Square antiprism

From Polytope Wiki
Jump to navigation Jump to search
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
Volume
Dihedral angles3–3:
 4–3:
Height
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[edit | edit source]

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

Representations[edit | edit source]

A square antiprism has the following Coxeter diagrams:

General variant[edit | edit source]

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 and its circumradius by .

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 and side edges of length .

In vertex figures[edit | edit source]

A square antiprism with base edges of length 1 and side edges of length occurs as the vertex figure of the small prismatotetracontoctachoron. One using side edges of length occurs as vertex figures of the great distetracontoctachoron.

Related polyhedra[edit | edit source]

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 where is the dihedral angle between the square and the triangle for the uniform square antiprism.

References[edit | edit source]

External links[edit | edit source]