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where the vectors Po ..... pn "generate" the simplex S. In the non-euclidean case we do not have such an elementary formula for the volume of S. Nevertheless, ...
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Math. Ann. 285, 541-569 (1989)

9 Springer-Verlag1989

On the volume of hyperbolic polyhedra Ruth Kellerhals Max-Plaack-Institut f'tir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany

Introduction

The problem of calculating volumes of convex polytopes in euclidean, spherical or hyperbolic space is a very difficult one. However, every convex polytope admits a simplical subdivision, and for n-simplices S (n > 3) the above problem is considerably simpler. In the euclidean case the volume is explicitly given by Voln (S) = l ldet (Po ..... Pn) l, where the vectors Po..... pn "generate" the simplex S. In the non-euclidean case we do not have such an elementary formula for the volume of S. Nevertheless, in 1852, Schl/ifli gave a simple description for the volume differential d Voln (S) as a function of the dihedral angles Wjk (0 < j < k < n) formed by the faces Sj, S~, o r s (see 1-16, p. 227ff]): dVoln(S)=--I n

~

1 j.k=o

VjkdWjk,

(1)

j0} =

{x~R~+l t -

2

X o + X

21 +

...

+ x

2_

n --

_

1,Xo>0}.

If we interpret this in real projective n-space p n we see that H ~ is the interior of P~ with respect to the quadric QI.,:= { [ x ] ~ P ~ I ( x , x ) = 0}: n ~ =:IQ1,, = { [ x ] ~ P * l ( x , x ) < 0}.

The closure H n of H n in P~ represents the natural compactification of H ~. Points of the boundary gH ~ = H n - H" are called points at infinity of H ~. We shall also consider points [ x ] e P ~ with ( x , x ) > 0 which lie outside the absolute quadric. These points are called ideal points of H n relative to QI.~, and the set of all such points is denoted by AQI,~. A projective k-plane is a k-dimensional projective subspace of Pn. For k = n - 2 or n - 1 we use the terms hyperlines or hyperplanes. A hyperbolic k-plane E k is the intersection of a projective k-plane F k with H ~. We call F k the projectively closed or hyperbolic k-plane in P~ corresponding to E ~ and use the notation Fk=/~k.

544

R. Kellerhals

To every point in P" corresponds a hyperplane in P" and vice versa: Let

P=[x]~P", A point [y]~P~ is said to be con]u~Tate to [x] relative to QI,. iff ( x , y ) = O holds. The set of all points which are conjugate to P = Ix] form a projective hyperplane Fj.:= { [y]eP"l ( x , y ) = 0}, the polar hyperplane to P. P is called the pole to Fe.

Notations pol (P):= polar hyperplane Fe to P, pol(Fp):= pole P to Fe. The quadric QI,, induces a bijection from the points of P" onto its hyperplanes, the polar hyperplanes to the points of P" relative t o Ql,w This map pole~-~polar hyperplane realizes the duality principle of the projective space P" (see [8, Sect. 4E] ).

Properties (see [8, Sect. 4]) (a) The polar hyperplane/~e to PeP" respectively intersects, touches or avoids the quadric QI,., iff PeAQI.., PeQt,. or P~IQI... (b) If two lines g, h in p2 intersect at S:= g c~h, then pol (S) is the line determined by pol (g), pol (h). (c) If a line g in P: contains the point pol (h) of the line h, then g Z h holds. The hyperbolic distance d(P, Q) between two points P = [x] and Q = [ y ] ~ H ~ with ( x , x ) = (y, y ) = - 1 is given by

coshd(P,Q)=i(x,y)[,

O >O,O< i < n + 2} contains a non-empty open subset of H A. For the projectively dosed hyperplanes Hi:=He,={[x]ePnl(x, el)=O} belonging to e~, we find that

HiZHj

for

h~:i-l,i,i+l

(Indicesmod.n+3).

Relative to the quadric Ql,n there are the following possibilities [9]: 1. One hyperplane, say Hn+ 2, lies outside the quadric QLn. 2. Two successive hyperplanes, say H~+ 1, Hn+2, lie outside the quadric QI.,. 3. All hyperplanes intersect Q1,, and are hyperbolic. Other cases do not occur, since out of two perpendicular hyperplanes at least one must intersect the quadric QI.,. Now, the hyperbolic hyperplanes H o. . . . . Hn+d, 0 < d < 2, in the cases 1.-3. bound a convex polytope in H ~ (see [17, Sect. 2, Proposition 2.2]). Choosing a suitable orientation of their normal vectors e~, (e~, e~> > 0, we define: Definition. The hyperbolic potytope n+d

(3 n?, O_ 0, i.e., for truncated orthoschemes, we define (also [17, Sect. 4]): Definition. The convex polytope in P"

R:= ~ H; ~= N {[x]~e*l~O} i=0

t=0

is called the ideal orthoscheme to/~.

