Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 57 (2010), 159-169

Andrei VESNIN

Volumes and Normalized Volumesof Right-Angled Hyperbolic Polyhedra

To Massimo Ferri and Carlo Gagliardion the occasion of their 60-anniversary

Abstract. We present some recent results on a structure of theset of volumes of right-angled polyhedra in hyperbolic space, suchas the initial list of smallest volumes. Also, we discuss normalizedvolumes of some classes of hyperbolic polyhedra.

Key Words: Hyperbolic polyhedron, Right-angled polyhedron,Volume.

Mathematics Subject Classification (2000): 51M10, 57M25.

1. Introduction

The class of right-angled polyhedra in a hyperbolic space Hn is themost studied class of Coxeter polyhedra. Basic facts on polyhedra inspaces of constant curvature, their existence and volume calculations canbe found in [1]. In this survey we present some recent results on a structureof the set of volumes of bounded right-angled hyperbolic polyhedra.

These results can be useful not only for studying polyhedra, but alsothe corresponding 3-manifolds. The simplest and smallest bounded poly-hedron in H3 with all dihedral angled π/2 is the dodecahedron. The second

Paper presented at Computational and Geometric Topology – A conference inhonour of Massimo Ferri and Carlo Gagliardi on their 60-th birthday, Bertinoro(Italy), 17–19 June, 2010Work performed under the auspices of the Russian Foundation for Basic Re-search (grants 10-01-00642 and 10-01-91056), and the grant SO RAN – UrORAN.

160 A. VESNIN [2]

smallest is the 14-hedron, eight copies of which were used by Lobell in 1931to construct the first example of a closed orientable hyperbolic 3-manifold[6]. Its generalizations, referred as Lobell manifolds, were introduced in[19]. An explicit formula for volumes of Lobell manifolds was obtained in[20] and was used in [7] to estimate the complexity for these manifolds.As shown in [19], for any bounded right-angled polyhedron in H3 a four-coloring of its faces defines a closed orientable hyperbolic 3-manifolds. Itis descibed in [9] how topological properties of these manifolds depend oncolorings. Thus, results on volumes of right-angled polyhedra can by ap-plied to a wide class of hyperbolic 3-manifolds. We recall that by Mostowrigidity theorem the volume of a closed hyperbolic 3-manifold is its topo-logical invariant.

In Section 2 we recall some results about existence of hyperbolic poly-hedra. In Section 3 we discuss a structure of the set of volumes of poly-hedra. In Theorem 3.2 we give the volume formula for Lobell polyhedrain terms of the Lobachevsky function. Volume formulae for hyperbolicpolyhedra are usually complicated to understand even simple number-theoretical properties.

Problem 1. Does there exists a pair of bounded right-angled hyper-bolic polyhedra such that the ratio of their volumes is irrational?

In Theorem 3.3 we present the initial list of smallest volume boundedright-angled hyperbolic polyhedra. A structure of the set of volumes ofideal polyhedra is not described yet.

Problem 2. Describe the initial list of volumes of ideal finite-volumeright-angled hyperbolic polyhedra.

In Sections 4 and 5 we discuss the normalized volume of a hyperbolicpolyhedron defined as the ratio of its volume to its number of vertices. Wepresent calculations of normalized volumes for some classes of polyhedra.

2. Right-angled polyhedra in Hn

There are strong combinatorial restrictions on existence of right-angled polyhedra in n-dimensional hyperbolic space Hn. For a polyhe-dron P let ak(P ) be the number of its k-dimensional faces and a.

k =1

ak

∑dim F=k a.(F ) be the average number of 4-dimensional faces in a k-

dimensional polyhedron. It was shown by Nikulin [11] that

a.k < Cn−k

n−.

C.[n2 ] + C.

[n+12 ]

Ck[n2 ] + Ck

[n+12 ]

for 4 < k ![

n2

]. From this result we see, in particular, that a1

2, theaverage number of sides in a 2-dimensional face, satisfies to the following

[3] VOLUMES OF RIGHT-ANGLED POLYHEDRA 161

inequality:

a12 <

4(n− 1)n− 2 , if n even,4n

n− 1 , if n odd.

But any hyperbolic right-angled polygon has at least five sides: a12 " 5.

Thus, it follows from the Nikulin inequality that there exist no boundedright-angled polyhedra in Hn for n > 4.

