# Digonal gyrobicupolic ring (EntityTopic, 17)

### From Hi.gher. Space

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.*