Digonal antiprism

In geometry, a disphenoid (from Greek sphenoeides, "wedgelike") is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are sphenoid,bisphenoid,isosceles tetrahedron,equifacial tetrahedron,almost regular tetrahedron, and tetramonohedron.

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. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths.

If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with T_dtetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. In this case it has D_2ddihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D₂ dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra: as well as being congruent to each other, all of their faces are symmetric to each other.

It is not possible to construct a disphenoid with right triangle or obtuse triangle faces. When right triangles are glued together in the pattern of a disphenoid, they form a flat figure (a doubly-covered rectangle) that does not enclose any volume. When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by Alexandrov's uniqueness theorem) but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles.

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