The Icosahedron Phases of the Jitterbug

By Greg -

The isotropic vector matrix may be described as a regular distribution of VE's and octahedra which combine to fill all-space. Jitterbugging into and out of this ground state, the matrix seems to reach maximum disequilibrium (i.e. maximum expansion) when the contracting VE's and expanding octahedra both describe regular icosahedra. However, because regular icosahedra do not combine to fill all space, the contracting VE's and expanding octahedra cannot pass through their icosahedral phases simultaneously. In fact, at maximum expansion they both describe an irregular icosahedron which does combine, along with a complementary irregular tetrahedron, to fill all space. 

The all-space filling irregular icosahedron is only 1.5 percent larger than the non-all-space filling regular icosahedron, an enticingly small difference.

Eight of its twenty faces remain equilateral triangles (all angles 60°), while the remaining twelve, corresponding to six open square faces of the VE, are isosceles triangles whose angles are (approximately) 48.2°, 65.9°, and 65.9°.

These edge lengths and angles may be irrational. The the all-space-filling complement to the regular icosahedron, however, does appear to have rational (and I would say beautiful), whole-number angles of 36°, 72°, and 72°.

Because matrix-maximum (maximum expansion of the isotropic vector matrix) and matrix minimum (vector equilibrium) are equally significant and deserving of close observation and accounting, the irrational angles and edge lengths and matrix maximum, and the entirely rational angular accounting of the matrix immediately before and after, suggests a model for the quantum leap between the real, physical states.

Comments

Post a Comment

Popular posts from this blog

My Stacy's are Soaking Wet

By Greg -
It's the 50th anniversary of one of the great coming-of-age songs: "Sitting on the dock of the Bay watching my time slip away." But for me, it was Tom Traubert's Blues by Tom Waits, and the line, "No one speaks English and everything's broken, and my Stacy's are soaking wet." If I was a few years older I might have separated myself from my feelings like Bob Dylan when he asked, "How does it feel to be on your own, like a complete unknown, like a rolling stone?" There was also Cat Stevens: "I keep on wondering if sleep too long, will I always wake up the same, or so? And if I make it to the waterside, will I even find me a boat, or so?" There was Catcher in the Rye before popular music got serious. And there was Danny Boy before literature acknowledged that generations were coming of age in America, and we were all doing it alone. For most of us, there was no apprenticeship, no ceremony, no mentors. There was onl...
Read more

Jitterbug from Sphere to Concave Octahedron?

By Greg -
Image
If the jitterbug were modeled as a sphere collapsing to a concave octahedron, the counterclockwise-rotating triangles would transform from a convex spherical triangle to a flat concave triangle, while the clockwise-rotating triangles would simply invert themselves, exposing the concave surface of the same spherical triangle. This may be difficult to animate in SketchUp. But that's not what happens. The spheres transform into concave VEs, the concave VEs expand to spheres (or convex VEs), and the concave octahedra simply turn themselves inside out. I'm working on an illustration of this.
Read more

The Rhombic Dodecahedron and the Closest Packing of Spheres

By Greg -
Image
The rhombic dodecahedron is basically a cube turned inside out and has been shown mathematically to be a projection of the tesseract (the four-dimensional cube) into three-dimensional space. It is, like the cube, an all-space-filling polyhedron and directly corresponds to the closest packing of spheres, enclosing the sphere and the space surrounding it. Rotations of the rhombic dodecahedra demonstrate its projection of the tesseract and show the potential of the isotropic vector matrix as a model of four-dimensional space.
Read more