Volumes of right-angled hyperbolic antiprisms and Löbell polyhedra

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Numbers

$P$

$n$

$\operatorname{Vol}(P)$

antiprism

3.663862376708876060218414059729536443096597497126688537065992478487052079105019077917426051704460425

comment: The regular ideal right-angled octahedron.

antiprism

6.023046020047188823634189314616797115498029024722499424079908837981719280845647550840509397222871580

antiprism

8.137885077568508032750990909375463897191649457101962939230939542700247723837384384856402788780009889

antiprism

10.14941606409653625021202554274520285941689307530299792017489106776597476258244022136470354228256695

antiprism

12.10629799233147351209208560778742534745395532911132890973961296640057691458017275318899518536843413

antiprism

14.03046119144703575798614008609586396425812479874568131557922592773212236700143427597383049624854890

antiprism

15.93338567247750071929326062224656462439038264702847478133059739981074455101238445553433588835777895

antiprism

10:

17.82170471596165597645395029576823772954114120719989070207713744597323916340061361025982590842355435

antiprism

11:

19.69953311272616168831484640920154051640711929511279637829010147276509870133545702919726098474769310

antiprism

12:

21.56956471350594531140084612976240489791068832030523749543887892713731488619816466060251966502014180

antiprism

13:

23.43363967756761095668299495604947368567714682921435074908333583040937251370024590373324110030179859

antiprism

14:

25.29305909149939550332191858017989419543371631498668141655020977451086997147879537730404510581857013

antiprism

15:

27.14876955376390304937132547107646267072666930739679780048982666903748811343358328975923615294446272

antiprism

16:

29.00147655858978888476476202774499108922684282614529102927974247823152907386826188869930098004601058

antiprism

17:

30.85171686539086958588654389251840563322103762472652565618322224565502440493002134197568966741608109

antiprism

18:

32.69990621740032937702371132870923042302148785493707538067294473737408781917685503558599903483333800

antiprism

19:

34.54637169939548039929110143407412376448463544095341943595043139343396020122896824009084598325271180

antiprism

20:

36.39137422041238372008174064323450980359551357406623988345899257631499089815163311048725831593308810

antiprism

21:

38.23512447322396491738117803323386222647046539583537221030493536187997122338996697283681278982600637

antiprism

22:

40.07779448030930313097291401931034948461963689509981504319442177213718025378693460613598559638068054

antiprism

23:

41.91952608957905562513123424453542897486647848714852706533234399643394596758011349386561495580054543

antiprism

24:

43.76043732158301929214598688525859134226497745249624497848594989823136434826072136977730210781398154

antiprism

25:

45.60062717724717368239364208061492253791778099527112827782787393590490620587538104481740799985221431

antiprism

26:

47.44017932533167801030787477029752623062807914217140320069564327343195632211603285354877399932954899

antiprism

27:

49.27916496309821523700237892073638173490380627982251242903476158570827584435716223458113016985758348

antiprism

28:

51.11764505888740666558129507525311039420711762761196928182854527504584234096323917617720941989196237

antiprism

29:

52.95567212714060468503115693834464370940858327429206967961009401842587664088387742256052160312414659

antiprism

30:

54.79329164587333606675641168686672294655268055720360574008674033301458108471131157534537588676782960

lobell

4.306207600730808652919837159842209449241185235412934114224949117819904343308238335156704866571730974

comment: The compact regular right-angled dodecahedron.

lobell

6.023046020047188823634189314616797115498029024722499424079908837981719280845647550840509397222871580

lobell

7.563249091443052227816922590217274093440690532148578248390116274626455382631151265493080298157058599

lobell

9.019052727442401789887731416575456186092354352156748456638883781530984955745994714033369264243072644

lobell

10.42605221624301648292919669991531522383957555351901459132016404636694673217311713794756363136786380

lobell

10:

11.80174745427738409296940496910500034028824695422865147629693202118729980294240346277382884048447031

lobell

11:

13.15596623173094178122681290005315309453637655997179838782230484455986079451781429056800504441634218

lobell

12:

14.49472769512293639543857869232161955127343018070328053079440343172031821755453827286344587987849402

lobell

13:

