# Square duoantiprism

Square duoantiprism
File:Square duoantiprism.png
Rank4
TypeIsogonal
Notation
Bowers style acronymSquiddap
Coxeter diagrams8o2s8o ()
Elements
Cells32 tetragonal disphenoids, 16 square antiprisms
Faces128 isosceles triangles, 16 squares
Edges64+64
Vertices32
Vertex figureGyrobifastigium
Measures (based on squares of edge length 1)
Edge lengthsLacing (64): ${\displaystyle {\sqrt {2-{\sqrt {2}}}}\approx 0.76537}$
Edges of squares (64): 1
Central density1
Related polytopes
ArmySquiddap
RegimentSquiddap
DualSquare duoantitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(I2(8)≀S2)/2, order 256
ConvexYes
NatureTame

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:${\displaystyle {\sqrt {\frac {2+{\sqrt {2}}}{2}}}}$ ≈ 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:

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