# Square biantiprismatic ring (EntityTopic, 19)

### From Hi.gher. Space

The **square biantiprismic ring** is a member of the infinite family of biantiprismatic rings, which are 4D CRF polychora. It consists of a 3-membered ring containing two square antiprisms and a cube, flanked by a circle of 4 tetrahedra and 4 square pyramids.

## Cartesian coordinates

Cube:

(±1, ±1, ±1, 0)

Square:

(±√2, 0, 0, H) (0, ±√2, 0, H)

where H = √(2√2 - 1).

## Incidence matrix

Dual: (dual of square biantiprismatic ring)

# | TXID | Va | Vb | Ea | Eb | Ec | Ed | 4a | 4b | 3a | 3b | 3c | 4c | C1a | C2a | C3a | C4a | Type | Name |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | Va | = point | ; end | ||||||||||||||||

1 | Vb | = point | ; mid | ||||||||||||||||

2 | Ea | 2 | 0 | = digon | ; end | ||||||||||||||

3 | Eb | 2 | 0 | = digon | ; side | ||||||||||||||

4 | Ec | 1 | 1 | = digon | ; lace | ||||||||||||||

5 | Ed | 0 | 2 | = digon | ; mid | ||||||||||||||

6 | 4a | 4 | 0 | 4 | 0 | 0 | 0 | = square | ; end | ||||||||||

7 | 4b | 4 | 0 | 2 | 2 | 0 | 0 | = square | ; side | ||||||||||

8 | 3a | 2 | 1 | 1 | 0 | 2 | 0 | = triangle | ; lace end | ||||||||||

9 | 3b | 2 | 1 | 0 | 1 | 2 | 0 | = triangle | ; lace side | ||||||||||

10 | 3c | 1 | 2 | 0 | 0 | 2 | 1 | = triangle | ; lace mid | ||||||||||

11 | 4c | 0 | 4 | 0 | 0 | 0 | 4 | = square | ; mid | ||||||||||

12 | C1a | 4 | 4 | 4 | 0 | 8 | 4 | 1 | 0 | 4 | 0 | 4 | 1 | = square antiprism | ; | ||||

13 | C2a | 8 | 0 | 8 | 4 | 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | = cube | ; | ||||

14 | C3a | 4 | 1 | 2 | 2 | 4 | 0 | 0 | 1 | 2 | 2 | 0 | 0 | = square pyramid | ; | ||||

15 | C4a | 2 | 2 | 0 | 1 | 4 | 1 | 0 | 0 | 0 | 2 | 2 | 0 | = tetrahedron | ; | ||||

16 | H4.1a | 8 | 4 | 8 | 4 | 16 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 2 | 1 | 4 | 4 | = square biantiprismatic ring |
; |

## Usage as facets

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