# Square antiprism

Square antiprism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSquap
Coxeter diagrams2s8o ()
Elements
Faces8 triangles, 2 squares
Edges8+8
Vertices8
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{4+\sqrt2}{8}} ≈ 0.82267}$
Volume${\displaystyle \frac{\sqrt{4+3\sqrt2}}{3} ≈ 0.95700}$
Dihedral angles3–3: ${\displaystyle \arccos\left(\frac{1-2\sqrt2}{3}\right) ≈ 127.55160°}$
4–3: ${\displaystyle \arccos\left(\frac{\sqrt3-\sqrt6}{3}\right) ≈ 103.83616°}$
Height${\displaystyle \frac{\sqrt[4]{8}}{2} ≈ 0.84090}$
Central density1
Number of external pieces10
Level of complexity4
Related polytopes
ArmySquap
RegimentSquap
DualSquare antitegum
ConjugateSquare retroprism
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
ConvexYes
NatureTame

The square antiprism, or squap, is a prismatic uniform polyhedron. It consists of 8 triangles and 2 squares. Each vertex joins one square and three triangles. As the name suggests, it is an antiprism based on a square.

## Vertex coordinates

A square antiprism of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4}\right).}$

## Representations

A square antiprism has the following Coxeter diagrams:

## General variant

The square antiprism has a general isogonal variant of the form xo4ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by ${\displaystyle \sqrt{l^2-b^2\frac{2-\sqrt2}{2}}}$ and its circumradius by ${\displaystyle \sqrt{\frac{2b^2+4l^2+b^2\sqrt2}{8}}}$.

The bases of the square antiprism are rotated from each other by an angle of 45°. If this angle is changed the result is more properly called a square gyroprism.

A notable case occurs as the alternation of the uniform octagonal prism. This specific case has base edges of length ${\displaystyle \sqrt{2+\sqrt2}}$ and side edges of length ${\displaystyle \sqrt2}$.

## In vertex figures

A square antiprism with base edges of length 1 and side edges of length ${\displaystyle \sqrt2}$ occurs as the vertex figure of the small prismatotetracontoctachoron. One using side edges of length ${\displaystyle \sqrt{2-\sqrt2}}$ occurs as vertex figures of the great distetracontoctachoron.

## Related polyhedra

A square pyramid can be attached to a base of the square antiprism to form the gyroelongated square pyramid. If a second square pyramid is attached to the other base, the result is the gyroelongated square bipyramid.

Two non-prismatic uniform polyhedron compounds are composed of square antiprisms:

There are also an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of squares.

Thirteen square antiprisms can be joined in a ring at their triangular faces if the faces are very slightly distorted.[1] This is because ${\displaystyle 2\pi / (2 \theta - \pi) \approx 13.0094}$ where ${\displaystyle \theta = \arccos \left( \frac{\sqrt{3} - \sqrt{6}}{3} \right)}$ is the dihedral angle between the square and the triangle for the uniform square antiprism.