Snub cube | |
---|---|
Type | Archimedean solid |
Faces | 38 |
Edges | 60 |
Vertices | 24 |
Symmetry group | Rotational octahedral symmetry |
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. 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 , and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol .
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]}
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 for the vertices of a snub cube are all the even permutations of
This snub cube has edges of length , a number which satisfies the equation
For a snub cube with edge length , its surface area and volume are:^{[3]}
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 .^{[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 . The dual polyhedron of a snub cube is pentagonal icositetrahedron, a Catalan solid.^{[7]}^{[page needed]}
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{3^{1,1}} |
t{3,4} t{3^{1,1}} |
{3,4} {3^{1,1}} |
rr{4,3} s_{2}{3,4} |
tr{4,3} | sr{4,3} | h{4,3} {3,3} |
h_{2}{4,3} t{3,3} |
s{3,4} s{3^{1,1}} |
= |
= |
= |
= or |
= or |
= | |||||
Duals to uniform polyhedra | ||||||||||
V4^{3} | V3.8^{2} | V(3.4)^{2} | V4.6^{2} | V3^{4} | V3.4^{3} | V4.6.8 | V3^{4}.4 | V3^{3} | V3.6^{2} | V3^{5} |
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 | ||||||||
---|---|---|---|---|---|---|---|---|
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 | ||||||||
---|---|---|---|---|---|---|---|---|
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 | |
---|---|
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]}