All right triangles are isosceles2/17/2024 Īnother characterization states that if d 1, d 2 and d 3 are the common perpendiculars of AB and CD AC and BD and AD and BC respectively in a tetrahedron ABCD, then the tetrahedron is a disphenoid if and only if d 1, d 2 and d 3 are pairwise perpendicular. We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide. The phyllic disphenoid similarly has faces with two shapes of scalene triangles.ĭisphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms.Ī tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled. The digonal disphenoid has faces with two different shapes, both isosceles triangles, with two faces of each shape. Two more types of tetrahedron generalize the disphenoid and have similar names. 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. 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. It is not possible to construct a disphenoid with right triangle or obtuse triangle faces. īoth tetragonal disphenoids and rhombic disphenoids are isohedra: as well as being congruent to each other, all of their faces are symmetric to each other. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. In this case it has D 2d dihedral symmetry.Ī sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D 2 dihedral symmetry. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with T d tetrahedral symmetry, although this is not normally called a disphenoid. Further information: Tetrahedron § Isometries of irregular tetrahedra
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