Hexagonal-truncated tetrahedral duoalterprism

Hexagonal-truncated tetrahedral duoalterprism
File:Hexagonal-truncated tetrahedral duoalterprism.png (OFF file)
Rank	5
Type	Isogonal
Notation
Bowers style acronym	Hatuta
Coxeter diagram
Elements
Tera	48 triangular cupofastegiums, 8 triangular-hexagonal duoprisms, 12 hexagonal antiprismatic prisms, 6 digonal-hexagonal duoantiprisms, 12 truncated tetrahedral alterprisms
Cells	144 digonal disphenoids, 72 tetragonal disphenoids, 144 wedges, 48 triangular prisms, 96 triangular cupolas, 24 hexagonal prisms, 24 hexagonal antiprisms, 12 truncated tetrahedra
Faces	288+288 isosceles triangles, 48 triangles, 144+144 rectangles, 48 ditrigons, 24 hexagons
Edges	72+144+144+288
Vertices	144
Vertex figure	Digonal-augmented triangular prismatic orthowedge
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Hexagonal-triakis tetrahedral duoaltertegum
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	(I2(12)×(A3×2))/2, order 576
Convex	Yes
Nature	Tame

The hexagonal-truncated tetrahedral duoalterprism, or hatuta, is a convex isogonal polyteron that consists of 12 truncated tetrahedral alterprisms, 6 digonal-hexagonal duoantiprisms, 12 hexagonal antiprismatic prisms, 8 triangular-hexagonal duoprisms, and 48 triangular cupofastegiums. 1 digonal-hexagonal duoantiprism, 1 triangular-hexagonal duoprism, 2 truncated tetrahedral cupoliprisms, 2 hexagonal antiprismatic prisms, and 4 trianguar cupofastegiums join at each vertex. It can be formed by tetrahedrally alternating the dodecagonal-small rhombicuboctahedral duoprism, so that all the small rhombicuboctahedra turn into truncated tetrahedra. However, it cannot be made scaliform.

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 when it is a hull of 2 uniform hexagonal-truncated tetrahedral duoprisms.

Vertex coordinates

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

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

with all permutations and odd changes of sign of the first three coordinates, and

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

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