Digonal-square duoantiprism

Digonal-square duoantiprism
File:Digonal-square duoantiprism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Disdap
Coxeter diagram	s4o2s8o ()
Elements
Cells	16 digonal disphenoids, 8 tetragonal disphenoids, 4 square antiprisms
Faces	32+32 isosceles triangles, 4 squares
Edges	8+16+32
Vertices	16
Vertex figure	Augmented triangular prism
Measures (based on polygons of edge length 1)
Edge lengths	Lacing (32): $\sqrt{\frac{3-\sqrt2}{2}} ≈ 0.89045$
	Digons (8): 1
	Edges of squares (16): 1
Circumradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Central density	1
Related polytopes
Army	Disdap
Regiment	Disdap
Dual	Digonal-square duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×I2(8))/2, order 64
Convex	Yes
Nature	Tame

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:$\sqrt{\frac{6+2\sqrt2}{7}}$ ≈ 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:

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

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

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