Triaugmented hexagonal prism

Triaugmented hexagonal prism
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Tauhip
Elements
Faces	6+6 triangles, 3 squares, 2 hexagons
Edges	6+6+6+12
Vertices	3+12
Vertex figures	3 squares, edge length 1
	12 irregular tetragons, edge lengths 1, 1, √2, √3
Measures (edge length 1)
Volume	${\frac {{\sqrt {2}}+3{\sqrt {3}}}{2}}\approx 3.30518$
Dihedral angles	3–4: $\arccos \left(-{\sqrt {\frac {7+2{\sqrt {6}}}{12}}}\right)\approx 174.73561^{\circ }$
	3–6: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	3–3: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	4–6: 90°
Central density	1
Number of external pieces	17
Level of complexity	10
Related polytopes
Army	Tauhip
Regiment	Tauhip
Dual	Laterotritruncated hexagonal tegum
Conjugate	Triaugmented hexagonal prism
Abstract & topological properties
Flag count	120
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×A1, order 12
Convex	Yes
Nature	Tame

The triaugmented hexagonal prism is one of the 92 Johnson solids (J₅₇). It consists of 6+6 triangles, 3 squares, and 2 hexagons. It can be constructed by attaching square pyramids to three mutually non-adjacent square faces of the hexagonal prism.

Vertex coordinates

Coordinates for a triaugmented hexagonal prism of edge length 1 are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm 1,\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {3}}}{4}},\,0\right),$
• $\left(0,\,-{\frac {{\sqrt {2}}+{\sqrt {3}}}{2}},\,0\right).$

External links

• Klitzing, Richard. "tauhip".
• Quickfur. "The Triaugmented Hexagonal Prism".

• Wikipedia contributors. "Triaugmented hexagonal prism".
• McCooey, David. "Triaugmented Hexagonal Prism"