Synergetics coordinates from a tetrahedron

• To: mathgroup at smc.vnet.net
• Subject: [mg57697] Synergetics coordinates from a tetrahedron
• From: "Clifford J. Nelson" <cjnelson9 at verizon.net>
• Date: Sat, 4 Jun 2005 03:04:45 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```You could make a coordinate system from the whole to the parts, based on
the closest packing of spheres, instead of building up from axioms or
reference vectors: rack up a triangle of pool balls on a pool table and
put a smaller triangle of balls on top of the big triangle of balls and
then a smaller triangle on that one, etc., to make a tetrahedron of pool
balls with five balls on each of the six edges, thirty five balls
altogether. Bisect the edges by removing pool balls to make an
octahedron and bisect the edges of the octahedron to make a
cuboctahedron of thirteen balls. The four planes that defined the
tetrahedron could move inward one layer of balls and meet at the origin
of the coordinate system (4D) which is at the center ball of the
cuboctahedron. Three of the four planes cut the bottom fourth plane if
the fourth plane doesn't move from the origin to make triangles in the
plane (3D) and two of the four planes define signed line segments (2D)
if the other two planes do not move from the origin of the coordinate
system. Look up closest packing of spheres on Google to see how everyone
else starts with a coordinate system in mind before they do some packing
and they are very confusing (CCP, FCC, HCP).

Somebody's got to rack'm up and show the process in graphics on the net
with nice gleaming spheres and four translucent planes.

I don't know how to show that with Mathematica.

Cliff Nelson