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In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890. [1]
The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point.
The polyhedron 𝚷 3 = {p, q} is divided by its planes of symmetry into g quadrirectangular tetrahedra (see 5.43), which join the centre 𝚶 3 to the simplicially subdivided faces. Analogously, the general regular polytope 𝚷 n is divided into a number of congruent simplexes ([orthoschemes]) which join the centre 𝚶 n to the simplicially ...
A subset of edges of this compound polyhedron can generate a compound regular skew polygon, with 3 skew squares. Each tetrahedron contains one skew square. This regular compound polygon containing the same symmetry as the uniform polyhedral compound.
In Conway polyhedron notation this represents an ortho operation to a cube or octahedron. The deltoidal icositetrahedron (dual of the small rhombicuboctahedron) is tightly related to the disdyakis dodecahedron (dual of the great rhombicuboctahedron).
An object of C 3v symmetry under one of the 3-fold axes gives rise under the action of T d to an orbit consisting of four such objects, and T d corresponds to the set of permutations of these four objects. T d is a normal subgroup of O h. See also the isometries of the regular tetrahedron. T h, (3*2) [3 +,4] 2/m 3, m 3 order 24: pyritohedral ...
The 3-fold axes are now S 6 (3) axes, and there is a central inversion symmetry. T h is isomorphic to T × Z 2 : every element of T h is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D 2h (that of a cuboid ), of type Dih 2 × Z 2 = Z 2 × Z 2 × Z 2 .
In this context, a simplex in -dimensional Euclidean space is the convex hull of + points that do not all lie in a common hyperplane.For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space).