Square duoantiprism

Square duoantiprism
File:Square duoantiprism.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Squiddap
Coxeter diagram	s8o2s8o ()
Elements
Cells	32 tetragonal disphenoids, 16 square antiprisms
Faces	128 isosceles triangles, 16 squares
Edges	64+64
Vertices	32
Vertex figure	Gyrobifastigium
Measures (based on squares of edge length 1)
Edge lengths	Lacing (64): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	Edges of squares (64): 1
Circumradius	1
Central density	1
Related polytopes
Army	Squiddap
Regiment	Squiddap
Dual	Square duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(8)≀S2)/2, order 256
Convex	Yes
Nature	Tame

The square duoantiprism or squiddap, also known as the square-square duoantiprism, the 4 duoantiprism or the 4-4 duoantiprism, is a convex isogonal polychoron that consists of 16 square antiprisms and 32 tetragonal disphenoids. 4 square antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the octagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal square dihedral swirlchora and also the third in an infinite family of isogonal digonal prismatic swirlchora, the other being the digonal double tetraswirlprism.

The square duoantiprism contains the vertices of 2 tesseracts, seen as square duoprisms where the base squares of one are in dual orientation to the squares of the other.

The square duoantiprism can be vertex-inscribed into a bitetracontoctachoron, in much the same way that each tesseract can be vertex-inscribed into an icositetrachoron.

The ratio between the longest and shortest edges is 1:${\sqrt {\frac {2+{\sqrt {2}}}{2}}}$ ≈ 1:1.30656.

Vertex coordinates

The vertices of a square duoantiprism based on squares of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),$
• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,0\right).$