Volumes of the hyperbolic Coxeter simplices

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Numbers

$n$

symbol

$\mathrm{Vol}$

[3, 3, 6]:

0.04228923360040223437588343976143834524757038781376249133406204611569156151076016758901959809284402896

comment: Kozma and Szirmai write this simplex as $\overline{V}_3$. Its volume is $\Lambda(\pi/3)/8$.

[3, 6, 3]:

0.1691569344016089375035337590457533809902815512550499653362481844627662460430406703560783923713761158

comment: Kozma and Szirmai write this simplex as $\overline{Y}_3$. Its volume is $\Lambda(\pi/3)/2$, which is the covolume of $\mathrm{PSL}_2(\mathbb{Z}[\omega])$ in the Bianchi covolume table.

[6, 3^{[3]}]:

0.5074708032048268125106012771372601429708446537651498960087445533882987381291220110682351771141283475

comment: Kozma and Szirmai write this simplex as $\overline{VP}_3$. Its volume is $3\Lambda(\pi/3)/2$.

[3^{[3, 3]}]:

1.014941606409653625021202554274520285941689307530299792017489106776597476258244022136470354228256695

comment: Kozma and Szirmai write this simplex as $\widehat{PP}_3$. Its volume is $3\Lambda(\pi/3)=\mathrm{Cl}_2(\pi/3)$.

[3, 3^{[3]}]:

0.08457846720080446875176687952287669049514077562752498266812409223138312302152033517803919618568805791

comment: Kozma and Szirmai write this simplex as $\overline{P}_3$. Its volume is $\Lambda(\pi/3)/4$.

[6, 3, 6]:

0.2537354016024134062553006385686300714854223268825749480043722766941493690645610055341175885570641737

comment: Kozma and Szirmai write this simplex as $\overline{Z}_3$. Its volume is $3\Lambda(\pi/3)/4$.

[4, 3, 6]:

0.1057230840010055859397085994035958631189259695344062283351551152892289037769004189725489952321100724

comment: Kozma and Szirmai write this simplex as $\overline{BV}_3$. Its volume is $5\Lambda(\pi/3)/16$.

[4, 3^{[3]}]:

0.2114461680020111718794171988071917262378519390688124566703102305784578075538008379450979904642201448

comment: Kozma and Szirmai write this simplex as $\overline{BP}_3$. Its volume is $5\Lambda(\pi/3)/8$.

[6, 3^{1, 1}]:

0.2114461680020111718794171988071917262378519390688124566703102305784578075538008379450979904642201448

comment: Kozma and Szirmai write this simplex as $\overline{DV}_3$. Its volume is $5\Lambda(\pi/3)/8$.

[3^{[]x[]}]:

0.4228923360040223437588343976143834524757038781376249133406204611569156151076016758901959809284402896

comment: Kozma and Szirmai write this simplex as $\overline{DP}_3$. Its volume is $5\Lambda(\pi/3)/4$.

[(3, 6)^{[2]}]:

0.8457846720080446875176687952287669049514077562752498266812409223138312302152033517803919618568805791

comment: Kozma and Szirmai write this simplex as $\widehat{VV}_3$. Its volume is $5\Lambda(\pi/3)/2$.

[3, 4, 4]:

0.07633046618143491792121695957769867589784578119013934452220817663514691831468789745661304274384292552

comment: Kozma and Szirmai write this simplex as $\overline{R}_3$. Its volume is $\Lambda(\pi/4)/6$.

[3, 4^{1, 1}]:

0.1526609323628698358424339191553973517956915623802786890444163532702938366293757949132260854876858510

comment: Kozma and Szirmai write this simplex as $\overline{O}_3$. Its volume is $\Lambda(\pi/4)/3$.

[(3^2, 4^2)]:

0.3053218647257396716848678383107947035913831247605573780888327065405876732587515898264521709753717021

comment: Kozma and Szirmai write this simplex as $\widehat{BR}_3$. Its volume is $2\Lambda(\pi/4)/3$, which is the covolume of $\mathrm{PSL}_2(\mathbb{Z}[i])$ in the Bianchi covolume table.

