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octahedron

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Note the duality between cube and octahedron, and between dodecahedron and icosahedron. en.wikipedia.org

Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. en.wikipedia.org

The tetrahedron, octahedron, icosahedron, and great icosahedron appear commonly in non-prismatic uniform polychora, but not in their snub constructions. en.wikipedia.org

The basic structure forms stacked sheets of linked octahedrons of aluminium hydroxide. en.wikipedia.org

Its shape is derived from a stellated octahedron, but the tips of the vertices are truncated to form small triangles. en.wikipedia.org

As the name suggests, it can be constructed by gyroelongating an octahedron (square bipyramid) by inserting a square antiprism between its congruent halves. en.wikipedia.org

In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors. en.wikipedia.org

The "tetragonal trisoctahedron" is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron. en.wikipedia.org

So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). en.wikipedia.org

The octahedron shares two of its opposite faces with that of neighbouring vanadinite units, forming a continuous chain of octahedrons. en.wikipedia.org

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