Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 57 (2010), 159-169
Andrei VESNIN
Volumes and Normalized Volumes
of Right-Angled Hyperbolic Polyhedra
To Massimo Ferri and Carlo Gagliardi
on the occasion of their 60-anniversary
Abstract. We present some recent results on a structure of the
set of volumes of right-angled polyhedra in hyperbolic space, such
as the initial list of smallest volumes. Also, we discuss normalized
volumes 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 the
most studied class of Coxeter polyhedra. Basic facts on polyhedra in
spaces of constant curvature, their existence and volume calculations can
be found in [1]. In this survey we present some recent results on a structure
of the set of volumes of bounded right-angled hyperbolic polyhedra.
These results can be useful not only for studying polyhedra, but also
the corresponding 3-manifolds. The simplest and smallest bounded polyhedron in H3 with all dihedral angled π/2 is the dodecahedron. The second
Paper presented at Computational and Geometric Topology – A conference in
honour of Massimo Ferri and Carlo Gagliardi on their 60-th birthday, Bertinoro
(Italy), 17–19 June, 2010
Work performed under the auspices of the Russian Foundation for Basic Research (grants 10-01-00642 and 10-01-91056), and the grant SO RAN – UrO
RAN.
160
A. VESNIN
[2]
smallest is the 14-hedron, eight copies of which were used by Löbell in 1931
to construct the first example of a closed orientable hyperbolic 3-manifold
[6]. Its generalizations, referred as Löbell manifolds, were introduced in
[19]. An explicit formula for volumes of Löbell 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 fourcoloring of its faces defines a closed orientable hyperbolic 3-manifolds. It
is descibed in [9] how topological properties of these manifolds depend on
colorings. Thus, results on volumes of right-angled polyhedra can by applied to a wide class of hyperbolic 3-manifolds. We recall that by Mostow
rigidity theorem the volume of a closed hyperbolic 3-manifold is its topological invariant.
In Section 2 we recall some results about existence of hyperbolic polyhedra. In Section 3 we discuss a structure of the set of volumes of polyhedra. In Theorem 3.2 we give the volume formula for Löbell polyhedra
in terms of the Lobachevsky function. Volume formulae for hyperbolic
polyhedra are usually complicated to understand even simple numbertheoretical properties.
Problem 1. Does there exists a pair of bounded right-angled hyperbolic polyhedra such that the ratio of their volumes is irrational?
In Theorem 3.3 we present the initial list of smallest volume bounded
right-angled hyperbolic polyhedra. A structure of the set of volumes of
ideal polyhedra is not described yet.
Problem 2. Describe the initial list of volumes of ideal finite-volume
right-angled hyperbolic polyhedra.
In Sections 4 and 5 we discuss the normalized volume of a hyperbolic
polyhedron defined as the ratio of its volume to its number of vertices. We
present calculations of normalized volumes for some classes of polyhedra.
2. Right-angled polyhedra in Hn
There are strong combinatorial restrictions on existence of rightangled polyhedra in n-dimensional hyperbolic space Hn . For a polyhe.
dron
& P let ak (P ) be the number of its k-dimensional faces and ak =
1
dim F =k a. (F ) be the average number of 4-dimensional faces in a kak
dimensional polyhedron. It was shown by Nikulin [11] that
a.k
<
n−k
Cn−.
C .n + C .n+1
[2]
[ 2 ]
C kn + C kn+1
[2]
[ 2 ]
C D
for 4 < k ! n2 . From this result we see, in particular, that a12 , the
average number of sides in a 2-dimensional face, satisfies to the following
[3]
VOLUMES OF RIGHT-ANGLED POLYHEDRA
inequality:

 4(n − 1) , if n even,
1
n−2
a2 <
 4n ,
if n odd.
n−1
161
But any hyperbolic right-angled polygon has at least five sides: a12 " 5.
Thus, it follows from the Nikulin inequality that there exist no bounded
right-angled polyhedra in Hn for n > 4.
Concerning other classes of polyhedra in n-dimensional hyperbolic
space the following results are known: there exist no bounded Coxeter
polyhedra for n > 29 [21] and examples are know up to n = 8 only; there
exist no finite volume right-angled polyhedra for n > 12 [4] and examples
are know up to n = 8 only; there exist no finite volume Coxeter polyhedra
for n > 995 [13] and examples are known up to n = 21 only.
We are interested in acute-angled and, especially, right-angled polyhedra in three-dimensional hyperbolic space H3 . The following uniqueness
result was obtained by Andreev in [2]: bounded acute-angled polyhedron
in H3 is uniquely determined, up to isometry, by its combinatorial type
and dihedral angles.
