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Scalene isosceles triangle3/28/2023 ![]() ![]() ![]() Some tetragonal disphenoids will form honeycombs. Two edges have dihedral angles of 90°, and four edges have dihedral angles of 60°. Honeycombs and crystals Ī space-filling tetrahedral disphenoid inside a cube. The bimedians are perpendicular to the edges they connect and to each other. The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid. If the four faces of a tetrahedron have the same area, then it is a disphenoid. If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid. The volume of a disphenoid with opposite edges of length l, m and n is given by V = ( l 2 + m 2 − n 2 ) ( l 2 − m 2 + n 2 ) ( − l 2 + m 2 + n 2 ) 72. ![]() They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints. The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, and the tetrahedra in which the angular defects of all four vertices equal π. On a disphenoid, all closed geodesics are non-self-intersecting. The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. 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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