Disphenoid

In geometry, a disphenoid (from Greek sphenoeides 'wedgelike') is a tetrahedron whose four faces are congruent acute-angled triangles.

A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D2 dihedral symmetry.

[4] 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.

The phyllic disphenoid similarly has faces with two shapes of scalene triangles.

[6] The volume of a disphenoid with opposite edges of length l, m and n is given by:[12] The circumscribed sphere has radius[12] (the circumradius): and the inscribed sphere has radius:[12] where V is the volume of the disphenoid and T is the area of any face, which is given by Heron's formula.

As Gibb (1990) describes, it can be folded without cutting or overlaps from a single sheet of a4 paper.

The rotation of the six disphenoids with opposite edges of length l, m and n (without loss of generality n≤l, n≤m) is physically realizable if and only if[16]