# Triangular duoantiprism

Triangular duoantiprism Rank4
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymTriddap
Coxeter diagrams6o2s6o (       )
Elements
Cells18 tetragonal disphenoids, 12 triangular antiprisms
Faces72 isosceles triangles, 12 triangles
Edges36+36
Vertices18
Vertex figureGyrobifastigium
Measures (based on triangles of edge length 1)
Edge lengthsLacing edges (36): $\frac{\sqrt6}{3} ≈ 0.81650$ Edges of triangles (36): 1
Circumradius$\frac{\sqrt6}{3} ≈ 0.81650$ Central density1
Related polytopes
ArmyTriddap
RegimentTriddap
DualTriangular duoantitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(G2≀S2)/2, order 144
ConvexYes
NatureTame

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:

## 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.