Schläfli orthoscheme

They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries.

[1] 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,[2] and later rediscovered by Coxeter.

Coxeter identifies various orthoschemes as the characteristic simplexes of the polytopes they generate by reflections.

The characteristic simplex is chiral (it comes in two mirror-image forms which are different), and the polytope is dissected into an equal number of left- and right-hand instances of it.

We proceed to describe the "simplicial subdivision" of a regular polytope, beginning with the one-dimensional case.

The polygon 𝚷2 = {p} is divided by its lines of symmetry into 2p right-angled triangles, which join the center 𝚶2 to the simplicially subdivided sides.

The polyhedron 𝚷3 = {p, q} is divided by its planes of symmetry into g quadrirectangular tetrahedra (see 5.43), which join the centre 𝚶3 to the simplicially subdivided faces.

Analogously, the general regular polytope 𝚷n is divided into a number of congruent simplexes ([orthoschemes]) which join the centre 𝚶n to the simplicially subdivided cells.