# Augmented truncated cube

Augmented truncated cube
Rank3
TypeCRF
Notation
Bowers style acronymAutic
Coxeter diagramxxoox4oxwwx&#xt
Elements
Faces
Edges4+4+4+4+4+4+8+8+8
Vertices4+4+4+8+8
Vertex figures4 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${\displaystyle 8{\frac {3+2{\sqrt {2}}}{3}}\approx 15.54247}$
Dihedral angles3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {3+2{\sqrt {2}}}{6}}}\right)\approx 170.26439^{\circ }}$
3–8 join: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 135°
3–8 tic: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
8–8: 90°
Central density1
Number of external pieces22
Level of complexity24
Related polytopes
ArmyAutic
RegimentAutic
DualRhombirhombistellated triakis octahedron
ConjugateAugmented quasitruncated hexahedron
Abstract & topological properties
Flag count192
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
Flag orbits24
ConvexYes
NatureTame

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:

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

Plus the following additional vertices:

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