Concerning other classes of polyhedra in n-dimensional hyperbolicspace the following results are known: there exist no bounded Coxeterpolyhedra for n > 29 [21] and examples are know up to n = 8 only; thereexist no finite volume right-angled polyhedra for n > 12 [4] and examplesare know up to n = 8 only; there exist no finite volume Coxeter polyhedrafor n > 995 [13] and examples are known up to n = 21 only.

We are interested in acute-angled and, especially, right-angled poly-hedra in three-dimensional hyperbolic space H3. The following uniquenessresult was obtained by Andreev in [2]: bounded acute-angled polyhedronin H3 is uniquely determined, up to isometry, by its combinatorial typeand dihedral angles.

For the case of right-angled polyhedra necessary and sufficient con-ditions were done by Pogorelov [12]: a polyhedron can be realized in H3

as a bounded right-angled polyhedron if and only if (1) any vertex is in-cident to 3 edges; (2) any face has at least 5 sides; (3) any simple closedcurve on the surface of the polyhedron which separate some two faces ofit (prismatic circuit), intersects at least 5 edges. Figure 1 presents twopolyhedra: the left, known as Greenbergs polyhedron, satisfies the aboveconditions, and the right satisfies conditions (1) and (2), but doesn’t sat-isfy condition (3), since it has a closed circuit which separates two 6-gonalfaces, but intersects 4 edges only.

! "

#

$ %

&

%$Fig. 1. Two polyhedra.

162 A. VESNIN [4]

3. The structure of the set of volumes

Let us denote by R the set of all bounded right-angled polyhedra inH3. Inoue [5] defined two operations on the set R. First is a composition,and its inverse is a decomposition. Let R1, R2 ∈ R; suppose F1 ⊂ R1 andF2 ⊂ R2 be a pair of k-gonal faces. A composition is defined as a union ofR1 and R2 along F1 and F2: R = R1 ∪F1=F2 R2. It is shown in [5] thatcomposition of two polyhedra from R belongs to R.

The second operation is an edge surgery. This is a combinatorial movefrom R to R − e. If R ∈ R and e is such that n1 and n2 are at least 6sides each and e is not a part of prismatic 5-circuit, then (R− e) ∈ R (seeFig. 2). The inverse move, from (R− e) to R, we will call an edge adding.

n 2

n 1

n 3 n 4e

polyhedron R

n 2 1

n 1 1

n 3 + n 4 4

polyhedron R e−

−

−

−

Fig. 2. Edge surgery operation.

The role of defined above operations is clear from the following theorem.

Theorem 3.1. ([5]) For any P0 ∈ R there exists a sequence of unionsof right-angled hyperbolic polyhedra P1, . . . , Pk such each set Pi is obtainedfrom Pi−1 by decomposition or edge surgery, and Pk consists of Lobellpolyhedra. Moreover,

vol(P0) " vol(P1) " vol(P2) " . . . " vol(Pk).

Lobell polyhedra Rn were defined in [19] for any n " 5 as right-angledhyperbolic polyhedra having 2n+2 faces: two n-gonal and 2n pentagonalmanaged similar to the lateral surface of a dodecahedron. The dodeca-hedron R5 and 14-hedron R6, used by Lobell in [6] to construct the firstclosed orientable hyperbolic 3-manifold, are presented in Fig. 3. An ex-plicit formula for volumes of Lobell polyhedra was obtained by the authorin terms of the Lobachevsky function

Λ(x) = −x∫

0

log |2 sin(t)|dt.

[5] VOLUMES OF RIGHT-ANGLED POLYHEDRA 163

Fig. 3. Polyhedra R5 and R6.

Theorem 3.2. ([20]) For any n " 5 the following formula holds forvolumes of Lobell polyhedra

vol(Rn) =n

2

(2Λ(θn) + Λ

(θn +

π

n

)+ Λ

(θn −

π

n

)+ Λ

(π

2− 2θn

)),

where θn = π2 − arccos( 1

2 cos(π/n) ).

Another formula for volumes of Lobell polyhedra can be found in [10].It is easy to see from Theorem 3.2 that the volume function volRn is

a monotonic increasing function of n (see, for example, [7] for the proof).Thus, Theorems 3.1 and 3.2 give a base to describe the ordering of the setof volumes of right-angled hyperbolic polyhedra. It is shown in [5] that R5and R6 are the first and the second smallest volume compact right-angledhyperbolic polyhedra. To describe the initial list of the set of volumes ofpolyhedra from R, we denote by R61

1, R621, R62

2 polyhedra presented inFig. 4, and by R63

1, R632, R63

3 – polyhedra presented in Fig. 5.

