EMAIL: chris@68k.org
NAME: Christopher Prince
TOPIC: Elements
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
TITLE: Polyhedral Elements
COUNTRY: USA
WEBPAGE: http://www.68k.org/~chris
RENDERER USED: POV-Ray for Windows v3.02
TOOLS USED: Moray (to import udo's), plot2dxf (see below), 3D Win
RENDER TIME: 1h14'58"
HARDWARE USED: 400 MHz PentiumII

IMAGE DESCRIPTION:
Five polyhedra (from left to right: cube, icosahedron, dodecahedron, tetrahedron, and octahedron) are hovering above marble pedestals.

When I first saw the topic, I thought immediately of Euclid's Elements, one of the most influential bodies of work in mathematics, behind perhaps only Newton's Principia.  I thought it would be neat to do something related to that, so when I found a great deal of proofs on these five polyhedra in Book 13 (the final book), I had a starting point. [1]

These polyhedra are referred to as being "perfect," since these are the only solids which can be constructed from regular, identical polygonal sides and having equal vertex angles.

There is another "elemental" connection to these shapes, though.  They are also called the "Platonic Solids," as it was Plato who popularised the connection between the solids and the classical elements of fire, air, earth, water, and spirit.  The correspondence is as follows:

 Tetrahedron: Fire
        Cube: Earth
  Octahedron: Air
Dodecahedron: Water
 Icosahedron: Spirit   [2]

Mathematically these solids can also be seen as elemental, as truncating, stellating, and otherwise altering these basic shapes gives rise to whole families of solids of interest.

DESCRIPTION OF HOW THIS IMAGE WAS CREATED:
First I created the solids in Maple V, a popular mathematics program.  The geom3d package was used to define the solids mathematically.  Because edges of a whole solid can be a little difficult to see, especially with large vertex angles, I used the cutout statement to remove the innermost 7/8 of each face.  The result is a hollow thick-stick model of the polyhedra.  Plot structures were created from the objects, which are then used to create AutoCAD .dxf files using a small utility I wrote for Maple called plot2dxf.  The .dxf files were then converted to .udo files using 3D Win.

If you are interested in obtaining the plot2dxf utility, if can be found at my home page as listed in the header of this file.  It can be used on a variety of 3D plot structures (everything I've tried it hasn't choked on yet...)

After I had made the polyhedra, it was a simple matter of positioning them above pedestals (contructed of a large cylinder minus smaller outer cylinders and a few squashed superellipsoids), laying down a marble floor, and finally adding an atmosphere and some colored lights inside the polyhedra for an eerie lantern effect.  When I did the first test render I first saw the shadows cast by the solids on the atmosphere and thought "oops, better go turn shadows off," but when the image finished it looked pretty darn spiffy, so I left it on.

REFERENCES:
[1] http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
[2] http://www.smartlink.net/~dem/temenos/theory/polyhedr.txt

COPYRIGHT HOOPLA:
Except as noted in the standard raytracing competition copyright, 
the accompanying image is (C)1998 by Christopher Prince