[4, 4, 4]:

0.2289913985443047537636508787330960276935373435704180335666245299054407549440636923698391282315287766

comment: Kozma and Szirmai write this simplex as $\overline{N}_3$. Its volume is $\Lambda(\pi/4)/2$.

[4^{1, 1, 1}]:

0.4579827970886095075273017574661920553870746871408360671332490598108815098881273847396782564630575531

comment: Kozma and Szirmai write this simplex as $\overline{M}_3$. Its volume is $\Lambda(\pi/4)$.

[4^{[4]}]:

0.9159655941772190150546035149323841107741493742816721342664981196217630197762547694793565129261151062

comment: Kozma and Szirmai write this simplex as $\widehat{RR}_3$. Its volume is $2\Lambda(\pi/4)$, which is Catalan's constant $G$.

[5, 3, 6]:

0.1715016612824999469000279964800349893678040756955395984715202632150696956462308486829963126912896437

comment: Kozma and Szirmai write this simplex as $\overline{HV}_3$.

[5, 3^{[3]}]:

0.3430033225649998938000559929600699787356081513910791969430405264301393912924616973659926253825792874

comment: Kozma and Szirmai write this simplex as $\overline{HP}_3$.

[(3^3, 6)]:

0.3641071003648810077218511179625890780335435014490911457147265838487516441831766328583240759519059958

comment: Kozma and Szirmai write this simplex as $\widehat{AV}_3$.

[(3, 4, 3, 6)]:

0.5258402692379140577613335798917157905882920634862270030080661680012334531885097372803152791544391707

comment: Kozma and Szirmai write this simplex as $\widehat{BV}_3$.

[(3, 5, 3, 6)]:

0.6729858045482070864989016868337961076494172338060998040665663070759617338022679831662452322170737629

comment: Kozma and Szirmai write this simplex as $\widehat{HV}_3$.

[(3, 4^3)]:

0.5562821155610392060029590597531612500704676674573281874254955747898259766273202377781400625263246846

comment: Kozma and Szirmai write this simplex as $\widehat{CR}_3$.

Definition

Let $P$ be a finite-volume hyperbolic Coxeter simplex [3] in $\mathbb{H}^n$, with dihedral angles of the form $\pi/m$. Listed is the hyperbolic volume of $P$ for the metric of curvature $-1$.

Parameters

$n$

— dimension (an integer with $3\leq n\leq9$)

symbol

— Coxeter symbol (a finite-volume hyperbolic Coxeter simplex)

Formulas

(1)

The Lobachevsky function is $\Lambda(\theta)=\frac12\operatorname{Im}\operatorname{Li}_2(e^{2i\theta}) =-\int_0^\theta\log|2\sin t|\,dt$ [1]. It is $\frac12\mathrm{Cl}_2(2\theta)$, where $\mathrm{Cl}_2$ is the Clausen function.

(2)

If $[p,q,r]$ is a complete hyperbolic Coxeter orthoscheme and $\alpha=\pi/p$, $\beta=\pi/q$, $\gamma=\pi/r$, set $\tan\theta=\sqrt{\cos^2\beta-\sin^2\alpha\sin^2\gamma}/(\cos\alpha\cos\gamma)$. Then $\mathrm{Vol}([p,q,r])=\frac14\{\Lambda(\alpha+\theta)-\Lambda(\alpha-\theta) +\Lambda(\gamma+\theta)-\Lambda(\gamma-\theta) +\Lambda(\pi/2+\beta-\theta)+\Lambda(\pi/2-\beta-\theta) +2\Lambda(\pi/2-\theta)\}$, the Kellerhals formula as used by [2].

(3)

$\mathrm{Vol}([3,3,6])=\Lambda(\pi/3)/8$ and $\mathrm{Vol}([3,4,4])=\Lambda(\pi/4)/6$. These are one quarter of the $D=-3$ and $D=-4$ entries of the Bianchi covolume table. Consequently $\mathrm{Vol}([3,6,3])$ and $\mathrm{Vol}([(3^2,4^2)])$ are those two Bianchi covolumes.

