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Feel free to surf to my site - online psychic; mouse click for source, 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. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The only other chiral Archimedean solid is the snub dodecahedron.

Dimensions

For a snub cube with edge length 1, its surface area is $\scriptstyle {6+8{\sqrt {3}}}$ and its volume is ${\sqrt {\tfrac {613t+203}{9(35t-62)}}}$, where t is the tribonacci constant ${\tfrac {1}{3}}\scriptstyle {\left(1+{\sqrt[{3}]{19-3{\sqrt {33}}}}+{\sqrt[{3}]{19+3{\sqrt {33}}}}\right)\approx 1.83929}$.

If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths ${\tfrac {1}{\sqrt {t+1}}}\scriptstyle {\approx 0.593465}$ and ${\tfrac {1}{2}}\scriptstyle {{\sqrt {t+1}}\approx 0.842509}$.

Cartesian coordinates

Cartesian coordinates for the vertices 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

$\xi ^{3}+\xi ^{2}+\xi =1,\,$

which can be written

$\xi ={\frac {1}{3}}\left({\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{-17+3{\sqrt {33}}}}-1\right)$

or approximately 0.543689. ξ is the reciprocal of the tribonacci constant. 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

$\alpha ^{6}-4\alpha ^{4}+16\alpha ^{2}-32=0,\,$

and can be written as

$\alpha ={\sqrt {{\frac {4}{3}}-{\frac {32}{6{\sqrt[{3}]{2}}\beta }}+{\frac {6{\sqrt[{3}]{2}}\beta }{9}}}}\approx 1.60972$

$\beta ={\sqrt[{3}]{13+3{\sqrt {33}}}}$

For a snub cube with unit edge length, use all the even permutations of

$(\pm C_{1},\pm C_{2},\pm C_{3})$

having an even number of plus signs, along with all the odd permutations having an odd number of plus signs.

$C_{1}={\sqrt {\frac {4-c_{1}+c_{2}}{12}}}\approx 0.337754$

$C_{2}={\sqrt {\frac {2+c_{1}-c_{2}}{12}}}\approx 0.621226$

$C_{3}={\sqrt {\frac {4+c_{3}+c_{4}}{12}}}\approx 1.14261$

$c_{1}={\sqrt[{3}]{3{\sqrt {33}}+17}}$

$c_{2}={\sqrt[{3}]{3{\sqrt {33}}-17}}$

$c_{3}={\sqrt[{3}]{199+3{\sqrt {33}}}}$

$c_{4}={\sqrt[{3}]{199-3{\sqrt {33}}}}$

Orthogonal projections

The snub cube has two special orthogonal projections, centered, on two types of faces: triangles, and squares, correspond to the A₂ and B₂ Coxeter planes.

Orthogonal projections

Centered by	Face Triangle	Face Square	Edge
Image
Projective symmetry	[3]+	[4]+	[2]

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 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.

Cube

Rhombicuboctahedron (Expanded cube)

Snub cube

It can also be constructed as an alternation of a nonuniform omnitruncated cube, deleting every other vertex and creating new triangles at the deleted vertices. A properly proportioned (nonuniform) 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.

A "improved" snub cube, with a slightly smaller square face and slightly larger triangular faces compared to Archimedes' uniform snub cube, is useful as a spherical design.

Related polyhedra and tilings

The snub cube is one of a family of uniform polyhedra related to the cube and regular octahedron.

This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram Template:CDD. These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

The snub cube is second in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. Template:Snub4 table

See also

• Truncated cube
• Snub square tiling

References

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External links

• Template:Mathworld2
• Template:KlitzingPolytopes
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Editable printable net of a Snub Cube with interactive 3D view