Mathematical art (Fractals and Splines)

I've been delving into the realms of fractals and formulaic art form within second life, and thought I'd share some of the things I've come up with as a result of these experiments:

These are splines imposed on a polar coordinate system: the x,y, and z positions are random functions of sin(t) or cos(t) that will complete anywhere between 2 and 16 full "Cycles" upon the individual axis for each sculpture generation. the rotation of the building block follows a similar suit, rotating the x,y,z axes as random functions of sin(t) cos(t) and tan(t). the size of the blocks follows a random |cos(t)|

the generator is not infinitely variable, but there are a total of 20447232 possible combinations for various values and blocks. It is interesting to note, that quite a few of the more "Common" combinations will retrace the exact same path in space, only rotated along one of the primary axis, resulting in 6 congruent patterns for 6 different combinations of values









The spline generator sculpture can be seen here.


I will ammend this post with my fractal generator photos once the asset server decides I have an inventory...

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amazing & beautiful

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Definitely a very interesting dip into 3D fractals, i am impressed.. thank you for sharing that.

Awesome

Your sculptures in The Future are what eventually led to my spline generator, as it was originally intended to be a device for making mobius strips.


What follows are some fractals I've been working on. If anyone recognizes me from the scripter's group, this is the project spawned the various work-arounds for the grey-goo fence. it's still nowhere near sim-friendly, but it can create some very interesting stuff, at one point I put together a random generator like with the spline curves, but I have since lost that particular version and am too lazy to rewrite it at this time.

for these photos, I'm using a spiral defined about the relative <1,1,1> axis that makes PI/4 radians in revolutions (orbits) for each iteration. the spiral has a block for each point (defining the spirals with 8 points), and each point spawns a new spiral (iteration) of relatively smaller size. I am only going 4 iterations deep, as it is already at a hefty 4,000 prims and I have filled up a sandbox with prims more than once in preparing these photos >_>.

the end result, when viewed from the side resembles a tree more or less, but from the top down takes on the properties of some of the more well known fractals such as the "dragon" and "strange attractors" and you can more easily see the spirals within spirals.

overhead view. That's a lotta shiny objects!


one of the spiral arms that has a low number of sub spirals


closeup of a spiral arm that has 4 sub spirals following a very close and similar overall spiral


overall structure view. with various values for number of orbits. Nhere is an overall cross shape with clusters of sub-spirals that appears. it's hard to make out, but can become recognizable after seeing 3 or 4 distinctly different values for the number of orbits


Size reference photo. My avatar's height is 65 on the slider
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as an aside, I've created a "chaos game" fractal generator that yields the classic Pascal's Triangle and the Snow Flake fractals as well, but due to the limitations of the client rendering engine, they do not look all that well, even with 7,000 iterations/points of reference, so I'm working on alternate ways to create these fractals.

Have you tried using prisms or cubes instead of spheres? The effect you get (particularly if you're playing with the rotations as well) can be cool, and it hits the graphics card a lot lighter.

These are fantastic, Senuka! I am going to visit them a.s.a.p.

As soon as I saw them I thought to myself, "Seifert Surface would love these."


!!!! Wow. Even better when you watch them being constructed in 3D. Amazing piece of work. It always amazes me how such organic and flowing shapes can come from such simple mathematical sources. Thanks for sharing this!

Beautiful Senuka

the potential to do things like this (and those done by Seifert and Vlad amongs others ) that makes SL such a special place. Will also visit when I can get back into SL

I've refined my fractal generation code some more, and have been able to produce a fractal that falls into the well known "Dragon" class of fractals. Probably most well known in mainstream from decorating the Chapter Titles in the Novel "Jurassic Park" the dragon fractal is created using a paper folding formula. Since Prims don't fold that well, I went ahead and decided to use the rotational similarity of the classic Dragon fractal to generate one using prims.

Overhead view:



close up of one of the sub-arms:







Also, when tinkering with the generator for this, I discovered how to make a Sierpinski "Sponge". Rather than make the typical "Cube with holes in it" I decided to draw just the holes of the fractal. Here is the fractal in progress. I do not have pictures of the fully completed piece since my methods for generation are not sim friendly and would have an excess of 20,000 prims at this resolution.


Side view:



Close up of one of the clusters:



Due to the high prim nature of these fractals I do not (currently) have plans for a large scale fractal display in world yet, but there is a small fractal generator (only 3-4 iterations deep) behind the desk near my spline generator here.

can the textures be animated? for example random colour changing.. or multiple frame textures? 0r would one need a script for every prim?

it can be done relatively easily (the block is the same piece rezzed over and over again, so commands and what-not can be passed via the on_rez() event), but I refrained from adding any further kind of client rendering to the blocks since with well over 6,000 prims most computers will see a massive drop in FPS near the fractal builds