Comments

(4)

Each entry is the volume of the simplex itself, equivalently the covolume of the full reflection group generated by its facets. The orientation-preserving subgroup has index $2$ and covolume twice the value listed here.

(5)

A Coxeter simplex is noncompact when it has at least one ideal vertex. Such simplices are also called Koszul-type [1], and in $\mathbb{H}^3$ there are $23$ of them.

(6)

Rows are labelled by Coxeter symbols [4]: $[p,q,r]$ is a linear diagram, $[(p,q,r,s)]$ a cycle, a superscript such as $1,1$ a fork, and a bracketed superscript such as $[3]$ a ring. The empty-bracket product in $[3^{[\,]\times[\,]}]$ is written $[3^{[]x[]}]$ in the row label. Each entry comment gives the Witt symbol of [1], Table 1, and, where that table gives one, the closed form; that table labels $\overline{DV}_3$ as $[4,3^{[1,1]}]$, while its diagram is $[6,3^{1,1}]$. The row order follows the commensurability classes of [1], Table 1.

(7)

The entry $[3^{[3,3]}]$ is the regular ideal tetrahedron. Its volume is $3\Lambda(\pi/3)=\mathrm{Cl}_2(\pi/3)$, the Gieseking constant held in the table of Clausen values.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.real_arb import RealBallField
RB = RealBallField(numberdb.bits(80, losing=80))
CB = ComplexBallField(numberdb.bits(80, losing=80))
pi = RB.pi()
def Lambda(theta):
    theta = RB(theta)
    if (theta / pi).contains_integer():
        return RB(0)
    z = (CB(0, 2) * CB(theta)).exp()
    return z.polylog(2).imag() / 2
Lambda(pi / 3) / 8        # Vol([3,3,6])

References

[1]

R. T. Kozma and J. Szirmai, Optimal horoball packing densities for Koszul-type tilings in hyperbolic 3-space, Acta Math. Hungar. 177 (2025), 419-449. (arXiv) (doi)

[2]

N. W. Johnson, R. Kellerhals, J. G. Ratcliffe and S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transform. Groups 4 (1999), 329-353. (doi)

Links

[3]

Wikipedia: Coxeter group

[4]

Wikipedia: Paracompact uniform honeycombs

Similar tables

Covolumes of the Bianchi groups — $[3,3,6]$ and $[3,4,4]$ have volumes one quarter of the $D=-3$ and $D=-4$ Bianchi covolumes

Values of the Clausen functions at rational multiples of $\pi$ — the Lobachevsky values in the formulas are half of Clausen values

Growth rates of hyperbolic Coxeter simplex groups — growth rates for the same finite-volume family; it holds the nine compact tetrahedral groups, while this table holds the noncompact ones

Hyperbolic volumes of the prime knots with at most ten crossings — hyperbolic 3-manifold volumes rather than reflection-orbifold covolumes

Volumes of the closed hyperbolic 3-manifolds of the Hodgson-Weeks census — closed hyperbolic 3-manifold volumes rather than Coxeter simplex volumes

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the $23$ noncompact Coxeter simplices in $\mathbb{H}^3$ and no compact tetrahedra or simplices in dimensions $4$ to $9$)

How they were obtained:

The generator evaluates the Lobachevsky function as $\frac12\operatorname{Im}\operatorname{Li}_2(e^{2i\theta})$ in arb ball arithmetic and applies (2) and the orthoscheme decompositions of [1]. It writes $100$ digits.

All $23$ rows were checked against the source values in [1]. The $[3^{[3,3]}]$ and $[4^{[4]}]$ rows were checked against the stored Clausen values. The $[3,3,6]$ and $[3,4,4]$ rows were checked as quarters of the $D=-3$ and $D=-4$ Bianchi covolumes; the $[3,6,3]$ and $[(3^2,4^2)]$ rows were checked as those covolumes themselves.