652

RUTH KELLERHALS

That these schemes are the only 2-asymptotic ones, is easily seen using list 14.14 in Coxeter's classification of regular star-honeycombs of finite densities (see [C2, §14]).

1 . 3 . Let ^(X) denote the и-th scissors congruence group of polytopes in X (see [Sa, §1]). Then, for « ^ 2, ^(Я")^8_isomorphic to ^(Я") (see [DS, Theorem 2.1, p. 162]), and, for d >3 odd, ^{H"^) is generated by the classes of 2-asymptotic orthoschemes (see [Sa, Remark 3.10 and p. 199]). This latter property was proved by Debrunner [D, p. 125] using a certain dissection of a öf-orthoscheme into d -h 1 orthoschemes {d>2 arbitrary). This dissection process will be helpful later (cf. 1.4, 3.2).

1 . 4 . Consider a five-dimensional 2-asymptotic orthoscheme R = Pq- • P^ with vertices Pq, .. , ,Ps and with graph

ai «2 »3 a4 «5

Zt ( I\ ) '. о--------- о -------- о--------- о-------- о -------- о ,

It is characterized by three independent dihedral angles a2, аз, a4, say, while aj, aj are given by the relations (cf. 1.1)

ai X2 «3 «4 (X2 аз a4 oc^ ^Ql (o -------- О -------- О -------- О -------- о) = det (о -------- о -------- О -------- о -------- о) = 0. (9)

An angle а, ( 1 :^ / < 5) is formed by the facet orthoschemes Д _ , = Я, _ i n i^ = Po"Pi~\ " ' Ps and R, = H,nR=Po- • • P,- • Psl it is attached to the apex orthoscheme F, = Д _ , n Д = Pq ' * ^»- 1Л ' * ' ^s» and, by the orthogonality ditions (5), can be seen as planar or spatial angle (cf. Figure 1). Moreover, the following angular relation will be of use later.

LEMMA . Let

I ( R ) : o - ^

denote the graph of a 1-asymptotic hyperbolic S-orthoscheme R. Then,

tan a, tan «2 = tan «4 tan aj. (10)

Proof . Denote by Pq, ... ,Ps the vertices of R satisfying (5). Consider the 1-asymptotic face orthoscheme Po^i^2^3 of dimension three and its spherical