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...
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.
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.
Comments