Post on 09-May-2022
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