Digonal gyrobicupolic ring (EntityTopic, 17)

From Hi.gher. Space

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<[#ontology [kind topic] [cats 4D Bicupolic_ring Segmentochoron]]>
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<[#ontology [kind topic] [cats Bicupolic_ring]]>
{{STS Shape
{{STS Shape
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| image=<[#img [hash DKVC00GZ95TX8Q6D2HCYGZ4EMJ]]><br />[[Petrie polygon]]
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| 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]]
}}}}
}}}}
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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.
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The '''digonal gyrobicupolic ring''' is a special member of the set of [[bicupolic ring]]s. Unlike the others, only the gyro- form is possible, since ortho- or magna- forms would be degenerate. 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.
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[[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 ==
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;'
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|<[#img [hash FQJDJCGRXBSBP94RHQHKJEA2MQ]]><br/>Square pyramid and<br/>opposite triangle highlighted
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|<[#embed [hash DKVC00GZ95TX8Q6D2HCYGZ4EMJ]]><br/>[[Petrie polygon]]<br/>&nbsp;
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|<[#img [hash E1T3GH5F04TPYTM0RR4485SQF4]]><br/>Tetrahedron and<br/>opposite square highlighted
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|<[#embed [hash FQJDJCGRXBSBP94RHQHKJEA2MQ]]><br/>Square pyramid and<br/>opposite triangle highlighted
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|<[#img [hash 0NF1CPV2F19RM64G3F450VANMY]]><br/>Triangular prism and<br/>opposite digon highlighted
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|<[#embed [hash E1T3GH5F04TPYTM0RR4485SQF4]]><br/>Tetrahedron and<br/>opposite square highlighted
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|<[#embed [hash 0NF1CPV2F19RM64G3F450VANMY]]><br/>Triangular prism and<br/>opposite digon highlighted
|}
|}
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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.
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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.
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== 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).
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:<[#embed [hash FKA3YMF1E5MNG97ABCHSJ25QFC]]>
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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).
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== Software models ==
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*[[Polyview]] [http://hddb.teamikaria.com/dl/4GSV209FK7E00Y3EJA12WVC7F0.def .def file]
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*[[Stella4d]] [http://hddb.teamikaria.com/dl/JDKBGG51Q0HD2R9TNZ8STRM7FF.off .off file]
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:<[#img [hash 2TCT0MYSHNJBPK594VWD8RYRSG]]>
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<[#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:

(image)
Petrie polygon
 
(image)
Square pyramid and
opposite triangle highlighted
(image)
Tetrahedron and
opposite square highlighted
(image)
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).

(image)

Software models

Incidence matrix

Dual: K4.8 dual

#TXIDVaVbEaEbEcEd3a3b3c4a4bC1aC2aC3aTypeName
0 Va = point ;
1 Vb = point ;
2 Ea 20 = digon ;
3 Eb 11 = digon ;
4 Ec 02 = digon ;
5 Ed 02 = digon ;
6 3a 211200 = triangle ;
7 3b 120210 = triangle ;
8 3c 030021 = triangle ;
9 4a 400040 = square ;
10 4b 221201 = square ;
11 C1a 42440120012 = triangular prism ;
12 C2a 23142112101 = square pyramid ;
13 C3a 04004200400 = tetrahedron ;
14 H4.1a 44484244414241 = K4.8 ;

Usage as facets

This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.