# Square antiprism

Square antiprism
Rank3
TypeUniform
Notation
Bowers style acronymSquap
Coxeter diagrams2s8o ()
Conway notationA4
Elements
Faces8 triangles, 2 squares
Edges8+8
Vertices8
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {2}}}{8}}}\approx 0.82267}$
Volume${\displaystyle {\frac {\sqrt {4+3{\sqrt {2}}}}{3}}\approx 0.95700}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
4–3: ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces10
Level of complexity4
Related polytopes
ArmySquap
RegimentSquap
DualSquare antitegum
ConjugateSquare retroprism
Abstract & topological properties
Flag count64
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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right).}$

## 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 ${\displaystyle {\sqrt {l^{2}-b^{2}{\frac {2-{\sqrt {2}}}{2}}}}}$ and its circumradius by ${\displaystyle {\sqrt {\frac {2b^{2}+4l^{2}+b^{2}{\sqrt {2}}}{8}}}}$.

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 ${\displaystyle {\sqrt {2+{\sqrt {2}}}}}$ and side edges of length ${\displaystyle {\sqrt {2}}}$.

## In vertex figures

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