Ursatope (EntityClass, 8)
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The other ursahedra, such as the square ursahedron and the pentagonal ursahedron (which is a diminished dodecahedron), are not CRF. | The other ursahedra, such as the square ursahedron and the pentagonal ursahedron (which is a diminished dodecahedron), are not CRF. | ||
| - | In 4D, there are three known CRF ursatopes: | + | In 4D, there are three known CRF ursatopes: |
| + | * The tetrahedral ursachoron (5 tetrahedra, 1 octahedron, 4 tridiminished icosahedra) | ||
| + | * The octahedral ursachoron (1 octahedron, 1 cuboctahedron, 8 tridiminished icosahedra, 6 square pyramids) | ||
| + | * The icosahedral ursachoron (1 icosahedron, 1 icosidodecahedron, 20 tridiminished icosahedra, 12 pentagonal pyramids) | ||
| + | |||
| + | Other ursachora are possible, but it is believed that they are not CRF. | ||
In 5D, there are believed to be the following CRF ursatopes: | In 5D, there are believed to be the following CRF ursatopes: | ||
Revision as of 05:10, 24 August 2012
The ursatopes are bistratic polytopes that may be constructed from any polytope P with triangular faces. P with unit edge length, P scaled by the golden ratio, and rectified P with unit edge length, are placed in three parallel hyperplanes, the relative heights of which are adjusted such that vertices from the three layers line up to form pentagons. The convex hull of these three layers of vertices will be a polytope that has P as the top cell, rectified P as the bottom cell, and lower-dimensional ursatopes and pyramids lacing the two. If the resulting edge lengths are equal, the result will be a CRF (convex regular-faced) polytope.
In 2D, there is one ursatope: the digonal ursagon, which is the same as a pentagon.
In 3D, there is one CRF ursatope:
- The trigonal ursahedron, which is the same as the tridiminished icosahedron.
The other ursahedra, such as the square ursahedron and the pentagonal ursahedron (which is a diminished dodecahedron), are not CRF.
In 4D, there are three known CRF ursatopes:
- The tetrahedral ursachoron (5 tetrahedra, 1 octahedron, 4 tridiminished icosahedra)
- The octahedral ursachoron (1 octahedron, 1 cuboctahedron, 8 tridiminished icosahedra, 6 square pyramids)
- The icosahedral ursachoron (1 icosahedron, 1 icosidodecahedron, 20 tridiminished icosahedra, 12 pentagonal pyramids)
Other ursachora are possible, but it is believed that they are not CRF.
In 5D, there are believed to be the following CRF ursatopes:
- The 5-cell ursateron: 6 5-cells, 1 rectified 5-cell, and 5 tetrahedral ursachora.
- The 16-cell ursateron: 1 16-cell, 1 24-cell, 16 tetrahedral ursachora, and 8 octahedral pyramids.
- The 24-cell ursateron: 1 24-cell, 1 rectified 24-cell, 24 octahedral ursachora, and 24 cubical pyramids.
- The 600-cell ursateron: 1 600-cell, 1 rectified 600-cell, 600 icosahedral ursachora, and 120 icosahedral pyramids.
In all higher dimensions, only 2 CRF ursatopes are believed possible: the n-simplex ursatope, the n-cross ursatope.