15.82195377019068805449868598441621208041084634154657895320054809490605305937781856520775031379621373

lobell

14:

17.14032239820113033891574992067580781949374356849701787270568761867662506839341638638986699888304692

lobell

15:

18.45173080431920535782739397700514009894770558297703250413333027636516903077689625536980415230531099

lobell

16:

19.75756392713212042941632483338725934604330123959558448673781884806787147792507740398256992208464554

lobell

17:

21.05885816792124400680902290320703462983763941278863415883583400454285328388081815788451774644271516

lobell

18:

22.35640555826288467996424435975231277526649047833855162504835825966557672261696771259338736292568739

lobell

19:

23.65082237877387000828485034134060715285913848203490087381229885753855092657806522469933304247873216

lobell

20:

24.94259571737793831091575859496215925120461150416652970429353497744061828594944463087736776205572639

lobell

21:

26.23211587287366904531569647466943331169410929374299554233137029132150641716923077749723110527865263

lobell

22:

27.51969941160361946852660645654963994543618902558213982186659640705300231796624317524873696250495388

lobell

23:

28.80560589551361491571540150905931426104864569980030173653535355848573611152083275730019057013626935

lobell

24:

30.09005022976959574106106456993627437938653609785888393540713023173799122461446130606541269926131863

lobell

25:

31.37321191850647918576048278630373062017967447289425676775027638537679565612656816595610341139682841

lobell

26:

32.65524209972738472880017151284878905110825253370775374818806772523541156061542443483585244093418626

lobell

27:

33.93626895968344892094484797679226914452611283690812407989966993366182201284339266459574710159921933

lobell

28:

35.21640194781134289471072603236596928588805688679038218867998332772323512506164783525756540754134307

lobell

29:

36.49573509230170680950015472225094332294156061156154315640083163754217271685688263359034860461595340

lobell

30:

37.77434963325451608402110959581686398450184404441718190807338623172884616544695855606808929285915089

Definition

Listed is the volume $\operatorname{Vol}(P)$ in hyperbolic three-space of sectional curvature $-1$, where $P=A_n$ is the ideal right-angled $n$-antiprism [2] or $P=L_n$ is the compact right-angled Löbell polyhedron [1]. The index $n$ counts the sides of either base.

Parameters

$P$

— Polyhedron family ($A_n$ (antiprism) or $L_n$ (Löbell polyhedron).)

$n$

— Number of sides of either base ($n\geq3$ for $A_n$ and $n\geq5$ for $L_n$.)

Formulas

(1)

$\Lambda(x)=-\int_0^x\log|2\sin t|\,dt=\tfrac12\operatorname{Cl}_2(2x)=\tfrac12\operatorname{Im}\operatorname{Li}_2(e^{2ix})$, where $\operatorname{Cl}_2$ is the Clausen function and $\operatorname{Li}_2$ is the principal dilogarithm.

(2)

$\operatorname{Vol}(A_n)=2n\bigl[\Lambda(\pi/4+\pi/(2n))+\Lambda(\pi/4-\pi/(2n))\bigr]$ for $n\geq3$ [2], equation (1).

(3)

$\operatorname{Vol}(L_n)=\tfrac n2\bigl[2\Lambda(\theta_n)+\Lambda(\theta_n+\pi/n)+\Lambda(\theta_n-\pi/n)-\Lambda(2\theta_n-\pi/2)\bigr]$, where $\theta_n=\arcsin(1/(2\cos(\pi/n)))\in(0,\pi/2)$, for $n\geq5$ [1], Theorem 4.1.

(4)

$F(A_n)=F(L_n)=2n+2$, $V(A_n)=2n$ and $V(L_n)=4n$, where $F(P)$ and $V(P)$ count the faces and vertices of $P$.

Comments

(5)

All dihedral angles are $\pi/2$. The antiprism $A_n$ has two $n$-gonal bases and $2n$ triangular lateral faces. A pentagonal flower consists of an $n$-gon surrounded by $n$ pentagons, each adjacent to the base and to its two cyclic neighbours. Joining two such flowers along their boundaries to form a trivalent sphere, as in [1], Section 4, defines $L_n$ uniquely up to isometry. It has two $n$-gonal bases and $2n$ pentagonal lateral faces.

