Square antiprism

Square antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Squap
Coxeter diagram	s2s8o ()
Conway notation	A4
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	${\sqrt {\frac {4+{\sqrt {2}}}{8}}}\approx 0.82267$
Volume	${\frac {\sqrt {4+3{\sqrt {2}}}}{3}}\approx 0.95700$
Dihedral angles	3–3: $\arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }$
	4–3: $\arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }$
Height	${\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090$
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

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:

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

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

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

In vertex figures

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

• 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".