For the case of right-angled polyhedra necessary and sufficient conditions 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 incident to 3 edges; (2) any face has at least 5 sides; (3) any simple closed
curve on the surface of the polyhedron which separate some two faces of
it (prismatic circuit), intersects at least 5 edges. Figure 1 presents two
polyhedra: the left, known as Greenbergs polyhedron, satisfies the above
conditions, and the right satisfies conditions (1) and (2), but doesn’t satisfy condition (3), since it has a closed circuit which separates two 6-gonal
faces, but intersects 4 edges only.
!
"
&
$
#
%
$
%
Fig. 1. Two polyhedra.
162
[4]
A. VESNIN
3. The structure of the set of volumes
Let us denote by R the set of all bounded right-angled polyhedra in
H3 . 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 and
F2 ⊂ R2 be a pair of k-gonal faces. A composition is defined as a union of
R1 and R2 along F1 and F2 : R = R1 ∪F1 =F2 R2 . It is shown in [5] that
composition of two polyhedra from R belongs to R.
The second operation is an edge surgery. This is a combinatorial move
from R to R − e. If R ∈ R and e is such that n1 and n2 are at least 6
sides each and e is not a part of prismatic 5-circuit, then (R − e) ∈ R (see
Fig. 2). The inverse move, from (R − e) to R, we will call an edge adding.
n1
n3 e
n1 − 1
n4
n3 + n4 − 4
n2
n2 − 1
polyhedron R
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 unions
of right-angled hyperbolic polyhedra P1 , . . . , Pk such each set Pi is obtained
from Pi−1 by decomposition or edge surgery, and Pk consists of Löbell
polyhedra. Moreover,
vol(P0 ) " vol(P1 ) " vol(P2 ) " . . . " vol(Pk ).
Löbell polyhedra Rn were defined in [19] for any n " 5 as right-angled
hyperbolic polyhedra having 2n + 2 faces: two n-gonal and 2n pentagonal
managed similar to the lateral surface of a dodecahedron. The dodecahedron R5 and 14-hedron R6, used by Löbell in [6] to construct the first
closed orientable hyperbolic 3-manifold, are presented in Fig. 3. An explicit formula for volumes of Löbell polyhedra was obtained by the author
in terms of the Lobachevsky function
Λ(x) = −
8x
0
log |2 sin(t)| dt.
[5]
163
VOLUMES OF RIGHT-ANGLED POLYHEDRA
Fig. 3. Polyhedra R5 and R6.
Theorem 3.2. ([20]) For any n " 5 the following formula holds for
volumes of Löbell polyhedra
5
5
5π
66
n5
π6
π6
vol(Rn) =
2Λ(θn ) + Λ θn +
+ Λ θn −
+Λ
− 2θn ,
2
n
n
2
where θn =
π
2
1
− arccos( 2 cos(π/n)
).
Another formula for volumes of Löbell polyhedra can be found in [10].
It is easy to see from Theorem 3.2 that the volume function vol Rn 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 set
of volumes of right-angled hyperbolic polyhedra. It is shown in [5] that R5
and R6 are the first and the second smallest volume compact right-angled
hyperbolic polyhedra. To describe the initial list of the set of volumes of
polyhedra from R, we denote by R611 , R621 , R622 polyhedra presented in
Fig. 4, and by R631 , R632 , R633 – polyhedra presented in Fig. 5.
6
6
6
6
6
6
6
6
6
6
6
Fig. 4. Polyhedra R611 , R621 and R622 .
6
6
6
6
6
6
7
6
6
6
6
Fig. 5. Polyhedra R631 , R632 and R633 (= R711 ).
6
6
6
164
[6]
A. VESNIN
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 the
Löbell polyhedron Rn. Of cause, the same polyhedron can be obtained
from different Löbell polyhedra; for example, R633 = R711 . In Fig.4 and 5
six-gonal faces are marked by 6, seven-gonal faces are marked by 7, and
all other faces are pentagonal. By 2R5 we denote a polyhedron obtained
by a composition of two dodecahedra R5.
The initial list of smallest volume bounded right-angled hyperbolic
polyhedra is described in the following theorem. Geometric realizations of
these polyhedra in H3 can be obtained by the computer program developed
by Roeder [16].
Theorem 3.3. ([17]) The first eleven smallest volume bounded rightangled hyperbolic polyhedra and their volumes are as in the following table:
1
2
3
4
5
6
volume
notation
4.3062 . . .
6.0230 . . .
6.9670 . . .
7.5632 . . .
7.8699 . . .
8.0002 . . .
R5
R6
R611
R7
R621
R622
7
8
9
10
11
volume
notation
8.6124 . . .
8.6765 . . .
8.8608 . . .
8.9466 . . .
9.0190 . . .
