Trisecting an orthoscheme

AbstractA tetrahedron having two right angles at each of two vertices was investigated by Lobachevsky (who called it a “pyramid”), Schläfli (who called it an “orthoscheme”), Wythoff (who called it “double-rectangular”), and Schoute (who called its theory “polygonometry”). There is a simple procedure for dissecting such a tetrahedron into three smaller orthoschemes. The two cutting planes meet three of the four faces (which are right-angled triangles) along lines which can easily be described. When the tetrahedron is unfolded so as to put all the faces in one plane, the arrangement of lines suggests an interesting theorem of absolute geometry. When a particular spherical orthoscheme of known volume is dissected into three pieces, and the volumes of these smaller orthoschemes are expressed as definite integrals, the result is a peculiar identity which has not been verified directly. There is a one-parameter family of orthoschemes for which the three smaller orthoschemes are all congruent; the Euclidean member of this family turns out to be related to a very simply frieze pattern of integers