fractals are a set of fractals cousin to the mandelbrot set, created by running the m-set through a specific function and flipping some signs. as explained by the person who
made this awesome chart showing them off
"Complete Set of 12 Fractals, created by adding abs() commands to the standard form Mandelbrot fractal, and flipping the signage the imaginary side. The Asymmetric fractals such as Burning Ship and Buffalo have chiral pairs, created by reversing the signage on the imaginary side. Because these fractals create mirror images of each other, I have omitted them. The total including these dulpicates would make a set of 16. Negating the real side only generates duplicates of existing fractals with the needle pointing east. Those variants are also omitted."
whatever that means.
The point is: These fractals are very cool to zoom into, a lot of complicated structures can be found in them. One of those many many structures is S-Triangle like patterns.
A video showing off this beautiful structure in the burning ship fractal.
A video showing other kinds of unexpected structures in the perpendicular burning ship fractal.
Its possible to create the sierpinski triangle through hooking together logical operations. I couldnt find out much more on the specific details for using AND operations... However...
I did find something absolutely bizarre: using molecular assembly with DNA tiles to compute XOR functions, in order to generate a sierpinski triangle.
WHAT. THE. HELL. This is some crazy research stuff that intersects molecular biology, cellular automata, and logical operations. In other words, its WAAAY WAAAAAAAAAY over my head, but
the research paper is fun to look and and has pretty pictures.
Ive seen two examples of pretty S-triangle like patterns in real life biology.
I asked someone who was knowledgable on the subject of cellular automata for their input on why seashells look like this sometimes, and they generously gave this reply:
"Evidently, there's a row of pigment producing cells along the linear organ that secretes the growing shell at its leading edge, and these cells implement a one-dimensional cellular automata by having an internal state and exchanging signals with their neighbors. Such cellular automata can often produce Sierpinski patterns. The evolutionary purpose is unclear; it could be accident, or it could be that the patterns afford some measure of camouflage."
And the other example
The arrangement of Algae cells forming colonies
This one isnt super suprising, again the S-triangle is a pretty simple construction, so its not unreasonable that simple cell organisms ocasionally stumble upon the structure by chance. Still, its really cool to see it happen in reality!
WHEW, I AM BEAT AFTER WRITING THIS. My goal was to create a small compendium of information on the S-Triangle and a lot of places it shows up and i believe i achieved that goal. I hope you got some enjoyment out of reading, or just from looking at the pretty ascii pictures, no judgement :)
Did you like what you saw? Got something more to add to the list? Angry at me for a technical screwup or typo? Leave your feedback at smokey@sdf.org
Thank you!