# Digonal double antiprismoid

Digonal double antiprismoid
File:Digonal double antiprismoid.png
Rank4
TypeIsogonal
Notation
Bowers style acronymDidiap
Elements
Cells32 sphenoids, 8+16 tetragonal disphenoids
Faces16+32+64 isosceles triangles
Edges8+32+32
Vertices16
Vertex figureHexakis digonal-hexagonal gyrowedge
Measures (for variant with minimal differences in edge lengths)
Edge lengthsBase edges of antiprismatic disphenoids (8): 1
Lacing edges of side disphenoids (32): 1
Side edges of antiprismatic disphenoids (32): ${\displaystyle {\frac {\sqrt {5+{\sqrt {5}}}}{2}}\approx 1.34500}$
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Central density1
Related polytopes
ArmyDidiap
RegimentDidiap
DualDigonal double antitegmoid
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2+≀S2×2, order 64
ConvexYes
NatureTame

The digonal double antiprismoid or didiap is a convex isogonal polychoron and the first member of the double antiprismoid family. It consists of 24 tetragonal disphenoids of two kinds and 32 sphenoids. 6 disphenoids and 8 sphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal digonal-digonal duoantiprisms or by alternating the square ditetragoltriate. However, it cannot be made uniform. As such it is one of a number of polychora that can be obtained as the convex hull of two variant hexadecachora. It is the first in an infinite family of isogonal digonal antiprismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\frac {\sqrt {5+{\sqrt {5}}}}{2}}}$ ≈ 1:1.34500. This variant is formed from duoantiprisms with base digons of length ratio 1:${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ ≈ 1:1.61803.

## Vertex coordinates

The vertices of a digonal double antiprismoid, assuming that the two short edges have edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,0,\,\pm {\frac {1}{2}}\right).}$