If the faces are equilateral triangles, one obtains a regular tetrahedron, which is not normally considered a disphenoid.
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A regular tetrahedron is one in which all four faces are equilateral triangles, and is one of the Platonic solids.
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These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube.
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The icosahedron can be considered a snub tetrahedron, as snubification of a regular tetrahedron gives a regular icosahedron having chiral tetrahedral symmetry.
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A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces.
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In 1874 he accounted for the phenomenon of optical activity by assuming that the chemical bonds between carbonatoms and their neighbors were directed towards the corners of a regular tetrahedron.