6

6 6

6

6

6

6 6

6

6

6

Fig. 4. Polyhedra R611, R62

1 and R622.

6

6

6

6

6 6

66

6

6 7

66

6

Fig. 5. Polyhedra R631, R63

2 and R633 (= R71

1).

164 A. VESNIN [6]

Generally, we use notation Rnkm for a polyhedron which is m-th in the

list of polyhedra obtained by applying k edge adding operations to theLobell polyhedron Rn. Of cause, the same polyhedron can be obtainedfrom different Lobell polyhedra; for example, R63

3 = R711. In Fig.4 and 5

six-gonal faces are marked by 6, seven-gonal faces are marked by 7, andall other faces are pentagonal. By 2R5 we denote a polyhedron obtainedby a composition of two dodecahedra R5.

The initial list of smallest volume bounded right-angled hyperbolicpolyhedra is described in the following theorem. Geometric realizations ofthese polyhedra in H3 can be obtained by the computer program developedby Roeder [16].

Theorem 3.3. ([17]) The first eleven smallest volume bounded right-angled hyperbolic polyhedra and their volumes are as in the following table:

volume notation volume notation

1 4.3062 . . . R5 7 8.6124 . . . 2R52 6.0230 . . . R6 8 8.6765 . . . R63

3

3 6.9670 . . . R611 9 8.8608 . . . R63

1

4 7.5632 . . . R7 10 8.9466 . . . R632

5 7.8699 . . . R621 11 9.0190 . . . R8

6 8.0002 . . . R622

4. Volumes and normalized volumes

In studding volumes of hyperbolic polyhedra, manifolds, and orbifoldsit is very useful to use the Schlafli variation formula which shows how vol-ume changes if we change dihedral angles, but preserve combinatorics ofpolyhedra. Considering the class R of bounded right-angled polyhedra,we have another situation – dihedral angles are fixed, but combinatoricchanges. So, we are interested to see how volume depends of a combina-torial structure, for example of a number of vertices. The following resultwas obtained by Atkinson [3].

Theorem 4.1. ([3]) Let P be a compact right-angled hyperbolic poly-hedron with N vertices. Then

(N − 2) · v8

32! vol(P ) < (N − 10) · 5v3

8,

where v8 is the maximal octahedron volume, and v3 is the maximal tetra-hedron volume. There exists a sequence of compact right-angled polyhedraPi with Ni vertices such that vol(Pi)/Ni tends to 5v3/8 as i →∞.

[7] VOLUMES OF RIGHT-ANGLED POLYHEDRA 165

Recall that constants v3 and v8 in the theorem are

v3 = 3Λ(π/3) = 1.0149416064096535 . . .

andv8 = 8Λ(π/4) = 3.663862376708876 . . . .

The lower bound from Theorem 4.1 can be improved for V ! 54 andF ! 29.

Theorem 4.2. ([14]) Let P be a compact right-angled hyperbolic poly-hedron, with V vertices and F faces. If P is not a dodecahedron, then

vol(P ) " max{(V − 2) · v8

32, 6.023 . . .

}

andvol(P ) " max

{(F − 3) · v8

16, 6.023 . . .

}.

The behavior of a volume as a function of a number of vertices is interestingto study for other classes of polyhedra also.

Let P be a finite volume polyhedron in H3; denote by vol(P ) itsvolume, and by vert(P ) number of its vertices. Let us define a normalizedvolume of P as the following ratio:

ω(P ) =vol(P )vert(P )

.

It was demonstrated in [15] that, in general, the behavior of the normalizedvolume function ω(P ) under a sequence of edge surgeries is not prescribed:it increases or decreases depending on the initial polyhedron.

Now we consider the behavior of normalized volume for some classesof polyhedra.

Let P (α1, . . . ,αn), n ≥ 3, be an ideal pyramid in H3 with dihedralangles α1, . . . ,αn incident to the bottom, see Fig. 6. It is known [18], thatα1 + · · ·+ αn = π, and

vol(P (α1, . . . ,αn)) = Λ(α1) + · · ·+ Λ(αn).