2R5
R633
R631
R632
R8
4. Volumes and normalized volumes
In studding volumes of hyperbolic polyhedra, manifolds, and orbifolds
it is very useful to use the Schläfli variation formula which shows how volume changes if we change dihedral angles, but preserve combinatorics of
polyhedra. Considering the class R of bounded right-angled polyhedra,
we have another situation – dihedral angles are fixed, but combinatoric
changes. So, we are interested to see how volume depends of a combinatorial structure, for example of a number of vertices. The following result
was obtained by Atkinson [3].
Theorem 4.1. ([3]) Let P be a compact right-angled hyperbolic polyhedron with N vertices. Then
v8
5v3
(N − 2) ·
! vol(P ) < (N − 10) ·
,
32
8
where v8 is the maximal octahedron volume, and v3 is the maximal tetrahedron volume. There exists a sequence of compact right-angled polyhedra
Pi 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 = 3 Λ(π/3) = 1.0149416064096535 . . .
and
v8 = 8 Λ(π/4) = 3.663862376708876 . . . .
The lower bound from Theorem 4.1 can be improved for V ! 54 and
F ! 29.
Theorem 4.2. ([14]) Let P be a compact right-angled hyperbolic polyhedron, with V vertices and F faces. If P is not a dodecahedron, then
E
F
v8
vol(P ) " max (V − 2) · , 6.023 . . .
32
and
E
F
v8
vol(P ) " max (F − 3) · , 6.023 . . . .
16
The behavior of a volume as a function of a number of vertices is interesting
to study for other classes of polyhedra also.
Let P be a finite volume polyhedron in H3 ; denote by vol(P ) its
volume, and by vert(P ) number of its vertices. Let us define a normalized
volume of P as the following ratio:
ω(P ) =
vol(P )
.
vert(P )
It was demonstrated in [15] that, in general, the behavior of the normalized
volume 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 classes
of polyhedra.
Let P (α1 , . . . , αn ), n ≥ 3, be an ideal pyramid in H3 with dihedral
angles α1 , . . . , αn incident to the bottom, see Fig. 6. It is known [18], that
α1 + · · · + 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 ( nπ , . . . , nπ ))
n · Λ( nπ )
=
,
π
π
vert(P ( n , . . . , n ))
n+1
hence ωn → 0,
if n → ∞.
166
[8]
A. VESNIN
α
α
α
α
α3
α4
α2
α1
α
α
α
α
Fig. 6. Ideal pyramid P (α1 , α2 , α3 , α4 ) and ideal prism P4α .
Let Pnα 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
G 5
5
5
π6
π6
π 6H
vol(Pnα ) = n Λ α +
+Λ α−
− 2Λ α −
.
n
n
2
5 π6
cos
An ideal n-gonal prism Pnα has maximal volume if αn = arccos √2n . In
particular, the maximal volume ideal 4-gonal prism is the π/3-cube, and
its volume is equal to volmax (P4 ) = 10 Λ(π/6) = 5 v3 = 5.07 . . . . Since
αn → π4 as n → ∞, for the normalized volume of maximal volume ideal
n-gonal prism we have
ωn =
5π 6 v
n · 4Λ( π4 )
vol(Pnαn )
8
= 2Λ
= ,
αn →
vert(Pn )
2n
4
4
n → ∞.
Let An (α) be an ideal n-antiprism in H3 with dihedral angles α incident to
the bottom and to the top (see Fig. 7 for An (α), where left and right sides
assumed to be identified). Denote by β dihedral angles between lateral
triangles. Since the antiprism is ideal, we have 2α + 2β = 2π.
α
α
β
β
β
β
β
α
α
α
β
β
α
Fig. 7. An ideal antiprism An (α).
It is known [8, 18], that
G 5α
5α
π6
π 6H
vol(An (α)) = 2n Λ
+
+Λ
−
.
2
2n
2
2n
[9]
VOLUMES OF RIGHT-ANGLED POLYHEDRA
167
An ideal n-antiprism An (α) is of maximal volume if αn = arccos(cos nπ − 12 )
(in particular, the maximal volume ideal 3-antiprism is the regular ideal
π
2 -octahedron). Therefore, for normalized volume we have
ωn =
5π 6
volmax (An )
→ 2Λ
= v3 ,
vert(An )
6
if n → ∞.
5. Double limits for normalized volume functions
The following result demonstrates that value 5v3 /8 in Theorem 4.1
is a double-limit point for the normalized volume function ω(R), where
R ∈ R.
Theorem 5.1. ([14]) For each integer k " 1 there is a sequence of
bounded right-angled hyperbolic polyhedra kRn such that
lim ω(kRn) = lim
n→∞
n→∞
vol(kRn)
k
5v3
=
·
.
vert(kRn)
k+1 8
Polyhedron kRn in the theorem is a composition of k Löbell polyhedra
Rn 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 form R∞
were done by Atkinson in [3].
