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

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
.

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

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s and the other 32 are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s. It has 60 edge
Edge

Edge may refer to:...
s and 24 vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. It is a chiral
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 polyhedron, that is, it has two distinct forms, which are mirror image
Mirror Image

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

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

esian coordinates for the vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 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.






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

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
.

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

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s and the other 32 are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s. It has 60 edge
Edge

Edge may refer to:...
s and 24 vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. It is a chiral
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 polyhedron, that is, it has two distinct forms, which are mirror image
Mirror Image

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

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

Cartesian coordinates

Cartesian coordinates for the vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 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 constant
Generalizations of Fibonacci numbers

In mathematics, the Fibonacci numbers form a sequence defined recursion by:The Fibonacci sequence has been studied extensively and generalized in many ways....
. 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 outward
Expansion (geometry)

In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements ....
 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 triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s.

Hexahedron

Cube
Small Rhombicuboctahedron

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 (nonuniform
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
) 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 cube
    Truncated cube

    The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
  • Snub dodecahedron
    Snub dodecahedron

    The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles....
  • Snub square tiling
    Snub square tiling

    In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....
  • Snub hexagonal tiling
    Snub hexagonal tiling

    In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....


External links

  • The Encyclopedia of Polyhedra