Snub cube (original) (raw)

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Archimedean solid with 38 faces

Snub cube
Two different forms of a snub cube
Type Archimedean solid
Faces 38
Edges 60
Vertices 24
Symmetry group Rotational octahedral symmetry O {\displaystyle \mathrm {O} } {\displaystyle \mathrm {O} }
Dihedral angle (degrees) triangle-to-triangle: 153.23° triangle-to-square: 142.98°
Dual polyhedron Pentagonal icositetrahedron
Properties convex, chiral
Vertex figure
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. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi.[1] H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol s { 4 3 } {\displaystyle s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}} {\displaystyle s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol t { 4 3 } {\displaystyle t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}} {\displaystyle t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}.

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.[2]

Process of snub cube's construction by rhombicuboctahedron

The snub cube may also be constructed from a rhombicuboctahedron. It started by twisting its square face (in blue), allowing its triangles (in red) to be automatically twisted in opposite directions, forming other square faces (in white) to be skewed quadrilaterals that can be filled in two equilateral triangles.[3]

The snub cube can also be derived from the truncated cuboctahedron by the process of alternation. 24 vertices of the truncated cuboctahedron form a polyhedron topologically equivalent to the snub cube; the other 24 form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform.

Uniform alternation of a truncated cuboctahedron

Cartesian coordinates

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Cartesian coordinates for the vertices of a snub cube are all the even permutations of ( ± 1 , ± 1 t , ± t ) , {\displaystyle \left(\pm 1,\pm {\frac {1}{t}},\pm t\right),} {\displaystyle \left(\pm 1,\pm {\frac {1}{t}},\pm t\right),}with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where t ≈ 1.83929 {\displaystyle t\approx 1.83929} {\displaystyle t\approx 1.83929} is the tribonacci constant.[4] 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. Taking them together yields the compound of two snub cubes.

This snub cube has edges of length α = 2 + 4 t − 2 t 2 {\displaystyle \alpha ={\sqrt {2+4t-2t^{2}}}} {\displaystyle \alpha ={\sqrt {2+4t-2t^{2}}}}, a number which satisfies the equation α 6 − 4 α 4 + 16 α 2 − 32 = 0 , {\displaystyle \alpha ^{6}-4\alpha ^{4}+16\alpha ^{2}-32=0,} {\displaystyle \alpha ^{6}-4\alpha ^{4}+16\alpha ^{2}-32=0,}and can be written as α = 4 3 − 16 3 β + 2 β 3 ≈ 1.609 72 β = 26 + 6 33 3 . {\displaystyle {\begin{aligned}\alpha &={\sqrt {{\frac {4}{3}}-{\frac {16}{3\beta }}+{\frac {2\beta }{3}}}}\approx 1.609\,72\\\beta &={\sqrt[{3}]{26+6{\sqrt {33}}}}.\end{aligned}}} {\displaystyle {\begin{aligned}\alpha &={\sqrt {{\frac {4}{3}}-{\frac {16}{3\beta }}+{\frac {2\beta }{3}}}}\approx 1.609\,72\\\beta &={\sqrt[{3}]{26+6{\sqrt {33}}}}.\end{aligned}}}To get a snub cube with unit edge length, divide all the coordinates above by the value α given above.

3D model of a snub cube

For a snub cube with edge length a {\displaystyle a} {\displaystyle a}, its surface area and volume are:[5] A = ( 6 + 8 3 ) a 2 ≈ 19.856 a 2 V = 8 t + 6 3 2 ( t 2 − 3 ) a 3 ≈ 7.889 a 3 . {\displaystyle {\begin{aligned}A&=\left(6+8{\sqrt {3}}\right)a^{2}&\approx 19.856a^{2}\\V&={\frac {8t+6}{3{\sqrt {2(t^{2}-3)}}}}a^{3}&\approx 7.889a^{3}.\end{aligned}}} {\displaystyle {\begin{aligned}A&=\left(6+8{\sqrt {3}}\right)a^{2}&\approx 19.856a^{2}\\V&={\frac {8t+6}{3{\sqrt {2(t^{2}-3)}}}}a^{3}&\approx 7.889a^{3}.\end{aligned}}}

The snub cube is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.[6] It is chiral, meaning there are two distinct forms whenever being mirrored. Therefore, the snub cube has the rotational octahedral symmetry O {\displaystyle \mathrm {O} } {\displaystyle \mathrm {O} }.[7][8] The polygonal faces that meet for every vertex are four equilateral triangles and one square, and the vertex figure of a snub cube is 3 4 ⋅ 4 {\displaystyle 3^{4}\cdot 4} {\displaystyle 3^{4}\cdot 4}. The dual polyhedron of a snub cube is pentagonal icositetrahedron, a Catalan solid.[9]

The graph of a snub cube

The skeleton of a snub cube can be represented as a graph with 24 vertices and 60 edges, an Archimedean graph.[10]

  1. ^ Conway, John H.; Burgiel, Heidi; Goodman-Struss, Chaim (2008). The Symmetries of Things. CRC Press. p. 287. ISBN 978-1-4398-6489-0.
  2. ^ Holme, A. (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN 978-3-642-14441-7.
  3. ^ Conway, Burgiel & Goodman-Struss (2008), p. 287–288.
  4. ^ Collins, Julian (2019). Numbers in Minutes. Hachette. p. 36–37. ISBN 978-1-78747-730-8.
  5. ^ Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
  6. ^ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
  7. ^ Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 49.
  8. ^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 386. ISBN 978-0-521-55432-9.
  9. ^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 85. ISBN 978-0-486-23729-9.
  10. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269