Schläfli orthoscheme

In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem. Orthoschemes, also called path-simplices in the applied mathematics literature, are a special case of a more general class of simplices studied by Fiedler, and later rediscovered by Coxeter. These simplices are the convex hulls of trees in which all edges are mutually perpendicular. In an orthoscheme, the underlying tree is a path. In three dimensions, an orthoscheme is also called a birectangular tetrahedron (because its path makes two right angles at vertices each having two right angles) or a quadrirectangular tetrahedron (because it contains four right angles). All 2-faces are right triangles. All facets of a d-dimensional orthoscheme are (d − 1)-dimensional orthoschemes. The dihedral angles that are disjoint from edges of the path have acute angles; the remaining dihedral angles are all right angles. The midpoint of the longest edge is the center of the circumscribed sphere. The case when is a generalized Hill tetrahedron. Every hypercube in d-dimensional space can be dissected into d! congruent orthoschemes. A similar dissection into the same number of orthoschemes applies more generally to every hyperrectangle but in this case the orthoschemes may not be congruent. Every regular polytope can be dissected radially into g congruent orthoschemes that meet at its center, where g is the order of the regular polytope's symmetry group.