Sierpinski and self-similar sets...

Wacław Franciszek Sierpiński was a polish Math-magician - aka Grand Master of set theory, Senior Warlock of number theory, dude who liked fractals - and school teacher.

He is best know for "Sierpinski's triangle" (in 3D its called Sierpiński tetrahedron)

 his carpet - also related to Sierpiński Sponge or Menger's Sponge when in 3D.

his arrowhead curve

 
and his pentagon O_O

*He also got to be on a coin, which is pretty cool.*


These beauties are known as iterated function systems or IFS, and we will be looking at how we can build some self-similar fractals using Softimage ICE in this post.
**All these IFS have fixed Attractors.**

Some fractal Terminology...

Iterated Function System:

A finite set of contraction maps for , 2, ..., , each with a contractivity factor , which map a compact metric space onto itself. It is the basis for fractal image compression techniques. 

Self-similarity:

An object is said to be self-similar if it looks "roughly" the same on any scale.

Attractor:

Each consists of affine transformations involving rotations, translation and scaling by a constant ratio, where applicable. It can also be divined as the smallest unit which itself cannot be decomposed into two or more attractors.

 L-Systems:

A Lindenmayer system, also known as an L-system, is a string writing system that can be used to generate fractals with dimensions between 1 and 2.




Sierpinski's triangle


To create this fractal you need the following things:
1. repeat on counter node
2. clone point node
3. basic kind of loop
*Just a note, this is a really render and ram intensive way of doing any fractal, but it is cool if you just want to get your head around the math of L-Systems*


In Image A you will see that I have two parts to my setup.  Part one is adding a single point to a pointcloud, and part two which I call the I.F.S is where that point gets cloned and repeated for the fractal.  This is the basic setup for all four of Sierpinski's fractals shown above.


Image A

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To start add a single point to an empty pointcloud at the origin. As shown in Image B.  Make that point into a cone or create your own triangle and use that as an instance shape.  The scale of the triangle/cone 10, 10, 10. *Its easier to see the small iterations when the first triangle is nice and big*

Image B - Attractor NODE

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After that you will need to setup the tree shown in Image C.  Here is where it gets interesting, to build the ICE tree shown in Image C you will have to build all the trees in images C1, D and E.
 In the cast of Sierpinski's triangle, the first Iteration on the Repeat with counter node will produce 3 clones of the original triangle - you want to turn on the delete source if cloned option - and then 9 triangles by the second iteration and so on.

Image C - IFS_Triangle NODE

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What the Log node - Image D - does is translate the new triangles so that they fit inside the original triangle.  We do this with the help of the point ID's and a select case node, run by a loop.


Image D - log 3 NODE

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The translate local point - Image C1 -  node makes sure that the previous point position gets calculated as well as the new point position.
*I got this useful little node from doing Chris Marshall's tutorial on a tree.*

Image C1 - Translate local points NODE

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In Image E we subtract the Point ID by one so that it starts with zero as a value just like the select case node.  Then we feed it through the modulo node to be modulated by the number 3 - number of clones in IFS.

Math behind the log node 



Image E - Loop Value Fractals NODE

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In Image F you can see the ram usage and render time if your on an Iteration step of 12 with 531 441 triangles in your scene.  I'm totally looking into ways of speeding this up.  My poor laptop can't really handle this much ram going to a render.




Image F

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Sierpinski's Triangle

Sierpinski's Tetrahedron

Sierpinski's Carpet
  


Once you've mastered the Triangle feel free to try and make his carpet as well.  Image AA and AB shows the log and scale for that. Start with a Box with a scale of 10 and then fit 8 new boxes to the side facing the X translation.

 Image AA

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Note the difference in the scale being 0.667 instead of 0.5 like with the triangle

 Image AB

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This one is even worse on the ram side but really nice looking.

Image AC

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Sierpinski's Carpet

Sierpinski's Sponge

Sierpinski's Arrowhead


The arrowhead is really interesting because you don't just work with the center point position like you did with the triangle and the carpet, you now work with the starting point 0, 0 - the root of the cylinder - and the end position 0, 1 - the tip of the cylinder, like with the triangle you will have a Log of 3 here, but now you also have to deal with the orientation of the cylinders.

Start with a single cylinder that is rotated to lay horizontally with a scale of  5, 10, 1.  Now just build Image BA and BB into the IFS node.


Image BA

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The scale for this one is also 0.5 just like the triangle.



Image BB

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This one becomes really heavy after 12 iterations couldn't even render it...


Image BC

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Sierpinski's Arrowhead

Sierpinski's Pentagon

To make the beautiful pentagon you will need to make a pentagon and the use the instance shape node to add it to your point in your pointcloud.  This one is a weird one... It has a log with 5 clones and a scale of 0.618 but it also has an offset of sorts.

Image CA

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The offset comes in with the 5 sides of the pentagon being in strange places : / See Image CB for the strangeness.

Image CB

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All of them seem to come in at around 2 to 4 min a frame with anything from 600mb to 1700mb of ram usage.

Image CC

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Sierpinski's Pentagon

End :D