# Digonal-square duoantiprism

Digonal-square duoantiprism
File:Digonal-square duoantiprism.png
Rank4
TypeIsogonal
Notation
Bowers style acronymDisdap
Coxeter diagrams4o2s8o ()
Elements
Cells16 digonal disphenoids, 8 tetragonal disphenoids, 4 square antiprisms
Faces32+32 isosceles triangles, 4 squares
Edges8+16+32
Vertices16
Vertex figureAugmented triangular prism
Measures (based on polygons of edge length 1)
Edge lengthsLacing (32): ${\displaystyle {\sqrt {\frac {3-{\sqrt {2}}}{2}}}\approx 0.89045}$
Digons (8): 1
Edges of squares (16): 1
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Related polytopes
ArmyDisdap
RegimentDisdap
DualDigonal-square duoantitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(B2×I2(8))/2, order 64
ConvexYes
NatureTame

The digonal-square duoantiprism or disdap, also known as the 2-4 duoantiprism or the 8-2 double step prism, is a convex isogonal polychoron that consists of 4 square antiprisms, 8 tetragonal disphenoids, and 16 digonal disphenoids. 2 square antiprisms, 2 tetragonal disphenoids, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the square-octagonal duoprism. However, it cannot be made uniform, as it generally has 3 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\sqrt {\frac {6+2{\sqrt {2}}}{7}}}}$ ≈ 1:1.12303.

## Vertex coordinates

The vertices of a digonal–square duoantiprism, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by:

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

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

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