Hyperbolic 5-orthoschemes and the Tnloganthm

661

Here , с denotes the constant of integration which can be computed by evaluating (36) in the degenerate case of an orthoscheme Ä^eg ^^ H^ satisfying (35) such that V0I5 (i^deg) = 0 For this, we consider the following class of orthoschemes R^^ a H^ given by

e я/2 — e 7i/2 — e e я/2 — e

Y о----------о--------------- о --------------- о ---------- о------------о

^е е

With О < е < 7г/6, в < е' < к/2 and sm^ e' = 2 sm^ e Then, property (35) is satisfied, and i^eg IS 2-asymptotic Since, for e->0, eXe)-^0 and 0^1(1^^) = — sin^8-»0, R^^ con\erges to an orthoscheme Ä^eg with vol5(JRdeg)=0 This implies that с = (3/16)C(3) which finishes the proof Q E D

3 2 The above Theorem combined with certain dissection properties of ortho- schemes (cf 1 4) enables us to compute explicitly the volumes of the three Coxeter orthoschemes (7) as well as the volumes of the characteristic simplices (8) associ- ated to certain regular star-honeycombs (being necessarily of infinite density) in H^ (cf 1 2)

The two Coxeter orthoschemes (Т2,(Тз (see (7)) satisfy the conditions of the Theorem Using 2 3, we get for their volumes V0I5 (crj, г = 2, 3

vols ((72) = 9^ C(3) ^ 0 000913, V0I5 ((Тз) = ^ C(3) c. 0 001826 (37)

Before we compute the volume V0I5 (gi) of the remaining Coxeter orthoscheme (т, (see (7)), which is 1-asymptotic, we make the following remark

REMARK Let a„ =arccos l/>/« e (0, я/2), « ^ 3, and consider the schemes

a a

p " (OL) о----------о----- -----о----------о о----------о----------о о----------о----- -----о----------о J

of order п + 1,1 е[0,п], which describe either sphencal, euchdean or compact hyperbolic «-orthoschemes if either a„ < a < 71 — a„, a = a„, or a„_, < a < a„ (see [D, (7 9)]) In the sphencal case, Schlafli (cf [S, p 270]) derived the following volume relations

vol „ (рГ(а)) = Г ) vol« (pS(a)), i e [0, n], (38)