TITLE: Polyhedral Elements
NAME: Christopher Prince
COUNTRY: USA
EMAIL: chris@68k.org
WEBPAGE: http://www.68k.org/~chris
TOPIC: Elements
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
JPGFILE: euclid.jpg
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