Digonal gyrobicupolic ring (EntityTopic, 17)

From Hi.gher. Space

(Difference between revisions)

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.

Digonal gyrobicupolic ring

Net space:	4
Bounding space:	4
Elements:	7, 17, 18, 8
Symmetry group:	Unknown
SSCN:	Unknown
SSC2:	Unknown
Circular packing value:	Unknown
Square packing value:	Unknown
[ Polytope ]
Symbol:	N/A
Facet layout:	Unknown
Petrie order:	8,0
Alternation:	N/A
Dual:	K4.8 dual
Bowers acronym:	N/A

@@ Line 22: / Line 22: @@
 |}
-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 [[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 ==