Snub cube

From Example Problems
Jump to navigation Jump to search
Snub cube
Snub cube, anticlockwise twistSnub cube, clockwise twist
Click on pictures for large version.
Click here or here for spinning versions.
Type Archimedean
Faces 32 triangles
6 squares
Edges 60
Vertices 24
Vertex configuration 3,3,3,3,4
Symmetry group octahedral (O)
Dual polyhedron pentagonal icositetrahedron
Properties convex, semi-regular (vertex-uniform), chiral

The snub cube, or snub cuboctahedron, is an Archimedean solid.

The snub cube has 38 faces, of which 6 are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

Canonical coordinates

Canonical coordinates for a snub cube are all the even permutations of (±1, ±ξ, ±1/ξ) with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where ξ is the real solution to ξ32+ξ=1, which can be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "":): {\displaystyle \xi = \frac{1}{3}\left(\sqrt[3]{17+\sqrt{297}} - \sqrt[3]{-17+\sqrt{297}} - 1\right)}

or approximately 0.543689. Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs gives a different snub cube, the mirror image.

Geometric relations

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

The snub cube should not be confused with the truncated cube.

File:Snub cube flat.png

See also

External links

nl:Stompe hexaëder ja:変形立方体