# Biaugmented truncated cube

Biaugmented truncated cube
Rank3
TypeCRF
Notation
Bowers style acronymBautic
Coxeter diagramxxooxx4oxwwxo&#xt
Elements
Faces
Edges4+8+8+8+16+16
Vertices8+8+16
Vertex figures8 isosceles trapezoid, edge length 1, 2, 2, 2
16 irregular tetragons, edge length 1, 2, 1, 2+2
8 isosceles triangles, edge lengths 1, 2+2, 2+2
Measures (edge length 1)
Volume${\displaystyle 3(3+2{\sqrt {2}})\approx 17.48528}$
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 pieces30
Level of complexity15
Related polytopes
ArmyBautic
RegimentBautic
DualRhombirhombibistellated triakis octahedron
ConjugateBiaugmented quasitruncated hexahedron
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×A1, order 16
Flag orbits15
ConvexYes
NatureTame

The biaugmented truncated cube (OBSA: bautic) is one of the 92 Johnson solids (J67). It consists of 8+8 triangles, 2+8 squares, and 4 octagons. It can be constructed by attaching square cupolas to two opposite octagonal faces of the truncated cube..

## Vertex coordinates

A biaugmented 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:

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