Snub cube


Snub cube
Snub cube
Snub cube
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 38, E = 60, V = 24 (χ = 2)
Faces by sides (8+24){3}+6{4}
Schläfli symbol s{4,3}
Wythoff symbol | 2 3 4
Coxeter-Dynkin CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Symmetry O, [4,3]+, (432)
Dihedral Angle
References U12, C24, W17
Properties Semiregular convex chiral
Snub cube color
Colored faces
Snub cube
3.3.3.3.4
(Vertex figure)
Pentagonalicositetrahedronccw.jpg
Pentagonal icositetrahedron
(dual polyhedron)
Snub cube Net
Net

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.

The snub cube has 38 faces, 6 of which are squares and the other 32 are 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.

Contents

Dimensions

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

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

ξ32+ξ=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

α6−4α4+16α2−32=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}}\approx1.60972
\beta = \sqrt[3]{13+3\sqrt{33}}

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

(\pm C_1,\pm C_2,\pm C_3)
C_1=\sqrt{\frac{1}{6}-\frac{1}{6c_1}+\frac{c_1}{12}}\approx0.621226
C_2=\sqrt{\frac{1}{3}-\frac{1}{6c_2}+\frac{c_2}{12}}\approx0.337754
C_3=\sqrt{\frac{1}{3}+\frac{1}{12c_3}+\frac{c_4}{12}}\approx1.14261 --This formula is wrong, but the value correct.[improper synthesis?]
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}}

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.

Hexahedron.png
Cube
Small rhombicuboctahedron.png
Rhombicuboctahedron
(Expanded 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.

Snubcubes in grCO.svg

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.[1]

Related polyhedra and tilings

This semiregular polyhedron is part of sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.p) and Coxeter-Dynkin diagram CDel node h.pngCDel p.pngCDel node h.pngCDel 3.pngCDel node h.png, which can be constructed as alternations of the omnitruncation sequence CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

232 332 432 532 632 732 832
Hexagonal prism.png
(4.6.4)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-33-t012.png
(4.6.6)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t012.png
(4.6.8)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t012.png
(4.6.10)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-63-t012.png
4.6.12
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 73-t012.png
4.6.14
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 83-t012.png
4.6.16
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Trigonal antiprism.png
(3.3.3.3.2)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-43-s012.png
(3.3.3.3.4)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-53-s012.png
(3.3.3.3.5)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
3.3.3.3.6
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 73-snub.png
3.3.3.3.7
CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 83-snub.png
3.3.3.3.8
CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png

See also

References

  1. ^ "Spherical Designs" by R. H. Hardin and N. J. A. Sloane
  • Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette 89 (514): 76–81. 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)

External links


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Look at other dictionaries:

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