P (α1, . . . ,αn) has the maximal volume if and only if it is regular: α1 =· · · = αn = π/n. In this case the volume is equal to n ·Λ(π/n). Therefore,normalized volume of the ideal regular pyramid is equal to

ωn =vol(P (π

n , . . . , πn ))

vert(P (πn , . . . , π

n ))=

n · Λ(πn )

n + 1, hence ωn → 0, if n →∞.

166 A. VESNIN [8]

1

2

34

α

α

α

αα

α

αα

α

αα

α

Fig. 6. Ideal pyramid P (α1, α2, α3, α4) and ideal prism Pα4 .

Let Pαn be an ideal n-prism in H3 with dihedral angles α incident to the

top as well as to the bottom as in Fig. 6. It is known [18], that

vol(Pαn ) = n

[Λ

(α +

π

n

)+ Λ

(α− π

n

)− 2Λ

(α− π

2

)].

An ideal n-gonal prism Pαn has maximal volume if αn = arccos

(cos π

n√2

). In

particular, the maximal volume ideal 4-gonal prism is the π/3-cube, andits volume is equal to volmax(P4) = 10Λ(π/6) = 5 v3 = 5.07 . . . . Sinceαn → π

4 as n → ∞, for the normalized volume of maximal volume idealn-gonal prism we have

ωn =vol(Pαn

n )vert(Pαn

n )→

n · 4Λ(π4 )

2n= 2Λ

(π

4

)=

v8

4, n →∞.

Let An(α) be an ideal n-antiprism in H3 with dihedral angles α incident tothe bottom and to the top (see Fig. 7 for An(α), where left and right sidesassumed to be identified). Denote by β dihedral angles between lateraltriangles. Since the antiprism is ideal, we have 2α + 2β = 2π.

α α α

α α α

β

β ββ

β ββ

Fig. 7. An ideal antiprism An(α).

It is known [8, 18], that

vol(An(α)) = 2n[Λ

(α

2+

π

2n

)+ Λ

(α

2− π

2n

)].

[9] VOLUMES OF RIGHT-ANGLED POLYHEDRA 167

An ideal n-antiprismAn(α) is of maximal volume if αn = arccos(cos πn−

12 )

(in particular, the maximal volume ideal 3-antiprism is the regular idealπ2 -octahedron). Therefore, for normalized volume we have

ωn =volmax(An)vert(An)

→ 2Λ(π

6

)= v3, if n →∞.

5. Double limits for normalized volume functions

The following result demonstrates that value 5v3/8 in Theorem 4.1is a double-limit point for the normalized volume function ω(R), whereR ∈ R.

Theorem 5.1. ([14]) For each integer k " 1 there is a sequence ofbounded right-angled hyperbolic polyhedra kRn such that

limn→∞

ω(kRn) = limn→∞

vol(kRn)vert(kRn)

=k

k + 1· 5v3

8.

Polyhedron kRn in the theorem is a composition of k Lobell polyhedraRn glued along n-gonal faces similar to a tower.

Let us denote by R∞ the set of all ideal (with all vertices at infinity)right-angled polyhedra in H3. Estimates of volumes of manifolds formR∞were done by Atkinson in [3].

Theorem 5.2. ([3]) Let P be an ideal right-angled hyperbolic polyhe-dron with N vertices. Then

(N − 2) · v8

4! vol(P ) ! (N − 4) · v8

2,

where v8 is the volume of the regular ideal octahedron. Both estimates be-came equalities if P is the regular ideal octahedron. There exists a sequenceof ideal right-angled polyhedra Pi with Ni vertices such that vol(Pi)/Ni

tends to v8/2 as i →∞.

The following result demonstrates that value v8/2 in the theoremis a double-limit point for the normalized volume function ω(R), whereR ∈ R∞.

Theorem 5.3. For each integer k " 1 there is a sequence of idealright-angled hyperbolic polyhedra kAn(π/2) such that

limn→∞

ω(kAn(π/2)) = limn→∞

vol(kAn(π/2))vert(kAn(π/2))

=k

k + 1· v8

2.

168 A. VESNIN [10]

Polyhedron kAn(π/2) in the theorem is a composition of k copies of idealn-gonal right-angled antiprisms An(π/2) glued along n-gonal faces similarto a tower.

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[11] VOLUMES OF RIGHT-ANGLED POLYHEDRA 169

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A. Vesnin:Sobolev Institute of MathematicsNovosibirsk 630090, RussiaandOmsk State Technical UniversityOmsk 644050, Russiavesnin@math.nsc.ru