Schläfli orthoscheme

In geometry, Schläfli orthoscheme is a type of simplex. They are defined by a sequence of edges ( v 0 v 1 ) , ( v 1 v 2 ) , … , ( v d − 1 v d ) 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 Fiedler (1957), and later rediscovered by Coxeter (1991). 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.