| Biaugmented truncated cube | |
|---|---|
 | |
| Type | Johnson J66 – J67 – J68 |
| Faces | 2×8 triangles 2+8 squares 4 octagons |
| Edges | 60 |
| Vertices | 32 |
| Vertex configuration | 8(3.82) 8(3.43) 16(3.4.3.8) |
| Symmetry group | D4h |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
In geometry, the biaugmented truncated cube is one of the Johnson solids (J67). As its name suggests, it is created by attaching two square cupolas (J4) onto two parallel octagonal faces of a truncated cube.
Similar to the augmented truncated cube, the biaugmented truncated cube is constructed by attaching two square cupolae onto the opposite pair of faces of a truncated cube; this makes it a composite polyhedron. Covering and replacing two octagons with cupola faces (ten squares and four equilateral triangles in total), the biaugmented truncated cube has sixteen equilateral triangles, ten squares, and four regular octagons as its faces. In total, it has thirty faces, sixty edges, and thirty-two vertices.
A convex polyhedron with regular polygonal faces is a Johnson solid, after American mathematician Norman W. Johnson who list 92 such polyhedra, excluding uniform polyhedra (i.e., Platonic solids, Archimedean solids, and the infinitely many families of prisms and antiprisms). The biaugmented truncated cube is enumerated as the sixty-seventh Johnson solid .
The biaugmented truncated cube has dihedral angles (angles between two polygonal faces), obtained from the angles of a square cupola and a truncated cube, as follows.
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