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A hexagon bisects the cube into two copies of a simple polyhedron with one hexagonal face, three isosceles right triangle faces, and three irregular pentagonal faces. It is not possible to form a simple polyhedron using only three triangles and three pentagons, without the added hexagon.
Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number ...
In geometry, the Rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has a total of 62 faces: 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, with 60 vertices, and 120 edges.
They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.
This decomposition is based on a Petrie polygon projection of an 8-cube, with 28 of 1792 faces. The list OEIS : A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.
A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct convex hexahedra, [1] one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined.
To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above ...
Because all the faces of the cC have an even number of sides and are centrally symmetric, it is a zonohedron: Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.) The chamfered cube is also the Goldberg polyhedron GP IV (2,0) or {4+,3} 2,0, containing square and hexagonal faces.