# Square antiprism (EntityTopic, 13)

### From Hi.gher. Space

The **square antiprism** is a special case of an antiprism where the base is a square.

## Cross-sections

- The planar cross-sections of a square antiprism in the [!z] plane are octagons, giving a regular octagon in the middle and degenerating to squares at the ends.
- The planar cross-sections of a square antiprism in the [!x] and [!y] planes are irregular pentagons, degenerating to trapeziums in the middle and at the ends.

## Incidence matrix

Dual: tetragonal trapezohedron

# | TXID | Va | Ea | Eb | 4a | 3a | Type | Name |
---|---|---|---|---|---|---|---|---|

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

1 | Ea | 2 | = digon | ; | ||||

2 | Eb | 2 | = digon | ; | ||||

3 | 4a | 4 | 4 | 0 | = square | ; | ||

4 | 3a | 3 | 1 | 2 | = triangle | ; | ||

5 | C1a | 8 | 8 | 8 | 2 | 8 | = square antiprism |
; |

## Usage as facets

- 2× 1-facets of a square biantiprismatic ring

Notable Trishapes
| |

Regular:
| tetrahedron • cube • octahedron • dodecahedron • icosahedron |

Direct truncates:
| tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |

Mesotruncates:
| stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |

Snubs:
| snub staurohedron • snub rhodohedron |

Curved:
| sphere • torus • cylinder • cone • frustum • crind |