Schläfli orthoscheme


In geometry, Schläfli orthoscheme is a type of simplex. They are defined by a sequence of edges 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. 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, and later rediscovered by. These simplices are the convex hulls of trees in which all edges are mutually perpendicular. In the orthoscheme, the underlying tree is a path. In three dimensions, an orthoscheme is also called a birectangular tetrahedron.

Properties

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.