Disphenoid
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In geometry, a disphenoid ( ) 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 isotetrahedron, 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.
In geometry, a disphenoid ( ) 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 isotetrahedron, 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. V = \sqrt{\frac{(l^2 + m^2 - n^2)(l^2 - m^2 + n^2)(-l^2 + m^2 + n^2)}{72}}. The circumscribed sphere has radius (the circumradius) R = \sqrt{\frac{l^2 + m^2 + n^2}{8}}, and the inscribed sphere has radius r = \frac{3V}{4T}, where V is the volume of the disphenoid and T is the area of any face, which is given by Heron's formula. There is also the following interesting relation connecting the volume and the circumradius: 16T^2 R^2 = l^2 m^2 n^2 + 9V^2. The squares of the lengths of the bimedians are \tfrac{1}{2}(l^2 + m^2 - n^2),\quad \tfrac{1}{2}(l^2 - m^2 + n^2),\quad \tfrac{1}{2}(-l^2 + m^2 + n^2).
Other uses
Six tetragonal disphenoids attached end-to-end in a ring construct a kaleidocycle, a paper toy that can rotate on 4 sets of faces in a hexagon. The rotation of the six disphenoids with opposite edges of length l , m and n (without loss of generality n\leq l , n\leq m ) is physically realizable if and only if -8(l^2 - m^2)^2 (l^2 + m^2) - 5n^6 + 11(l^2 - m^2)^2 n^2 + 2(l^2 + m^2) n^4 \ge 0.
References
Text derived from https://en.wikipedia.org/w/index.php?oldid=1335710075 under CC BY-SA 4.0; modifications have been made.