Interesting polytopes with different edge sizes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Interesting polytopes with different edge sizes

Postby Klitzing » Mon Dec 19, 2016 5:56 pm

Thought to make up this thread, 'cos there are lots of interesting polytopes out there, which do not qualify as Wythoffian, uniform, scaliform, CRF, etc.
Special examples probably are found in the Catalans, cf. this thread, or the perfect Gévay polytopes mentioned here.

 
Today I managed to wrap my mind around an interesting convex polychoron of this type. It just uses 2 different edge sizes. These are x (unity) and v (=1/f = 0.618...).
The cells of this polychoron are regular unit dodecahedra x5o3o and two further convex polyhedra, which use both edge types.

The first additional cell type could be described as x3v3o, i.e. a truncated tetrahedron with only semiregular hexagons. Here its semiregularity does not affect the angles between the sides, those still are all 120 degrees. Rather it affects the edge sizes, which would alternate. The triangle then are again regular, but using the smaller edge size v only.

The other additional polyhedron then is best understood, when refering to pero (pentagonal rotunda). That one clearly has regular pentagonal lacing faces, a further regular pentagon at the top and a regular decagon at the bottom. The remainder then are regular triangles. In the figure to be used here we only apply trigonal axial symmetry. Then the figure could be described accordingly as a triangular variant of pero, as vov3ofx&#xt = v3o || o3f || v3x (where f=1.618...).

The idea to the to be described here polychoron in fact was applying the Gévay ideas a bit beyond his usages. I tried to place other polyhedra into the positions of cells of well-known polychora in some symmetrical way and then apply the convex hull to this. So I started here with the f-scaled rectified icositetrachoron (rico) o3f4o3o. And then I placed vertex incident unit dodecahedra at the places of the former f-cubes. Here the dodecahedra at the one hand would break the full symmetry of the cubes down into a pyritohedral one only, and on the other hand will reach beyond. Therefore the other cell type of rico, the cuboctahedra, get augmented by some kind of cupola. Thus we will obtain these truncated tetrahedra atop the former cuboctahedra, will have these dodecahedra at the places of the former cubes, and additionally would need some further figure to fill in the remainder, esp. in order to match these regular unit pentagons, these smaller regular triangles and these semiregular hexagons. This is how this trigonal variant of the pentagonal rotunda comes into play. Here the medial vertex layer, the f-scaled triangle, then will be just the remainder of the "old" triangular faces of o3f4o3o. The extension thereatop resp. therebelow then are the lacing parts of these asked for cupolae, which happen here to be co-realmic, one side then attaching to the small v3o triangle of these truncated tetrahedra, while the other one would attach to the semiregular hexagons of those. The latteral regular pentagons then clearly attach to the regular unit dodecahedra, while the acute golden triangles ov&#x = o || v just happen to attach to further such cells.

The full incidence matrix then just is:
Code: Select all
96   * |   6   0   0 |   6   3  0  0 |  2  3  0
 * 288 |   2   1   2 |   3   2  2  1 |  1  3  1
-------+-------------+---------------+---------
 1   1 | 576   *   * |   2   1  0  0 |  1  2  0  x
 0   2 |   * 144   * |   2   0  2  0 |  1  2  1  x
 0   2 |   *   * 288 |   0   1  1  1 |  0  2  1  v
-------+-------------+---------------+---------
 2   3 |   4   1   0 | 288   *  *  * |  1  1  0  x5o
 1   2 |   2   0   1 |   * 288  *  * |  0  2  0  ov&#x
 0   6 |   0   3   3 |   *   * 96  * |  0  1  1  x3v
 0   3 |   0   0   3 |   *   *  * 96 |  0  1  1  v3o
-------+-------------+---------------+---------
 8  12 |  24   6   0 |  12   0  0  0 | 24  *  *  x5o3o
 3   9 |  12   3   6 |   3   6  1  1 |  * 96  *  vov3ofx&#x
 0  12 |   0   6  12 |   0   0  4  4 |  *  * 24  x3v3o


Would anyone have an idea on how to call that found polychoron?

--- rk
Klitzing
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