Augmented truncated cube

Augmented truncated cube
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Autic
Coxeter diagram	xxoox4oxwwx&#xt
Elements
Faces	4+4+4 triangles 1+4 squares 1+4 octagons
Edges	4+4+4+4+4+4+8+8+8
Vertices	4+4+4+8+8
Vertex figures	4 isosceles trapezoid, edge length 1, √2, √2, √2
	8 trapezoids, edge length 1, √2, 1, √2+√2
	4+4+8 isosceles triangles, edge lengths 1, √2+√2, √2+√2
Measures (edge length 1)
Volume	$8{\frac {3+2{\sqrt {2}}}{3}}\approx 15.54247$
Dihedral angles	3–4 join: $\arccos \left(-{\sqrt {\frac {3+2{\sqrt {2}}}{6}}}\right)\approx 170.26439^{\circ }$
	3–8 join: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	4–4: 135°
	3–8 tic: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	8–8: 90°
Central density	1
Number of external pieces	22
Level of complexity	24
Related polytopes
Army	Autic
Regiment	Autic
Dual	Rhombirhombistellated triakis octahedron
Conjugate	Augmented quasitruncated hexahedron
Abstract & topological properties
Flag count	192
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Flag orbits	24
Convex	Yes
Nature	Tame

The augmented truncated cube (OBSA: autic) is one of the 92 Johnson solids (J₆₆). It consists of 4+4+4 triangles, 1+4 squares, and 1+4 octagons. It can be constructed by attaching a square cupola to one of the octagonal faces of the truncated cube.

Vertex coordinates

An augmented truncated cube of edge length 1 has vertices given by all permutations of:

• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)$,

Plus the following additional vertices:

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

External links

• Klitzing, Richard. "autic".
• Quickfur. "The Augmented Truncated Cube".

• Wikipedia contributors. "Augmented truncated cube".
• McCooey, David. "Augmented Truncated Cube"