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.
|Bowers style acronym||Squap|
|Coxeter diagram||s2s8o ()|
|Faces||8 triangles, 2 squares|
|Vertex figure||Isosceles trapezoid, edge lengths 1, 1, 1, √2|
|Measures (edge length 1)|
|Number of external pieces||10|
|Level of complexity||4|
|Abstract & topological properties|
|Symmetry||(I2(8)×A1)/2, order 16|
A square antiprism of edge length 1 has vertex coordinates given by:
A square antiprism has the following Coxeter diagrams:
- s2s8o (alternated octagonal prism)
- s2s4s (alternated ditetragonal prism)
- xo4ox&#x (bases considered separately)
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 figuresEdit
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.
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 where is the dihedral angle between the square and the triangle for the uniform square antiprism.
- ↑ McNeill, Jim. "A Ring of Anti-prisms."
- Klitzing, Richard. "squap".
- Quickfur. "The Square Antiprism".
- Wikipedia Contributors. "Square antiprism".
- McCooey, David. "Square Antiprism"
- Hi.gher.Space Wiki Contributors. "Square antiprism".