(6)

Each volume is the covolume of the full group generated by reflections in the faces of one polyhedron. The orientation-preserving subgroup has twice that covolume. A classical Löbell manifold assembled from eight copies has eight times the listed volume [1].

(7)

The hyperbolic octahedron $A_3$ and dodecahedron $L_5$ use curvature $-1$; volumes of Euclidean Platonic solids use length or area normalizations instead.

(8)

The octahedron satisfies $\operatorname{Vol}(A_3)=4G$, where $G=\operatorname{Cl}_2(\pi/2)$ is Catalan's constant. The two families meet in volume at $\operatorname{Vol}(A_4)=\operatorname{Vol}(L_6)$, although these polyhedra are not isometric.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField
R = RealBallField(numberdb.bits(100, losing=80))
C = ComplexBallField(R.precision())
def Lambda(x):
    return (C(0, 2)*C(x)).exp().polylog(2).imag()/2
def volume(family, n):
    p = R.pi()
    if family == 'antiprism' and n >= 3:
        return 2*n*(Lambda(p/4+p/(2*n))+Lambda(p/4-p/(2*n)))
    if family == 'lobell' and n >= 5:
        t = (1/(2*(p/n).cos())).arcsin()
        return R(n)/2*(2*Lambda(t)+Lambda(t+p/n)+Lambda(t-p/n)+Lambda(p/2-2*t))
    raise ValueError('Use antiprism, n >= 3, or lobell, n >= 5')
volume('antiprism', 3)
volume('lobell', 5)

References

[1]

Taiyo Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebraic & Geometric Topology 8 (2008), 1523–1565. Section 4 and Theorem 4.1. (arXiv) (doi)

Links

[2]

D. D. Repovš and A. Yu. Vesnin, On a family of hyperbolic Brunnian links and their volumes (2024), Section 2

[3]

A. D. Mednykh and A. Yu. Vesnin, Löbell manifolds revised, Siberian Electronic Mathematical Reports 4 (2007), 605–609. Theorem 2.

Similar tables

Volumes of the hyperbolic Coxeter simplices — Volumes of simplex reflection chambers with the same curvature and full reflection-group covolume normalization.

Values of the Clausen functions at rational multiples of pi — The antiprism formula uses $\Lambda(\pi t)=\operatorname{Cl}_2(2\pi t)/2$ at rational $t$.

Volumes of Euclidean Platonic solids — Euclidean volumes of the Platonic solids, including the octahedron and dodecahedron.

Growth rates of Löbell reflection groups — The growth rates count distinct elements of the full face-reflection groups of the $L_n$ rows by minimum word length in the face reflections, using the same base index $n$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every $A_n$ with $3\leq n\leq30$ and every $L_n$ with $5\leq n\leq30$, covering consecutive small and moderate combinatorial sizes in both the ideal and compact families; the practical, noncanonical cutoff of 30 reaches 62 faces in each family and 60 or 120 vertices, respectively)

How they were obtained:

The generator evaluates the published volume formulas with Arb real and complex balls at 413 bits (100 decimal digits plus 80 guard bits), including the derived angles and principal dilogarithm. Every result is finite and positive; the worst relative radius is smaller than 10^(-122), so each stored 100-digit decimal interval contains its computed ball.

All 54 entries were compared with independent mpmath quadrature at 115 and 135 decimal digits. For antiprisms the checker integrates the log-sine definition; for Löbell polyhedra it uses the separate Schläfli integral of [3], Theorem 2, dividing the manifold volume by eight. Constant integration and two Catalan identities are controls, and the octahedron, dodecahedron and equality of the A_4 and L_6 volumes are additional checks. The quadrature is a numerical cross-check; the certified error bounds come from Arb.

The sign before Lambda(2 theta - pi/2) printed in equation (2) of [3] differs from Inoue's Theorem 4.1. The attached checks document that the printed plus sign contradicts the same paper's Theorem 2 integral and the known dodecahedron volume. This generator uses Inoue's minus sign.