Remarks (a) A complete orthoscheme /~ of degree 2 can be interpreted as an ideal orthoscheme/~ with two ideal principal verticesPo, Pn which is truncated by the polar hyperplanes pol (Po), pol (P,). Therefore, R is also called doubly truncated

with continuation R. (b) A complete orthoscheme /~ of degree 1 can be interpreted as an ideal orthoscheme R with one ideal principal vertex, say P0, which is truncated by pol (P0)./~ is called simply truncated with ideal vertex Po and continuation R. (c) The notions "dihedral angle of order k", "apex", etc. translate to complete orthoschemes according to 1.2. (d) Every l-face (2 < l ___n - 1) of a complete orthoscherne is itself a complete

548

R. Kellerhals

orthoscheme. This follows easily from the above definition using the corresponding properties of the associated ideal orthoscheme according to 1.2. (e) Let ff ~ . ~- ( ~ ~ i =+a Oo ~ i + be a complete orthoscheme of degree d, 0 < d < 2. Then, for two non-orthogonal bounding hyperplanes Hi, Hi+ 1 one has one of the following cases (see (3) and [17, Sect. 1]): (intersect on/~ at an angle < re/2. In H" , H~ and Hi+ 1~are parallel. admit a common perpendicular. Since the dihedral angles of order n - 1 of an orthoscheme are reducable to angles of second order at edges (see 1.2), for the ideal orthoscheme/~ corresponding to /~ we make the following Definition. /~ has finite edges (and/~ is of type A) iff every edge emanating from the principal vertices Po, P, intersects the absolute quadric QI.,, hence contains a hyperbolic segment. In the other case,/~ is said to be of type B.

Remark. If /~ is a complete orthoscheme of dimension n and of type B (no continuation with finite edges), then, for n ~ 4, /~ has m essential angles with n < m < n + 3. For n = 3, however, we have m = n = 3, since the Euler equation for compact polyhedrons implies that the number of degrees of freedom equals the number of dihedral angles. Furthermore, it is easy to show that for three-dimensional complete orthoschemes/~ only three configurations of type A and one configuration of type B can occur (see also [1]): A1. /~ = R is an ordinary orthoscheme. A2. /~ is a simplefrustum with ideal vertex Po, i.e.,/~ is simply truncated with ideal vertex Po and with continuation/~ that has finite edges. A3. /~ is a double frustum, i.e., /~ is doubly truncated with continuation/~ that has finite edges. B. /~ is a Lambert cube, i.e.,/~ is doubly truncated with continuation/~ whose hypotenuse (edge connecting the two principal vertices) lies outside the quadric Q~,a. Hence, R is a polyhedron with six Lambert quadrilaterals (i.e. quadrilaterals with one acute and three right angles) as bounding faces and with three essential angles at prescribed edges (opposite faces are congruent). Hence, to calculate volumes of three-dimensional complete orthoschemes, it suffices to consider the above four types A1-A3 and B. 1.4 Complete Coxeter orthoschemes Let X" be E ", S~ or H ~. A polytope in X ~ is called a Coxeter polytope iff it has /z

natural dihedral angles, i.e., angles of the form -, pEN, p ~ 2. P To every Coxeter polytope Pc corresponds a Coxeter graph ,F,(Pc) ([17, Sect. 5]). In particular, a complete Coxeter orthoscheme/~c of dimension n and degree d can be characterized as follows [9]:

549

On the volume of hyperbolic polyhedra

1. If/~c is an ordinary orthoscheme, the graph -~(Rc) consists of a linear chain of length n + 1. 2. If/~c is of degree 1, ~(/~c) consists of a linear chain of length n + 2. 3. If/~c is of degree 2, 2~(/~c) consists of a cycle of length n + 3. 2. The Scldiifli differential formula

Let ~ (n => 3) denote the set of all convex, compact n-polytopes in H" of 7~ combinatorial type x (see I-l,l) and dihedral angles not exceeding ~. In particular, let ~n denote the set of all compact n-simplices in H n. It is known that the congruence class of an element of ~ is uniquely determined by its dihedral angles ([1,l, Sect. 3, Uniqueness Theorem). Therefore the volume function Voln = V o l n l ~ on ~ is a function of the dihedral angles. On the set of spherical n-simplices, Schl/ifli established a formula for the differential of the volume depending on the dihedral angles 116]. Kneser proved the validity of this formula also in the case of hyperbolic n-simplices [1 l-l: Theorem (Schl~ifli's differential formula). Let S e ~ ~ (n >=2) have vertices Po..... P~ and dihedral angles Wjk= /_(Sj, Sk), 0 < j < k < n, of order n - 1 with apex Sjk:=SjC~Sk. Then the differential of the volume function Vol n on cfn can be represented by

dVol~(S)=

1 ~ Voln_2(Sjk)dWjk (Volo(Sjk):= 1). 1 - nj, k=o jwk,

1.0 0, k = 0, 1,2. Then we integrate 12 from Q:= (wo, wl, w2, ~)e0G to P. Since

Wh(Wo,Wt, W2,2)=O,

k = 0 , 1,2,

On the volume of hyperbolic polyhedra

565

we obtain the following antiderivative of O in G (see (37) and 3.5): W3

~:= ~ w3(wo,wl, w2,w3)dw3 x/2

] w3

s i n 2 w3 c o s 4 w3

= 4 ~!2 log (c~ 2 Wo _ cos2 w3)(cosZ wl -- cos 2 wa)(cos 2 w2 -- COS2 w3) dwa

= ~ k~=o{L(wk + w3) -- L(wk-- w3)} + L( 2-- w3) --89L(w3).

