Hexagonal-tetrahedral duoantiprism

Hexagonal-tetrahedral duoantiprism
File:Hexagonal-tetrahedral duoantiprism.png (OFF file)
Rank	5
Type	Isogonal
Notation
Bowers style acronym	Hatetdap
Coxeter diagram	s12o2s4o3o
Elements
Tera	48 triangular scalenes, 12 tetrahedral antiprisms, 6 digonal-hexagonal duoantiprisms
Cells	144 digonal disphenoids, 96 triangular pyramids, 72 tetragonal disphenoids, 12 tetrahedra, 12 hexagonal antiprisms
Faces	144+288 isosceles triangles, 48 triangles, 8 hexagons
Edges	48+72+144
Vertices	48
Vertex figure	Disphenoid-augmented triangular prism wedge
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Hexagonal-tetrahedral duoantitegum
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	(I2(12)×(A3×2))/2, order 576
Convex	Yes
Nature	Tame

The hexagonal-tetrahedral duoantiprism, or hatetdap, is a convex isogonal polyteron that consists of 12 tetrahedral antiprisms, 6 digonal-hexagonal duoantiprisms, and 48 triangular scalenes. 2 tetrahedral antiprisms, 3 digonal-hexagonal duoantiprisms, and 5 triangular scalenes join at each vertex. It can be obtained through the process of alternating the dodecagonal-cubic duoprism. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {10+4{\sqrt {3}}}{13}}}$ ≈ 1:1.14113. This occurs as the hull of 2 uniform hexagonal-tetrahedral duoprisms.

Vertex coordinates

The vertices of a hexagonal-tetrahedral duoantiprism, assuming that the edge length differences are minimized, centered at the origin, are given by:

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0,\,\pm 1\right),$
• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$

with all even changes of sign of the first three coordinates, and

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm 1,\,0\right),$
• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right),$

with all odd changes of sign of the first three coordinates.