Snub cube

Snub cube
Two different forms of a snub cube
Type	Archimedean solid
Faces	38
Edges	60
Vertices	24
Symmetry group	Rotational octahedral symmetry ${\displaystyle \mathrm {O} }$
Dihedral angle (degrees)	triangle-to-triangle: 153.23° triangle-to-square: 142.98°
Dual polyhedron	Pentagonal icositetrahedron
Properties	convex, chiral
Vertex figure
Net

3D model of a snub cube

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. 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 ${\displaystyle s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}$, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol ${\displaystyle t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}$.

Construction

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. ^[1]

Uniform alternation of a truncated cuboctahedron

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.

Cartesian coordinates

Cartesian coordinates for the vertices of a snub cube are all the even permutations of

$${\displaystyle \left(\pm 1,\pm {\frac {1}{t}},\pm t\right),}$$

with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where ${\displaystyle t\approx 1.83929}$ is the tribonacci constant. ^[2] 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 ${\displaystyle \alpha ={\sqrt {2+4t-2t^{2}}}}$, a number which satisfies the equation

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

and can be written as

$${\displaystyle {\begin{aligned}\alpha &={\sqrt {{\frac {4}{3}}-{\frac {16}{3\beta }}+{\frac {2\beta }{3}}}}\approx 1.609\,72\\\beta &={\sqrt[{3}]{26+6{\sqrt {33}}}}.\end{aligned}}}$$

To get a snub cube with unit edge length, divide all the coordinates above by the value α given above.

Properties

For a snub cube with edge length ${\displaystyle a}$, its surface area and volume are: ^[3]

$${\displaystyle {\begin{aligned}A&=\left(6+8{\sqrt {3}}\right)a^{2}&\approx 19.856a^{2}\\V&={\frac {8t+6}{3{\sqrt {2(t^{2}-3)}}}}a^{3}&\approx 7.889a^{3}.\end{aligned}}}$$

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. ^[4] It is chiral, meaning there are two distinct forms whenever being mirrored. Therefore, the snub cube has the rotational octahedral symmetry ${\displaystyle \mathrm {O} }$. ^{[5] [6]} The polygonal faces that meet for every vertex are four equilateral triangles and one square, and the vertex figure of a snub cube is ${\displaystyle 3^{4}\cdot 4}$. The dual polyhedron of a snub cube is pentagonal icositetrahedron, a Catalan solid. ^{[7] [ page needed ]}

Related polyhedra and tilings

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{3,4} s{31,1}
		=	=	=				= or	= or	=
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

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 . 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.

n32 symmetry mutations of snub tilings: 3.3.3.3.n v t e
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
	232	332	432	532	632	732	832	∞32
Snub figures
Config.	3.3.3.3.2	3.3.3.3.3	3.3.3.3.4	3.3.3.3.5	3.3.3.3.6	3.3.3.3.7	3.3.3.3.8	3.3.3.3.∞
Gyro figures
Config.	V3.3.3.3.2	V3.3.3.3.3	V3.3.3.3.4	V3.3.3.3.5	V3.3.3.3.6	V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.∞

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

4n2 symmetry mutations of snub tilings: 3.3.4.3.n v t e
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	242	342	442	542	642	742	842	∞42
Snub figures
Config.	3.3.4.3.2	3.3.4.3.3	3.3.4.3.4	3.3.4.3.5	3.3.4.3.6	3.3.4.3.7	3.3.4.3.8	3.3.4.3.∞
Gyro figures
Config.	V3.3.4.3.2	V3.3.4.3.3	V3.3.4.3.4	V3.3.4.3.5	V3.3.4.3.6	V3.3.4.3.7	V3.3.4.3.8	V3.3.4.3.∞

Snub cubical graph

Snub cubical graph
4-fold symmetry
Vertices	24
Edges	60
Automorphisms	24
Properties	Hamiltonian, regular
Table of graphs and parameters

In graph theory, a snub cubical graph is the graph of vertices and edges of the snub cube, one of the Archimedean solids. It has 24 vertices and 60 edges, and is an Archimedean graph. ^[8]

Orthogonal projection

See also

• Compound of two snub cubes
• Snub (geometry)
• Snub dodecahedron
• Snub square tiling
• Truncated cube

References

1. Holme, A. (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN   978-3-642-14441-7.
2. Collins, Julian (2019). Numbers in Minutes. Hachette. p. 36–37.
3. 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.
4. 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.
5. 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.
6. Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 386. ISBN   978-0-521-55432-9.
7. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc.
8. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

• Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81. doi:10.1017/S0025557200176818. S2CID   125675814.
• 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

• Weisstein, Eric W., "Snub cube" ("Archimedean solid") at MathWorld .
  • Weisstein, Eric W. "Snub cubical graph". MathWorld .
• Klitzing, Richard. "3D convex uniform polyhedra s3s4s - snic".
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Editable printable net of a Snub Cube with interactive 3D view