Augmented cube

Augmented cube
(3D model) (OFF file)
Rank	3
Type	Isohedral
Elements
Faces	24 triangles
Edges	12+24
Vertices	6+8
Vertex figure	8 triambi, 6 squares
Measures (edge length 1)
Circumradius	8 (6-fold): ${\frac {\sqrt {3}}{2}}\approx 0.8660254038$ 6 (4-fold): ${\frac {1+{\sqrt {2}}}{2}}\approx 1.207106781$
Volume	$1+{\sqrt {2}}\approx 2.414213562$
Number of external pieces	24
Related polytopes
Dual	Semi-uniform hypertruncated octahedron
Conjugate	Excavated cube
Convex hull	Triakis octahedron
Convex core	Semi-uniform-dual tetrakis cube
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

"Augmented cube" redirects here. For the Johnson solid, see Elongated square pyramid.

The augmented cube is an isohedral deltahedron. It can be formed by augmenting every face of the regular cube with a square pyramid. It is also a non-convex variant of the tetrakis cube.

An interesting property of this polyhedron is that its convex hull is the exact uniform dual triakis octahedron.

Vertex coordinates

Coordinates for the vertices of a cube of edge length 1, centered at the origin, are given by all sign changes and permutations of:

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

External links

• "Isohedral Deltahedra" on orchidpalms.com