A tetrahedron having two right angles at each of two vertices was investigated by Lobachevsky (who called it a "pyramid"), Schl~ifli (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 orthosehemes. 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 orthosehemes 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.
I. FRIEZE PATTERNS
Let us define a frieze pattern to be an arrangement of real numbers in staggered rows so that, if a and d are adjacent in one row, with b between them in the preceding row and c in the following row, then ad -bc = 1, as in a unimodular matrix rotated by 45 ยฐ. For instance, we might have an endless row of zeros followed by an endless row of ones and then a periodic row associated with a convex n-gon as follows. Let the n-gon be dissected, by means of n -3 diagonals, into n -2 triangles. Associate the successive vertices with positive integers which count the number of triangles occurring at the vertex. When such a cycle of integers is used for the third row of the frieze pattern (with zeros and ones in the preceding rows), the nth row will be found to consist (like the second) entirely of ones. No other choice of positive integers in the third row will have this remarkable effect [1, p. 181]. For instance, although there are several such frieze patterns when n > 5, there is essentially only one when n = 5, since the only way to triangulate a pentagon by means of two diagonals is when these diagonals share one vertex. This vertex belongs to all the 3 tri/~ngles; thus the cycle of numbers