Pascal's Pyramid Scheme

Fun with fractals, notably "Pascal's Pyramid" (hence the name). Pascal's Pyramid is very similar to the Serpinski Gasket (aka Pascal's Triangle), only 3D. There are four other fractals that are similar to this (the other platonic solids). Then there is a plant fractal, and some sphere tests.

I've done a lot of experimentation and learning with this program so some of the fractals may seem misplaced. The plant was add during an experiment for a CS class, the basis of it comes from the book "The Algorithmic Beauty of Plants". I plan to create a full 3D L-system generator to handle a lot more plants, but at this time I haven't yet done so.

I was curious about the challenge of simulating planets on MU*s. In such environments whole planets are a major challenge, if you're simulating planets at their real size the amount of terrain is huge. Even for someone with a great deal of bandwidth, downloading the terrain of a whole planet would take a long time. Worse if you allow FTL travel (which you must unless you confine everyone to one planet). The general category of "plasma" fractals work well for the terrain generation with very little data needing to be shared (less then 1K for a simple planet). The trick then becomes dividing things in a consistant way. Well, use a couple of the platonic solids for a basis and see how well they turn into planet-like spheres. Well, surprise, surprise, it works best to start with the solid that has the most sides. The icosehedron needs fewer sides to become more spherical, and the sides stay closest to equalateral triangles rather then being scalene.

Run the main program, the controls are described below.

Controls

ESCape key

Pressing the escape key will cause the program to exit at any time.

Mouse Button 1 Drag

Clicking on the first mouse button and dragging over the main window will rotate the object being displayed.

0-9 on keyboard

The number keys change the level of detail/number of iterations. In general increasing the level results in an exponential increase in work, so I recommend against increasing the level by more than one at a time. I recommend against starting above level 3 when switching between fractals, particularly #4 and #5 require a lot of work even at that level. As a rule level 0 displays the basic polyhedra or shape used as the basis of higher levels.

)!@#$%^&*( on keyboard (shift+0-9)

Detail/texturing level. A new feature simply to show off texturing. Draws a 2D variation of the full 3D fractals on the sides of the 3D fractal. This provides the illusion of higher quality while not needing to go as deep on the display lists.

T on keyboard New!

Switches between transparency textures and black textures. In transparency mode, the center of faces are discarded via alpha testing. In black mode, the centers are simply drawn black rather than colored.

F1 on keyboard

The tetrahedron fractal (aka Pascal's Pyramid, this program's namesake). A fun little fractal to see displayed. Level 5 or 6 can be used with decent response time.

F2 on keyboard

The hexahedron (cube) version of the main fractal. Level 3 is reasonable on most machines.

F3 on keyboard

The octahedron version of the main fractal. Level 3 is reasonable on most machines.

F4 on keyboard

The dodecahedron version of the main fractal. Level 2 will be accessible with reasonable response times. Be very cautious about exploring higher levels, as each additional level entails much more than an order of magnitude additional work.

F5 on keyboard

The icosahedron version of the main fractal. Level 2 will be accessible with reasonable response times. Be very cautious about exploring higher levels, as each additional level entails more than an order of magnitude additional work.

F6 on keyboard

A fractal (L-system) plant. Certainly the most pleasing of the fractals. Level 3 is the sweet spot, too much higher and the leaves become too small. This one actually suffers the least with each additional level due to a low branching factor, but has the most complex base shape.

F7 on keyboard

Tetrahedron-based sphere. Each additional level tessellates one more time. At levels 1-3 this looks like a very lumpy sphere. In order to reduce memory usage, the maximum level is capped at 5 (because heirarchical display lists don't work here, memory consumption grows fast).

F8 on keyboard

Octahedron-based sphere. Similar to directly above, each additional level tessellates one more time. At levels 1 and 2 though lumpy, this looks very much more spherical than the above. At level 3 the faces are already disappearing.

F9 on keyboard

Icosahedron-base sphere. Another sphere aproximation by tesselation. Due to the higher number of triangles at the base level we get an extra level's worth of triangles over the tetrahedron, and about half a level's worth over the octahedron. Note though that we get a much smoother sphere for the same number of sides because the triangles are closer to being equilateral and so they're close in size rather than the tetrahedron producing triangles that are quite different in size. Here level 2 looks very spherical and at level 3 the sides are getting very small.

Copyright © 2005 Elliott Mitchell

I make no guarentees that these will work on anyone else's system. I make no guarentees these will not damage anyone else's systems. I make no guarentees whatsoever about these projects.

I retain copyright on all code. You may download, unpack, and compile, but you are not permitted to retain code on a permanent basis. In many cases I may be willing to release this under GPL, but unless explicitly stated none of this available under GPL.

Last updated: $Date: 2005/07/14 05:59:52 $