Thursday, March 12, 2009

Forbidden Knowledge

Forbidden Knowledge

The Spanish Inquisition would prefer that you not gaze upon these shapes too intently, lest you start thinking in unorthodox (non-rectilinear) ways.

Here shown: three Sytes (the Lite, Bite and Rite) all assembled from the magnetized Mite at the center, a limit case space-filler in the sense of having only six edges (cubes have twice this many).

The Bite (top right) and Rite (bottom center) are also both space-filling tetrahedra, as is the 1/4 Rite (all identified by Sommerville though not named the same way).

In Synergetics, we assign the Mite a volume of 1/8, the Sytes a volume of 1/4. The 1/8 comes from the two As and one B module, each of volume 1/24. These latter come in left and right handed versions.

Those volume figures derive from our canonical "sculpture garden" somewhat popular in Asia, Portland's Old Town, but slow to catch on where proud Anglos hold court.

From the horse's mouth:
Three Self-Packing, Allspace-Filling Irregular Tetrahedra: There are three self-packing irregular tetrahedra that will fill allspace without need of any complementary shape (not even with the need of right- and left-hand versions of themselves). One, the Mite (A), has been proposed by Fuller and described by Coxeter as a tri-rectangular tetrahedron in his book Regular Polytopes, p.71. By joining together two Mites, two varieties of irregular tetrahedra, both called Sytes, can be formed. The tetragonal disphenoid (B), described by Coxeter, is also called the isosceles tetrahedron because it is bounded by four congruent isosceles triangles. The other Syte is formed by joining two Mites by their right-triangle faces (C). It was discovered by Fuller that the Mite has a population of two A quanta modules and one B quanta module (not noted by Coxeter). It is of interest to note that the B quanta module of the Mite may be either right- of left-handed (see the remarks of Arthur L. Loeb). Either of the other two self-packing irregular tetrahedra (Sytes) have a population of four A quanta modules and two B quanta modules, since each Syte consists of two Mites. [ excerpt from caption to Figure 950.12 ]
CubeIT! by Huntar, a 24-Mite cube, is available through Math 'n Stuff, other outlets.