Ursatope (EntityClass, 8)

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Revision as of 17:19, 23 June 2013

The ursatopes are bistratic polytopes that may be constructed from any polytope P. 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.

List of ursatopes

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 pyroursachoron, with 5 tetrahedra, one octahedron and 4 tridiminished icosahedra;
The aeroursachoron, with an octahedron, a cuboctahedron, 8 tridiminished icosahedra and 6 square pyramids;
The hydroursachoron, with an icosahedron, an icosidodecahedron, 20 tridiminished icosahedra and 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 pyroursateron, with 6 pyrochora, one rectified pyrochoron, and five pyroursachora.
The aeroursateron, with an aerochoron, a xylochoron, 16 pyroursachora, and 8 octahedral pyramids.
The xyloursateron, with a xylochoron, a rectified xylochoron, 24 aeroursachora, and 24 cubical pyramids.
The hydroursateron, with a hydrochoron, a rectified hydrochoron, 600 pyroursachora, and 120 icosahedral pyramids.

In all higher dimensions n > 5, only two CRF ursatopes are believed possible:

The pyroursatope: (n+1) pyro-(n-1)-topes, a rectified pyro-(n-1)-tope and n pyroursa-(n-1)-topes;
The aeroursatope: an aero-(n-1)-tope, a rectified aero-(n-1)-tope, 2^n-1 pyroursa-(n-1)-topes and (2n-2) aero-(n-2)-tope pyramids.

Expanded ursatopes

Expanded ursatopes may be constructed by expanding the facets of the top facet such that the ursatope facets are pushed apart. Prisms are inserted between the ursatope facets, causing the bottom facet to also expand, and the lacing pyramids to expand into cupolae. Expanded ursatopes may be used to construct higher-dimensional modified ursatopes in which the lacing facets are expanded ursatopes, etc.

Each of the three ursachora have corresponding expanded forms:

The expanded pyroursachoron, with a cuboctahedron, a truncated octahedron, 4 tridiminished icosahedra, 6 pentagonal prisms, 4 triangular prisms and 4 triangular cupola (to be verified);
The expanded aeroursachoron, with a rhombicuboctahedron, a truncated cube, 8 tridiminished icosahedra, 12 pentagonal prisms, 6 cubes and 6 square cupola;
The expanded hydroursachoron, with a rhombicosidodecahedron, a truncated dodecahedron, 20 tridiminished icosahedra, 30+12 = 42 pentagonal prisms and 12 pentagonal cupola (to be verified).

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 == List of ursatopes ==
-In 2D, there is one ursatope: the digonal ursagon, which is the same as a pentagon.
+In 2D, there is one ursatope: the digonal ursagon, which is the same as a [[pentagon]].
-In 3D, there is one CRF ursatope:
+In 3D, there is one CRF ursatope: the trigonal ursahedron, which is the same as the [[tridiminished icosahedron]].
-* 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 pyroursachoron, with 5 [[tetrahedra]], one [[octahedron]] and 4 tridiminished icosahedra;
-* The octahedral ursachoron (1 octahedron, 1 cuboctahedron, 8 tridiminished icosahedra, 6 square pyramids)
+* The aeroursachoron, with an octahedron, a [[cuboctahedron]], 8 tridiminished icosahedra and 6 [[square pyramid]]s;
-* The icosahedral ursachoron (1 icosahedron, 1 icosidodecahedron, 20 tridiminished icosahedra, 12 pentagonal pyramids)
+* The hydroursachoron, with an icosahedron, an [[icosidodecahedron]], 20 tridiminished icosahedra and 12 [[pentagonal pyramid]]s.
 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 pyroursateron, with 6 [[pyrochora]], one [[rectified pyrochoron]], and five pyroursachora.
-* The 16-cell ursateron: 1 16-cell, 1 24-cell, 16 tetrahedral ursachora, and 8 octahedral pyramids.
+* The aeroursateron, with an [[aerochoron]], a [[xylochoron]], 16 pyroursachora, and 8 [[octahedral pyramid]]s.
-* The 24-cell ursateron: 1 24-cell, 1 rectified 24-cell, 24 octahedral ursachora, and 24 cubical pyramids.
+* The xyloursateron, with a xylochoron, a [[rectified xylochoron]], 24 aeroursachora, and 24 [[cubical pyramid]]s.
-* The 600-cell ursateron: 1 600-cell, 1 rectified 600-cell, 600 tetrahedral ursachora, and 120 icosahedral pyramids.
+* The hydroursateron, with a [[hydrochoron]], a [[rectified hydrochoron]], 600 pyroursachora, and 120 [[icosahedral pyramid]]s.
-In all higher dimensions n>5, only 2 CRF ursatopes are believed possible:
+In all higher dimensions ''n'' > 5, only two CRF ursatopes are believed possible:
-* The (n-1)-simplex ursatope: n+1 (n-1)-simplexes, 1 rectified (n-1)-simplex, and n (n-1)-ursatopes;
+* The pyroursatope: (''n''+1) pyro-(''n''-1)-topes, a rectified pyro-(''n''-1)-tope and ''n'' pyroursa-(''n''-1)-topes;
-* The (n-1)-cross ursatope: 1 (n-1)-cross, 1 rectified (n-1)-cross, 2^(n-1) (n-1)-ursatopes, and 2n-2 (n-2)-cross pyramids.
+* The aeroursatope: an aero-(''n''-1)-tope, a rectified aero-(''n''-1)-tope, 2<sup>n-1</sup> pyroursa-(''n''-1)-topes and (2''n''-2) aero-(''n''-2)-tope pyramids.
 == Expanded ursatopes ==
-Expanded ursatopes may be constructed by expanding the facets of the top facet such that the ursatope facets are pushed apart. Prisms are inserted between the ursatope facets, causing the bottom facet to also expand, and the lacing pyramids to expand into cupolae.
+Expanded ursatopes may be constructed by expanding the facets of the top facet such that the ursatope facets are pushed apart. Prisms are inserted between the ursatope facets, causing the bottom facet to also expand, and the lacing pyramids to expand into cupolae. Expanded ursatopes may be used to construct higher-dimensional modified ursatopes in which the lacing facets are expanded ursatopes, etc.
-Expanded ursatopes may be used to construct higher-dimensional modified ursatopes in which the lacing facets are expanded ursatopes, etc..
+Each of the three ursachora have corresponding expanded forms:
+* The expanded pyroursachoron, with a cuboctahedron, a [[truncated octahedron]], 4 tridiminished icosahedra, 6 [[pentagonal prism]]s, 4 [[triangular prism]]s and 4 [[triangular cupola]] (to be verified);
-'''TBD''': the naming of these derived ursatopes have not been decided on yet.
+* The expanded aeroursachoron, with a [[rhombicuboctahedron]], a [[truncated cube]], 8 tridiminished icosahedra, 12 pentagonal prisms, 6 [[cube]]s and 6 [[square cupola]];
+* The expanded hydroursachoron, with a [[rhombicosidodecahedron]], a [[truncated dodecahedron]], 20 tridiminished icosahedra, 30+12 = 42 pentagonal prisms and 12 [[pentagonal cupola]] (to be verified).
-Three CRF 4D expanded ursatopes are known so far: the expanded tetrahedral ursachoron, the expanded octahedral ursachoron, and the expanded icosahedral ursachoron.