Digonal gyrobicupolic ring (EntityTopic, 17)
From Hi.gher. Space
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The '''digonal gyrobicupolic ring''', or '''K4.8''', is a member of the set of [[bicupolic ring]]s. Its cells are 1 [[tetrahedron]], 4 [[square pyramid]]s and 2 [[triangular prism]]s. Its faces are 1+4 [[square]]s and 4+4+4 [[triangle]]s. It has 2+4+4+8 edges and 4+4 vertices. | The '''digonal gyrobicupolic ring''', or '''K4.8''', is a member of the set of [[bicupolic ring]]s. Its cells are 1 [[tetrahedron]], 4 [[square pyramid]]s and 2 [[triangular prism]]s. Its faces are 1+4 [[square]]s and 4+4+4 [[triangle]]s. It has 2+4+4+8 edges and 4+4 vertices. | ||
[[Keiji]] studied it explicitly to try to understand more about the segmentochora. | [[Keiji]] studied it explicitly to try to understand more about the segmentochora. | ||
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== Construction == | == Construction == | ||
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|<[#embed [hash 0NF1CPV2F19RM64G3F450VANMY]]><br/>Triangular prism and<br/>opposite digon highlighted | |<[#embed [hash 0NF1CPV2F19RM64G3F450VANMY]]><br/>Triangular prism and<br/>opposite digon highlighted | ||
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This segmentochoron also arises from a bidiminishing of the [[3-pyrotomochoron]]. First, delete any vertex from the 3-pyrotomochoron. That forms the ''(mono)diminished 3-pyrotomochoron'', better known as the [[trigonal biantiprismatic ring]], or ''K4.6''; its cells are 1 triangular prism, 2 [[octahedra]], 3 square pyramids, and 3 tetrahedra. It can be constructed as ''trigonal prism || gyrated triangle''; if a vertex from the gyrated triangle is deleted, it will create a second triangular prism, thus resulting in this segmentochoron. | This segmentochoron also arises from a bidiminishing of the [[3-pyrotomochoron]]. First, delete any vertex from the 3-pyrotomochoron. That forms the ''(mono)diminished 3-pyrotomochoron'', better known as the [[trigonal biantiprismatic ring]], or ''K4.6''; its cells are 1 triangular prism, 2 [[octahedra]], 3 square pyramids, and 3 tetrahedra. It can be constructed as ''trigonal prism || gyrated triangle''; if a vertex from the gyrated triangle is deleted, it will create a second triangular prism, thus resulting in this segmentochoron. | ||
== Projections == | == Projections == | ||
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The following projection shows this segmentochoron from a viewpoint analogous to that of the other bicupolic rings, to show how the gyrated digons connect to each other via a tetrahedron (digon antiprism) and to the opposite square face via triangular prisms (digon cupolae). | The following projection shows this segmentochoron from a viewpoint analogous to that of the other bicupolic rings, to show how the gyrated digons connect to each other via a tetrahedron (digon antiprism) and to the opposite square face via triangular prisms (digon cupolae). | ||
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:<[#embed [hash FKA3YMF1E5MNG97ABCHSJ25QFC]]> | :<[#embed [hash FKA3YMF1E5MNG97ABCHSJ25QFC]]> | ||
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Revision as of 22:35, 16 February 2014
The digonal gyrobicupolic ring, or K4.8, is a member of the set of bicupolic rings. Its cells are 1 tetrahedron, 4 square pyramids and 2 triangular prisms. Its faces are 1+4 squares and 4+4+4 triangles. It has 2+4+4+8 edges and 4+4 vertices.
Keiji studied it explicitly to try to understand more about the segmentochora.
Construction
It is possible to construct the digonal gyrobicupolic ring from three different pairs of polytopes:
 Petrie polygon |  Square pyramid and opposite triangle highlighted |  Tetrahedron and opposite square highlighted |  Triangular prism and opposite digon highlighted |
This segmentochoron also arises from a bidiminishing of the 3-pyrotomochoron. First, delete any vertex from the 3-pyrotomochoron. That forms the (mono)diminished 3-pyrotomochoron, better known as the trigonal biantiprismatic ring, or K4.6; its cells are 1 triangular prism, 2 octahedra, 3 square pyramids, and 3 tetrahedra. It can be constructed as trigonal prism || gyrated triangle; if a vertex from the gyrated triangle is deleted, it will create a second triangular prism, thus resulting in this segmentochoron.
Projections
The following projection shows this segmentochoron from a viewpoint analogous to that of the other bicupolic rings, to show how the gyrated digons connect to each other via a tetrahedron (digon antiprism) and to the opposite square face via triangular prisms (digon cupolae).
Incidence matrix
Dual: K4.8 dual
| # | TXID | Va | Vb | Ea | Eb | Ec | Ed | 3a | 3b | 3c | 4a | 4b | C1a | C2a | C3a | Type | Name |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | ||||||||||||||
| 1 | Vb | = point | ; | ||||||||||||||
| 2 | Ea | 2 | 0 | = digon | ; | ||||||||||||
| 3 | Eb | 1 | 1 | = digon | ; | ||||||||||||
| 4 | Ec | 0 | 2 | = digon | ; | ||||||||||||
| 5 | Ed | 0 | 2 | = digon | ; | ||||||||||||
| 6 | 3a | 2 | 1 | 1 | 2 | 0 | 0 | = triangle | ; | ||||||||
| 7 | 3b | 1 | 2 | 0 | 2 | 1 | 0 | = triangle | ; | ||||||||
| 8 | 3c | 0 | 3 | 0 | 0 | 2 | 1 | = triangle | ; | ||||||||
| 9 | 4a | 4 | 0 | 0 | 0 | 4 | 0 | = square | ; | ||||||||
| 10 | 4b | 2 | 2 | 1 | 2 | 0 | 1 | = square | ; | ||||||||
| 11 | C1a | 4 | 2 | 4 | 4 | 0 | 1 | 2 | 0 | 0 | 1 | 2 | = triangular prism | ; | |||
| 12 | C2a | 2 | 3 | 1 | 4 | 2 | 1 | 1 | 2 | 1 | 0 | 1 | = square pyramid | ; | |||
| 13 | C3a | 0 | 4 | 0 | 0 | 4 | 2 | 0 | 0 | 4 | 0 | 0 | = tetrahedron | ; | |||
| 14 | H4.1a | 4 | 4 | 4 | 8 | 4 | 2 | 4 | 4 | 4 | 1 | 4 | 2 | 4 | 1 | = K4.8 | ; |
Usage as facets
This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.
