Digonal-triangular duoantiprism

Digonal-triangular duoantiprism
File:Digonal-triangular duoantiprism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Ditdap
Coxeter diagram	s4o2s6o ()
Elements
Cells	12 digonal disphenoids, 6 tetragonal disphenoids, 4 triangular antiprisms
Faces	24+24 isosceles triangles, 4 triangles
Edges	6+12+24
Vertices	12
Vertex figure	Augmented triangular prism
Measures (based on component polygons of edge length 1)
Edge lengths	Lacing edges (24): $\frac{\sqrt{30}}{6} ≈ 0.91287$
	Digons (6): 1
	Edges of triangles (12): 1
Circumradius	$\frac{\sqrt{21}}{6} ≈ 0.76376$
Central density	1
Related polytopes
Army	Ditdap
Regiment	Ditdap
Dual	Digonal-triangular duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×G2)/2, order 48
Convex	Yes
Nature	Tame

The digonal-triangular duoantiprism or ditdap, also known as the 2-3 duoantiprism, is a convex isogonal polychoron that consists of 4 triangular antiprisms, 6 tetragonal disphenoids, and 12 digonal disphenoids. 2 triangular antiprisms, 2 tetragonal disphenoids, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the square-hexagonal 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:$\frac{\sqrt{30}}{5}$ ≈ 1:1.09545. In this specific variant the digonal disphenoids become tetragonal disphenoids. A variant where the triangular antiprisms become fully regular octahedra also exists.

Vertex coordinates

The vertices of a digonal-triangular duoantiprism, assuming that the triangular antiprisms are regular octahedra of edge length 1, centered at the origin, are given by:

• $\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt6}{6},\,\frac{\sqrt6}{6}\right),$

with all even changes of sign, and

• $\left(±\frac12,\,\frac{\sqrt3}{6},\,\frac{\sqrt6}{6},\,\frac{\sqrt6}{6}\right),$

with all odd changes of sign except for the first coordinate.

An alternate set of coordinates where the digonal disphenoids become tetragonal disphenoids, centered at the origin, are given by:

• $\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),$

with all even changes of sign, and

• $\left(±\frac12,\,\frac{\sqrt3}{6},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$

with all odd changes of sign except for the first coordinate.

External links

• Klitzing, Richard. "ditdap".