4/12/2023 0 Comments Square octahedron templateThe uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. The regular octahedron has eleven arrangements of nets. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. Consequently, it is the only member of that group to possess, among its mirror planes, some that do not pass through any of its faces. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. The three faces interior to the octahedron are: a 45-45-90 triangle with edges √ 1, √ 1, √ 2, a 45-45-90 triangle with edges √ 0.5, √ 0.5, √ 1, and a right triangle with edges √ 0.5, √ 1, √ 1.5. The exterior face (the right triangle which is half an octahedron face) is a 30-60-90 triangle with edges √ 0.5, √ 1.5, √ 2. The orthoscheme has four dissimilar right triangle faces. The 3-edge path along orthogonal edges of the orthoscheme is √ 0.5, √ 0.5, 1, first along an octahedron edge to its midpoint, then turning 90° to the octahedron center, and finally turning 90° again to the fourth orthoscheme vertex. If the octahedron has edge length √ 2, its orthoscheme's six edges have lengths √ 2, √ 0.5, √ 1.5 (the exterior right triangle face), plus 1, 1, √ 0.5 (edges that are radii of the octahedron). The regular octahedron can be dissected (3 different ways) into 16 such characteristic tetrahedra, which all surround the same axis of the octahedron and meet at the octahedron's center. Each equilateral triangle face of the octahedron is divided into two 30°-60°-90° right triangles that are faces of adjacent mirror-image orthoschemes. One edge of the characteristic tetrahedron is also an edge of the octahedron. A left-handed orthoscheme and a right-handed orthoscheme meet at each of the octahedron's eight faces, forming a trirectangular tetrahedron: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron. It occurs in two chiral forms which are mirror images of each other. The octahedron's characteristic orthoscheme, the characteristic tetrahedron of the regular octahedron, is a quadrirectangular irregular tetrahedron. ![]() The octahedron and its dual polytope, the cube, have the same symmetry group, but different characteristic orthoschemes. The octahedron's symmetry group is denoted B 3. ![]() The faces of the octahedron's characteristic orthoscheme lie in the octahedron's mirror planes of symmetry. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. Like all regular convex polytopes, the octahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. This and the regular tessellation of cubes are the only such uniform honeycombs in 3-dimensional space. Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space. ![]() An icosahedron produced this way is called a snub octahedron. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. ![]() The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. The octahedron represents the central intersection of two tetrahedra If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is
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