ICN5D wrote:Well, that's interesting: hopf fibration of a digon. So, then what would be the HF of the triangle? Or square? But, let's not stop there! Is there any research on HF's of toratopes themselves?
The Hopf fibration applies only to the 2-sphere, so it cannot be applied to toroidal shapes, and it only applies to the 3D sphere (or its tilings thereof). (However, it does
produce toroidal shapes, like the duocylinder!
).
The digon I mentioned is actually the digonal tiling of the 2-sphere (i.e., two hemispheres). The Hopf mapping transforms the hemispheres into the two toroidal surfaces of the duocylinder.
You can, of course, also have a trigonal tiling of the 2-sphere (i.e., 3 lunes stretching from pole to pole). I'm not sure what kind of shape it will produce. It will certainly be interesting to find out, though!
The applications to various regular polyhedra are known, and correspond to certain cell groupings among the 4D polytopes. For example, in my duoprism page (linked in previous post) you see that the tesseract can be decomposed into two rings of 4 cubes each, which actually corresponds with the two toroids of the duocylinder. Similarly, the duoprisms exhibit the same kind of decomposition into two orthogonal rings. The 24-cell has two possible decompositions, corresponding with the Hopf fibration applied to the tetrahedral and cubical tilings of the 2-sphere. The 120-cell can be decomposed into 12 rings of 10 dodecahedra each, and they correspond with the Hopf fibration applied to the dodecahedral tiling of the 2-sphere. Similarly, the 600-cell can be decomposed into 20 rings of 30 tetrahedra each, corresponding with the Hopf fibration applied to the icosahedral tiling of the 2-sphere.
Oddly enough, none of the regular 4D polytopes correspond with the Hopf fibration applied to the octahedral tiling of the 2-sphere: that honor goes to the unusual biicositetradiminished 600-cell, aka
BXD, which consists of 8 rings of 6 tridiminished icosahedra each.
Spidrox, whose projections I gave earlier, has two sets of rings, as I mentioned, 12 rings of alternating prisms and antiprisms corresponding with the dodecahedral tiling of the 2-sphere, and 20 rings of square pyramids corresponding with the icosahedral tiling. You could say that, considered together, these rings correspond with the icosidodecahedral tiling of the 2-sphere.
The Hopf fibration mapping preserves the relationships between the elements of the 2-sphere tiling in the resulting structure, so generally it's most interesting when applied to objects of high symmetry, since the toroidal rings will likewise be symmetrical with each other. But it's also possible to apply the mapping to an irregular tiling of the 2-sphere, say a snub disphenoid, and you'd get an irregular bundle of interlocking toroids that exhibit the symmetries of the snub disphenoid.