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Other names for the same shape are isotetrahedron, [2] sphenoid, [3] bisphenoid, [3] isosceles tetrahedron, [4] equifacial tetrahedron, [5] almost regular tetrahedron, [6] and tetramonohedron. [ 7 ] All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles .
A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.
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This definition is used in the naming of two Johnson solids: the snub disphenoid and the snub square antiprism, and of higher dimensional polytopes, such as the 4-dimensional snub 24-cell, with extended Schläfli symbol s{3,4,3}, and Coxeter diagram .
An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with a cubic honeycomb, subdividing it at the planes =, =, and = (i.e. subdividing each cube into path-tetrahedra), then squashing it along the main diagonal until the distance between the points (0, 0, 0) and (1, 1, 1) becomes the same as the distance between the ...
The snub disphenoid can be visualized as an atom cluster surrounding a central atom, that is the dodecahedral molecular geometry. Its vertices may be placed in a sphere and can also be used as a minimum possible Lennard-Jones potential among all eight-sphere clusters. The dual polyhedron of the snub disphenoid is the elongated gyrobifastigium.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space.It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the ~ Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by ...