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Concept
Snub cube
Summary
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 s (or "enantiomorphs") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron.
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 .
For a snub cube with edge length 1, its surface area and volume are:
where t is the tribonacci constant
If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths
In general, the volume of a snub cube with side length can be found with this formula, using the t as the tribonacci constant above:
Cartesian coordinates for the vertices of a snub cube are all the even permutations of
(±1, ±1/t, ±t)
with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where t ≈ 1.83929 is 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. Taking all of them together yields the compound of two snub cubes.
This snub cube has edges of length , a number which satisfies the equation
and can be written as
To get a snub cube with unit edge length, divide all the coordinates above by the value α given above.
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.
The snub cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection.
Official source
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