(38)

Restricting to the hypersurface w3 = O(Wo,Wl, w2) in R 4, we can identify f~ with the volume Vol a (/~) of the Lambert cube R, since (see 3.6): (i) F o r w3 = 0:

f'

-- Wk(Wo, Wl, W2,0) = _ ~ V k ( w o ,

dWk

w2)

OVo13(~) - - ,

OWk

k--0,1,2.

n

(ii) F o r w 3 - - - we have f ' = 0. O n the other hand, L e m m a 2 in 3.4 shows that 2 0 = _n implies Vol3(/~) = 0. 2 Using 3.5, (b), we derive the following

Let R be a Lambert cube with essential angles Wk,0 ~-~Wk ~ ~, k = Then the volume Vol 3 (/~) of R is given by

Theorem III.

V o l 3 (R) =

O, 1, 2.

~ f L(w~ + O)- L(w o - O) + L(wx + O)- L(wl - O)

with 0 < 0 = arctan x/c~

Vi -- sin 2 Wo sin 2 w2 < _~ cos Wo cos w2 = 2"

Remarks.

(a) By means of hyperbolic trigonometry, the quantity cosh 2 Vx in the definition of 0 can be expressed as a function of the essential angles w o, wt, w2 as follows: cosh 2 V1 = 1 + 89

2 + (2B sin wl) 2 - A)

with A = cos 2 Wo + cos 2 w2 - B 2,

B = cos Wocos w 2

(40)

COS w 1

(b) In the limiting case w 1 = V1 = 0 (see 3.2, B), the formulas (33) and (39) for Vo13(/~ ) of Theorems II and III coincide. A p a r t from this special case, these two

566

R. K e l l e r h a l s

formulas are conceptually different, i.e., they cannot be related to each other by means of suitable functional equations for L(co). This can be proved by evaluating ~t both abstract formulas for the values Wo = wl = w2 = ~ using (40). By means of Theorem III we can explicitly calculate volumes of complete Coxeter orthoschemes of type B (see 1.4). Results are listed in the Appendix. Appendix

A. Let/~c be a three-dimensional complete Coxeter orthoscheme of type A with essential angles --, Pk > 2, k = 0, 1, 2. P~ A1. There are exactly 10 realizations of ordinary Coxeter orthoschemes/~c (see [7]) with graphs X(/~c) and volumes V(po,p,,p2): N

I~(Rc)

V(po,p,, ~) 8

v(3,3,6) = l.n(~) _.=0.0423

0 - - 0 - - 0 - - 0

o

o

o

5

o--o

o

o--o

V(3,5,3)

6 O - - O

V(3,6,3)

0 - - 0

5

0 ~ 0 - - 0 '

~

0

-

-

0

= 89

"~ 0.1692

V(4,3,5) ~" 0.0359

0

V(4,3,6) = ~sJ l ( ~,- ) _ 0.1057

5

, o - - o - - o - - o

o

0

5

6

. o--o--o

V(5,3,5)

0

6

0 ~ 0 - - 0

0 0763

--~ 0.0391

V(4,4,4) = ~ 3 I ( ~ )

5

_~

0

6 0

v ( 3 , 4 , 4 ) = ~.,'z(~)

V(5,3,6)-~

v ( 6 , 3, 6) = 89

~_ 0.2290

~ 0.0933

0.1715

_~ 0.2s37

A2. The simple Coxeter frustums/~c form an infinite class ofpolyhedra (see [9]): Pe

p~

p2

o ~coo - -P0 o ~ oI~- - o oo

1

1 >I

~I + ~I> ~ 1

I

1

1

O n the v o l u m e o f hyperbolic polyhedra

567

The v o l u m e is maximal for o

o

o

o

o

V(4, 4, 0) = dt

-~ 0.4560

The volume is minimal for the asymptotic Coxeter frustum 6

.......

V(,3,6) 3

1 = ~JI(~) _~0.~Z3 .

A3. The double Coxeter frustums/~c form an infinite class of polyhedra I-9]: iO . . . . . 9

O %

r4 or

~9

~

1

l

I

I

L § pa

p~

~,

I

I

p-~+ ~-~< ~. g + ~ , < ~-

'o

Pl

.-"~

7

O"

"O

o

r

*,9

9

9

,0\

o

1

k=l

or

2.

OO

/\:/\O .....

e

o

O

0

O\o /. .\ /Pl Fig. 4

The maximal v o l u m e is attained in the asymptotic limit case co O'

,.O

O

'O

The minimal v o l u m e is attained in the asymptotic limit case O - - 0 - - 0 - - 0

, ,

(j/

B. L e t / ~ c be a three-dimensional complete Coxeter orthoscheme of type B with essential angles --, Pk > 2, k = 0, 1, 2. These Coxeter polyhedra form an infinite Pk

568

R. Kellerhals

class (see [9]): Pl

i?~~ /

9

o

2