Triangular duoantiprism

Triangular duoantiprism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Triddap
Coxeter diagram	s6o2s6o ()
Elements
Cells	18 tetragonal disphenoids, 12 triangular antiprisms
Faces	72 isosceles triangles, 12 triangles
Edges	36+36
Vertices	18
Vertex figure	Gyrobifastigium
Measures (based on triangles of edge length 1)
Edge lengths	Lacing edges (36): $\frac{\sqrt6}{3} ≈ 0.81650$
	Edges of triangles (36): 1
Circumradius	$\frac{\sqrt6}{3} ≈ 0.81650$
Central density	1
Related polytopes
Army	Triddap
Regiment	Triddap
Dual	Triangular duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(G2≀S2)/2, order 144
Convex	Yes
Nature	Tame

The triangular duoantiprism or triddap, also known as the triangular-triangular duoantiprism, the 3 duoantiprism or the 3-3 duoantiprism, is a convex isogonal polychoron that consists of 12 triangular antiprisms and 18 tetragonal disphenoids. 4 triangular antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the hexagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal triangular dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt6}{2}$ ≈ 1:1.22474.

Vertex coordinates

The vertices of a triangular duoantiprism based on triangles of edge length 1, centered at the origin, are given by:

• $±\left(0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3}\right),$
• $±\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$
• $±\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$
• $±\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right).$

These coordinates show that a triangular duoantiprism can be obtained as the convex hull of two inversely oriented triangular duoprisms.

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Triangular antiprism (12): Hexagonal duotegum
• Tetragonal disphenoid (18): Triangular duoantiprism
• Triangle (12): Hexagonal duotegum
• Isosceles triangle (72): Hexagonal ditetragoltriate
• Edge (36): Hexagonal duoprism
• Edge (36): Triangular double gyroantiprismoid

Variations

The triangular duoantiprism has isogonal variants where the two base triangles are of different edge lengths., and the tetragonal disphenoids become digonal disphenoids while the antiprisms split into 2 sets of 6.

If the ratio of the sides of the two triangular bases is $1:\sqrt2 ≈ 1:1.41421$, the resulting polychoron has 6 regular octahedra, 6 octahedral variants formed from removing 2 vertices from a cube, and 18 disphenoids formed from removing two adjacent vertices from an octahedron. In fact this polychoron is nothing but a diminishing of the icositetrachoron created by removing 6 vertices that form an equatorial hexagon.

External links

• Klitzing, Richard. "triddap".