Snub cube

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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 symbols sr{4,3}
ht0,1,2{4,3}
Wythoff symbol | 2 3 4
Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Symmetry group O, ½BC3, [4,3]+, (432), order 24
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 3-3:153°14'04" (153.23°)
3-4:142°59'00" (142.98°)
References U12, C24, W17
Properties Semiregular convex chiral
Snub hexahedron.png
Colored faces
Snub cube
3.3.3.3.4
(Vertex figure)
Pentagonalicositetrahedronccw.jpg
Pentagonal icositetrahedron
(dual polyhedron)
Snub cube flat.svg
Net

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[edit]

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[edit]

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}}\approx1.60972
\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[edit]

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 Snub cube A2.png Snub cube B2.png Snub cube e1.png
Projective
symmetry
[3]+ [4]+ [2]

Geometric relations[edit]

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)
Snub hexahedron.png
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.

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[edit]

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

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{4,3}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg POV-Ray-Dodecahedron.svg

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 CDel node h.pngCDel n.pngCDel node h.pngCDel 3.pngCDel node h.png. 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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure
Spherical trigonal antiprism.png
3.3.3.3.2
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical snub cube.png
3.3.3.3.4
Spherical snub dodecahedron.png
3.3.3.3.5
Uniform tiling 63-snub.png
3.3.3.3.6
Uniform tiling 73-snub.png
3.3.3.3.7
Uniform tiling 83-snub.png
3.3.3.3.8
Uniform tiling i32-snub.png
3.3.3.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{2,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{4,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{5,3}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{6,3}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{7,3}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{8,3}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{∞,3}
Snub
dual
figure
Hexahedron.svg
V3.3.3.3.2
POV-Ray-Dodecahedron.svg
V3.3.3.3.3
Pentagonalicositetrahedroncw.jpg
V3.3.3.3.4
Pentagonalhexecontahedroncw.jpg
V3.3.3.3.5
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V3.3.3.3.6
Ord7 3 floret penta til.png
V3.3.3.3.7
V3.3.3.3.8 V3.3.3.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png

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

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
242
[2,4]+
342
[3,4]+
442
[4,4]+
542
[5,4]+
642
[6,4]+
742
[7,4]+
842
[8,4]+...
∞42
[∞,4]+
Snub
figure
Spherical square antiprism.png
3.3.4.3.2
Spherical snub cube.png
3.3.4.3.3
Uniform tiling 44-snub.png
3.3.4.3.4
Uniform tiling 54-snub.png
3.3.4.3.5
Uniform tiling 64-snub.png
3.3.4.3.6
Uniform tiling 74-snub.png
3.3.4.3.7
Uniform tiling 84-snub.png
3.3.4.3.8
Uniform tiling i42-snub.png
3.3.4.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{2,4}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{3,4}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{4,4}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{5,4}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{6,4}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{7,4}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{8,4}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{∞,4}
Snub
dual
figure
Tetragonal trapezohedron.png
V3.3.4.3.2
Pentagonalicositetrahedronccw.jpg
V3.3.4.3.3
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
V3.3.4.3.4
Order-5-4 floret pentagonal tiling.png
V3.3.4.3.5
V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png

See also[edit]

References[edit]

  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)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

External links[edit]