Digonal gyrobicupolic ring (EntityTopic, 17)
From Hi.gher. Space
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- | <[#ontology [kind topic] [cats | + | <[#ontology [kind topic] [cats Bicupolic_ring]]> |
{{STS Shape | {{STS Shape | ||
- | | image=<[# | + | | image=<[#embed [hash 2TCT0MYSHNJBPK594VWD8RYRSG] [width 150]]> |
| dim=4 | | dim=4 | ||
| elements=7, 17, 18, 8 | | elements=7, 17, 18, 8 | ||
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| dual=[[K4.8 dual]] | | dual=[[K4.8 dual]] | ||
}}}} | }}}} | ||
- | + | 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''' is a | + | |
[[Keiji]] studied it explicitly to try to understand more about the segmentochora. | [[Keiji]] studied it explicitly to try to understand more about the segmentochora. | ||
- | |||
== Construction == | == Construction == | ||
It is possible to construct the digonal gyrobicupolic ring from three different pairs of polytopes: | It is possible to construct the digonal gyrobicupolic ring from three different pairs of polytopes: | ||
{| style='text-align: center;' | {| style='text-align: center;' | ||
- | |<[# | + | |<[#embed [hash DKVC00GZ95TX8Q6D2HCYGZ4EMJ]]><br/>[[Petrie polygon]]<br/> |
- | |<[# | + | |<[#embed [hash FQJDJCGRXBSBP94RHQHKJEA2MQ]]><br/>Square pyramid and<br/>opposite triangle highlighted |
- | |<[# | + | |<[#embed [hash E1T3GH5F04TPYTM0RR4485SQF4]]><br/>Tetrahedron and<br/>opposite square highlighted |
- | |<[# | + | |<[#embed [hash 0NF1CPV2F19RM64G3F450VANMY]]><br/>Triangular prism and<br/>opposite digon highlighted |
|} | |} | ||
- | + | This segmentochoron also arises from a bidiminishing of the [[pyrorectichoron]]. First, delete any vertex from the pyrorectichoron. That forms the ''(mono)diminished pyrorectichoron'', 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 [[ | + | |
== Projections == | == 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). | ||
+ | :<[#embed [hash FKA3YMF1E5MNG97ABCHSJ25QFC]]> | ||
- | + | == Software models == | |
+ | |||
+ | *[[Polyview]] [http://hddb.teamikaria.com/dl/4GSV209FK7E00Y3EJA12WVC7F0.def .def file] | ||
+ | *[[Stella4d]] [http://hddb.teamikaria.com/dl/JDKBGG51Q0HD2R9TNZ8STRM7FF.off .off file] | ||
- | + | <[#polytope [id 60]]> |
Latest revision as of 13:07, 13 March 2016
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 pyrorectichoron. First, delete any vertex from the pyrorectichoron. That forms the (mono)diminished pyrorectichoron, 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).
Software models
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.