Snub cube

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol

s

{

4

3

}

{\displaystyle s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}

, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol

t

{

4

3

}

{\displaystyle t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}

.

The snub cube, like the snub dodecahedron, is chiral, which means it does not equal its mirror image; it has two equally valid forms.

Construction

The snub cube can be generated by taking the six faces of the cube, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles.

The snub cube may also be constructed from a rhombicuboctahedron. It started by twisting its square face (in blue), allowing its triangles (in red) to be automatically twisted in opposite directions, forming other square faces (in white) to be skewed quadrilaterals that can be filled in two equilateral triangles.

The snub cube can also be derived from the truncated cuboctahedron by the process of alternation. 24 vertices of the truncated cuboctahedron form a polyhedron topologically equivalent to the snub cube; the other 24 form its mirror-image.