Square antiprism

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.

Square antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Squap
Coxeter diagram	s2s8o ()
Elements
Faces	8 triangles, 2 squares
Edges	8+8
Vertices	8
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, √2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{4+\sqrt2}{8}} ≈ 0.82267$
Volume	$\frac{\sqrt{4+3\sqrt2}}{3} ≈ 0.95700$
Dihedral angles	3–3: $\arccos\left(\frac{1-2\sqrt2}{3}\right) ≈ 127.55160^\circ$
	4–3: $\arccos\left(\frac{\sqrt3-\sqrt6}{3}\right) ≈ 103.83616^\circ$
Height	$\frac{\sqrt[4]{8}}{2} ≈ 0.84090$
Central density	1
Number of external pieces	10
Level of complexity	4
Related polytopes
Army	Squap
Regiment	Squap
Dual	Square antitegum
Conjugate	Square retroprism
Abstract & topological properties
Flag count	64
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(8)×A1)/2, order 16
Convex	Yes
Nature	Tame

Vertex coordinates

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

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

Representations

A square antiprism has the following Coxeter diagrams:

• s2s8o (alternated octagonal prism)
• s2s4s (alternated ditetragonal prism)
• xo4ox&#x (bases considered separately)

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 $\sqrt{l^2-b^2\frac{2-\sqrt2}{2}}$ and its circumradius by $\sqrt{\frac{2b^2+4l^2+b^2\sqrt2}{8}}$.

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 $\sqrt{2+\sqrt2}$ and side edges of length $\sqrt2$.

In vertex figures

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

• Great snub cube (3)
• Great disnub cube (6)

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. This is because $2\pi / (2 \theta - \pi) \approx 13.0094$ where $\theta = \arccos \left( \frac{\sqrt{3} - \sqrt{6}}{3} \right)$ is the dihedral angle between the square and the triangle for the uniform square antiprism.

External links

• Klitzing, Richard. "squap".

• Quickfur. "The Square Antiprism".

• Wikipedia Contributors. "Square antiprism".
• McCooey, David. "Square Antiprism"

• Hi.gher.Space Wiki Contributors. "Square antiprism".