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An octahedron can be any polyhedron with eight faces. In a previous example, the regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. [26] There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11 ...
Vertices Dual name Picture Platonic solid: Cube: 6: 12: 8 Octahedron: Archimedean solid (dual Catalan solid) Cuboctahedron: 14: 24: 12 Rhombic dodecahedron: Truncated cube: 14: 36: 24 Triakis octahedron: Truncated octahedron: 14: 36: 24 Tetrakis hexahedron: Rhombicuboctahedron: 26: 48: 24 Deltoidal icositetrahedron: Truncated cuboctahedron: 26: ...
In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other.
Vertices [5] Point group [6] Truncated tetrahedron: 3.6.6: 4 triangles 4 hexagons: 18 12 T d: Cuboctahedron: 3.4.3.4: 8 triangles 6 squares: 24 12 O h: Truncated cube: 3.8.8: 8 triangles 6 octagons: 36 24 O h: Truncated octahedron: 4.6.6: 6 squares 8 hexagons 36 24 O h: Rhombicuboctahedron: 3.4.4.4: 8 triangles 18 squares 48 24 O h: Truncated ...
There are many relations among the uniform polyhedra. [1] [2] [3] Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The grouping below exhibit some of these relations.
The vertices can be seen in 3 hyperplanes, [at] with the 6 vertices of an octahedron cell on each of the outer hyperplanes and 12 vertices of a cuboctahedron on a central hyperplane. These vertices, combined with the 8 vertices of the 16-cell , represent the 32 root vectors of the B 4 and C 4 simple Lie groups.
The number of cubes in an octahedron formed by stacking centered squares is a centered octahedral number, the sum of two consecutive octahedral numbers. These numbers are These numbers are 1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, ...
The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron G IV (1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a ...