Hypertoon Scenario
Morphing Polyhedrons in Synergetics
The geometry my subculture is spreading overlaps a lot of pre-existing geometry that’s entirely conventional, in the sense of well known, meaning spotting our memes and therefore influence, may involve some sensitivity to subtleties.
However, in other cases, it’ll be obvious that we’re harkening back to 1900s futurism, with its geodesic domes and Dymaxion maps, and two published volumes from Macmillan: Synergetics and Synergetics 2.
These two volumes subsequently appeared on the Web as a single unified database, thanks to Dr. Bob Gray, with their numbered passages interleaved to form a single, consecutively enumerated work, complete with table of contents, pictures and an index.
The shape we start with in the following scenario was occasionally itself labeled a “dymaxion” by our geometer, RBF, and it appears in conjunction with sphere packing studies that Newton and Kepler would have found familiar, as they were likewise engaged in such sphere packing studies.
Twelve balls around a nuclear ball, each “kissing” (tangent to) its neighbors, may occur at the corners of a cuboctahedron. See Figure 222.01.
Given four inter-tangent unit radius balls (R=1) define our unit volume tetrahedron, said “dymaxion” has a volume of 20.
Halve those edges and the volume shrinks by a 3rd power, to one eighth of 20, or 2.5 — that’s the cuboctahedron we start with. It embeds in an octahedron of edges 2R, or D, ball diameter.
Jitterbug Transformation: the twist-contracting of the Cuboctahedron into the Icosahedron, for starters. The twist-contracting continues to an Octahedron phase, down to a tetrahedron, where we then talk about inside-outing through a pinch point and re-expanding.
However we see another way to transform (morph) between a cuboctahedron and an icosahedron, by means if “riding the rails” of the Octahedron. Have the vertices of the 2.5 Cuboctahedron all begin moving along the Octahedron’s edges, clockwise or counterclockwise, preserving the equilateral aspect of the triangle connecting them. In this transformation the edges elongate, whereas in the Jitterbug Transformation, they were preserved.
As the eight triangles all rotate by “riding the rails” of the Octahedron, they approach a new position as eight triangles of an icosahedron, the Icosahedron embedded in the Octahedron with eight of its faces flush thereto.
At this point, we reintroduce the classic Jitterbug Transformation. The Icosahedron twist-opens into a Cuboctahedron while keeping all edge lengths constant.
The ratio of these two volumes, Cuboctahedron to Icosahedron, same edge lengths, is called the S-Factor. We call it that because it’s also the ratio of S to E modules, tetrahedral wedges we’ve not introduced yet.
Starting from the Cuboctahedron of 2.5, two applications of the S-factor are sufficient to boost the volume to that of the Icosahedron Within. We call it that (IW) because it’s nested within the Octahedron of volume 4, faces flush.
A third application of the S-factor jitterbugs that Icosahedron to a Cuboctahedron that’s volumetrically bigger than the one we started with by S-factor to the 3rd power. This means it’s linearly bigger than the R-edged cuboctahedron by S-factor exactly. All the edges of this larger Cuboctahedron are now S-factor.
The S-factor edged Cuboctahedron jitterbugs down (twist contracts) to the IW, without changing its edge lengths, so the IW also has edges S-factor.
The round trip scenario would start and end with the Cubocta of volume 2.5. Have its vertices “ride the rails” of the Octahedron by inflating its volume (imagine pumping in air) by S-Factor to the 2nd power. Jitterbug from this Icosahedron back to a Cuboctahedron of volume 2.5 times S-Factor to the 3rd power. Then shrink all edges of the Cuboctahedron back to R, as it subsides back to volume 2.5. And repeat.
