## Tuesday, March 26, 2013

### Even more fractal fun....

"THEY NEVER END!" *runs around in a circle like her hairs on fire*

### Line replacement Fractals

Def : Draw a fractal based on iterated line replacement

Koch curve and snowflake

Levy Tapestry

Cross-stitch Curve

Cresaro Fractal

Minkowski Sausage
*Doesn't look anything like a sausage*

### Space-filling Curves

some variations on the Peano curve I made before.....

### 3D Fractals

Just some random 3D fractals I've been playing with.

End

## Tuesday, March 12, 2013

### some more Fractals.....

Just some more fractal fun I've been having...

Heighway Dragon Curve

Sierpinski's n-gon

Durers Pantagon

Peano Curve

This is my first space filling curve fractal... made pretty much in the same way as the Sierpinski ones.

Will be adding more post on fractals soon ;)

## Friday, March 1, 2013

### 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.

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...

# 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
 *click on image to make bigger* 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
 *click on image to make bigger* 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
 *click on image to make bigger*

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
*click on image to make bigger*
 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
 *click on image to make bigger* 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
 *click on image to make bigger*

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
 *click on image to make bigger*

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
 *click on image to make bigger* Note the difference in the scale being 0.667 instead of 0.5 like with the triangle
Image AB
 *click on image to make bigger* This one is even worse on the ram side but really nice looking. Image AC
 *click on image to make bigger* Sierpinski's Carpet Sierpinski's Sponge

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
 *click on image to make bigger*

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

Image BB
 *click on image to make bigger*

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

Image BC
 *click on image to make bigger*

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
 *click on image to make bigger*

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

Image CB
 *click on image to make bigger*

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
 *click on image to make bigger*

Sierpinski's Pentagon

End :D