Schläfli orthoscheme
In geometry, a Schläfli orthoscheme is a type of simplex. They are defined by a sequence of edges formula_1 that are mutually orthogonal. These were introduced by Ludwig Schläfli, who called them "orthoschemes" and studied their volume in the Euclidean, Lobachevsky and the spherical geometry. H. S. M. Coxeter later named them after Schläfli. Schoute studied the theory of orthoschemes which he called polygonometry. J.-P. Sydler and Børge Jessen studied them 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). Dissection into orthoschemes. Hugo Hadwiger conjectured in 1956 that every simplex can be dissected into finitely many orthoschemes. The conjecture has been proven in spaces of five or fewer dimensions, but  remains unsolved in higher dimensions. Hadwiger's conjecture implies that every convex polytope can be dissected into orthoschemes. Coxeter identifies various orthoschemes as the characteristic simplexes of the polytopes they generate by reflections.