Schläfli orthoscheme

A cube dissected into six orthoschemes.

• All 2-faces are right triangles.
• All facets of a d-dimensional orthoscheme are (d − 1)-dimensional orthoschemes.
• The midpoint of the longest edge is the center of the circumscribed sphere.
• The case when ${\displaystyle |v_{0}v_{1}|=|v_{1}v_{2}|=\cdots =|v_{d-1}v_{d}|}$ is a generalized Hill tetrahedron.
• In 3- and 4-dimensional Euclidean space, every convex polytope is scissor congruent to an orthoscheme.
• 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 orthoscheme can be trisected into three smaller orthoschemes.[1]
• In 3-dimensional hyperbolic and spherical spaces, the volume of orthoschemes can be expressed in terms of the Lobachevsky function, or in terms of dilogarithms.[5]
• The ${\displaystyle d}$ dihedral angles that are disjoint from edges of the path have acute angles; the remaining ${\displaystyle {\tbinom {d}{2}}}$ dihedral angles are all right angles.[3]

Hugo Hadwiger conjectured in 1956 that every simplex can be dissected into finitely many orthoschemes.[6] The conjecture has been proven in spaces of five or fewer dimensions,[7] but remains unsolved in higher dimensions.[8]

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.[9]

• 3-orthoscheme (tetrahedron with right-triangle faces)
• 4-orthoscheme (5-cell with right-triangle faces)
• Goursat tetrahedron
• Order polytope