Theorem 5.2. ([3]) Let P be an ideal right-angled hyperbolic polyhedron with N vertices. Then
v8
v8
(N − 2) ·
! vol(P ) ! (N − 4) · ,
4
2
where v8 is the volume of the regular ideal octahedron. Both estimates became equalities if P is the regular ideal octahedron. There exists a sequence
of 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 theorem
is a double-limit point for the normalized volume function ω(R), where
R ∈ R∞ .
Theorem 5.3. For each integer k " 1 there is a sequence of ideal
right-angled hyperbolic polyhedra kAn (π/2) such that
vol(kAn (π/2))
k
v8
=
· .
n→∞ vert(kAn (π/2))
k+1 2
lim ω(kAn (π/2)) = lim
n→∞
168
A. VESNIN
[10]
Polyhedron kAn (π/2) in the theorem is a composition of k copies of ideal
n-gonal right-angled antiprisms An (π/2) glued along n-gonal faces similar
to a tower.
References
[1] D. V. Alekseevskij – E. B. Vinberg – A. S. Solodovnikov , Geometry of spaces of constant curvature, In: “Geometry II: Spaces of Constant
Curvature”, Encyclopaedia of Mathematical Sciences, Vol. 29, Vinberg, E.
B. (Ed.) 1993, 6–138.
[2] E. M. Andreev, On convex polyhedra in Lobachevsky spaces, Math. USSR
Sbornik, 10 (3) (1970), 413–440.
[3] C. K. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebraic & Geometric Topology, 9 (2009), 1225–1254.
[4] G. Dufour, Notes on right-angled Coxeter polyhedra in hyperbolic spaces,
Geometriae Dedicata 147 (1) (2010), 277–282.
[5] T. Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebraic & Geometric Topology, 8 (2008), 1523–1565.
[6] F. Löbell, Beispiele geschlossene dreidimensionaler Clifford — Kleinischer Räume negative Krümmung, Ber. Verh. Sächs. Akad. Lpz., Math.Phys. Kl. 1931. V. 83. P. 168–174.
[7] S. Matveev – C. Petronio – A. Vesnin, Two-sided asymptotic bounds
for the complexity of some closed hyperbolic three-manifolds, Journal of the
Australian Math. Soc., 86 (2) (2009), 205–219.
[8] A. Mednykh – A. Vesnin, Hyperbolic volumes of the Fibonacci manifolds,
Siberian Math. J., 36 (2) (1995), 235–245.
[9] A. Mednykh – A. Vesnin, Three-dimensional hyperelliptic manifolds and
hamiltonian graphs, Siberian Math. J., 40 (4) (1999), 628–643.
[10] A. Mednykh – A. Vesnin, Löbell manifolds revised, Siberian Electronic
Mathematical Reports, 4 (2007), 605–609.
[11] V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986, 654–671.
[12] A. V. Pogorelov, Regular decomposition of the Lobacevskii space, Mat.
Zametki, 1 (1967), 3–8. (In Russian)
[13] M. N. Prokhorov, The absence of discrete reflection groups with a noncompact fundamental polyhedron of finite volume in Lobachevsky space of
large dimension, Mathematics of the USSR - Izvestiya 28 (2) (1987), 401–
411.
[14] D. Repovš – A. Vesnin, Two-sided bounds for the volume of right-angled
hyperbolic polyhedra, Math. Notes, 89 (1) (2011), 31–36.
[15] D. Repovš – A. Vesnin, Normalized volumes of right-angled hyperbolic
polyhedra, Abdus Salam School of Mathematical Sciences, Research Preprint
Series, No. 279, Lahore, Pakistan, February 2011, 10 pp.
[16] R. K.W. Roeder, Constructing hyperbolic polyhedra using Newton’s method,
Experimental Math., 16 (4) (2007), 463–492.
[17] K. Shmel’kov – A. Vesnin, The initial list of compact hyperbolic rightangled polyhedra, in progress.
[11]
VOLUMES OF RIGHT-ANGLED POLYHEDRA
169
[18] W. Thurston, Three Dimensional Geometry and Topology, Princeton
Math. Ser. 35, Princeton Univ. Press, 1997.
[19] A. Vesnin, Three-dimensional hyperbolic manifolds of Löbell type, Siberian
Math. J. 28 (5) (1987), 731–734.
[20] A. Vesnin,Volumes of three-dimensional hyperbolic Löbell manifolds, Math.
Notes 64 (1) (1998), 15–19.
[21] E. B. Vinberg, The absence of crystallographic groups of reflections in
Lobachevskij spaces of large dimension, Trans. Mosc. Math. Soc. (1985),
75–112.
A. Vesnin:
Sobolev Institute of Mathematics
Novosibirsk 630090, Russia
and
Omsk State Technical University
Omsk 644050, Russia
vesnin@math.nsc.ru