# Digonal-square duoantiprism

Digonal-square duoantiprism
File:Digonal-square duoantiprism.png
Rank4
TypeIsogonal
SpaceSpherical
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): ${\displaystyle \sqrt{\frac{3-\sqrt2}{2}} ≈ 0.89045}$
Digons (8): 1
Edges of squares (16): 1
Circumradius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
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:${\displaystyle \sqrt{\frac{6+2\sqrt2}{7}}}$ ≈ 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:

• ${\displaystyle \left(0,\,±\frac{\sqrt2}{2},\,\frac{\sqrt[4]{8}}{4},\,\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4},\,-\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,\frac{\sqrt[4]{8}}{4},\,\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4},\,-\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4},\,-\frac{\sqrt[4]{8}}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,-\frac{\sqrt[4]{8}}{4},\,\frac{\sqrt[4]{8}}{4}\right).}$

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

• ${\displaystyle \left(0,\,±\frac{\sqrt2}{2},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt2}{4},\,-\frac{\sqrt2}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt2}{4},\,-\frac{\sqrt2}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,\frac{\sqrt2}{4},\,-\frac{\sqrt2}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,-\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).}$