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Snub cube

 

 
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Snub cube
 
 
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The snub cube, or snub cuboctahedron, is an Archimedean solidArchimedean solid

In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more ty...
.

The snub cube has 38 faces, of which 6 are squareSquare (geometry)

In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides....
s and the other 32 are equilateral triangleEquilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have equal lengths....
s. It has 60 edges and 24 vertices. It has two distinct forms, which are mirror imageMirror Image

Mirror Image is an episode of the television series The Twilight Zone. ...
s (or "enantiomorphsChirality (mathematics) Summary

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to i...
") of each other.

Cartesian coordinates

Cartesian coordinates for the verticesFacts About Vertex (geometry)

In geometry, a vertex is a special kind of point....
 of 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
ξ3+ξ2+ξ=1,

which can be written
or approximately 0.543689. ξ is the reciprocal of the tribonacci constantGeneralizations of Fibonacci numbers

In mathematics, the Fibonacci numbers form a sequence defined recursively by:...
. 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.

This snub cube has edges of length α, a number which satisfies the equation
α6-4α4+16α2-32=0,

and can be written as

For a snub cube with unit edge length, use the following coordinates instead:

Geometric relations

The snub cube can be generated by taking the six faces of the cube, pulling them outwardExpansion (geometry)

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are forme...
 so they no longer touch. Then give them all a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangleFacts About Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have equal lengths....
s.


Cube
Rhombicuboctahedron
(Expanded cube)


It can also be constructed as an alternation of a nonuniform great rhombicuboctahedron, deleting every other vertex and creating new triangles at the deleted vertices. A properly proportioned great rhombicuboctahedron will create equilateral triangles at the deleted vertices. Depending on which set of vertices are alternated, the resulting snub cube can have a clockwise or counterclockwise twist.


See also

  • Truncated cubeTruncated cube

    The truncated cube, or truncated hexahedron, is an Archimedean solid....
  • Snub dodecahedronSnub dodecahedron

    The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid....
  • Snub square tilingSnub square tiling

    In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane....
  • Snub hexagonal tilingSnub hexagonal tiling

    In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane....


External links

  • The Encyclopedia of Polyhedra