Interior conformal radius of regular Euclidean and ideal hyperbolic polygons

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Numbers

geometry

$n$

$r(D,0)$

Euclidean

0.5660466803631597004496705504629791354940723413655169065007370945507907595693023060778393869923904321

Euclidean

0.7627597635018131880623259809636157932926292373480729152170715982644226929562561921954661469120229046

Euclidean

0.8514622702010604057102828548376625014003249113146211531772709852799538344536759065796590282875753249

Euclidean

0.8985430958715868687872100370824071082215485676608521340037788409756387069464986927149822523122311202

Euclidean

0.9264045228490812806683491751862562698171985521840274095746096811186983859581868783661248658395019356

Euclidean

0.9442171947845104184532388307498046067494740501878726200075767786528439932120410128723269191832694254

Euclidean

0.9562806525447942039591817398468131363848824993903474536587113217567247277484826952882400846996150110

Euclidean

10:

0.9648226302321035729356323479697867885883181563486882768034001700562682269004118048083412198010816627

Euclidean

11:

0.9710893442828940014586181613928079836777508828608042048716064186479150627653860253324544671182571167

Euclidean

12:

0.9758215093702986869909181466310022053283292454738148218590365467019478778118799627749999016355958603

Euclidean

13:

0.9794815610122843432854420648352036236504228297663183122388694456276379110427744470978598528976917685

Euclidean

14:

0.9823701786385645927498136828637484352560686743480356961504596516342730162428921101733392548252352478

Euclidean

15:

0.9846896641764671026877593917011759563861615359492838188234341369286952192374099644029977674561519688

Euclidean

16:

0.9865801547261162753160144611135150044926889327339060581620399257596000344789295753194793498142681949

Euclidean

17:

0.9881411969540614579060153196443071803150344803936614463828456166113658924244792089932703303148541475

Euclidean

18:

0.9894450710361057203655924654238993321298404551191554657457083507206550772489440033929560570639581217

Euclidean

19:

0.9905452805430341704205863084933622797955619237754975021313565129511028182047762654398603939098497394

Euclidean

20:

0.9914821125373944712741216890337792320776821278310935935411338848641608628236480177245604585844709139

Euclidean

21:

0.9922863683596216476836048755635047868613742183257728192300964267496170359119127324301417912202893721

Euclidean

22:

0.9929819216702745557677083474314208616053962104320744959011513570681893490067154691410564887481245326

Euclidean

23:

0.9935875067278473825895053739985370134496867005980126265166891532036855689768459271542338142577292607

Euclidean

24:

0.9941179905907459427297331716000688402349023196002063397983305783348354709744992242412420243352123869

Euclidean

25:

0.9945852926371868154597097223929271292626413621718584997338289651647955267349621899736005249363022705

Euclidean

26:

0.9949990588381847990997071444725275875655512090805944462684990229389744355846640676355764934149922983

Euclidean

27:

0.9953671627653791187639908734151849387624108328077548564604010884842330332182971427689973769217785208

Euclidean

28:

0.9956960823971119812371549800498854097039286709584667223357139468718048587707053232269501637552537300

Euclidean

29:

0.9959911866958598647276384320599287797929775136641636216984397434633657626079013248626522090706849311

Euclidean

30:

0.9962569558247591193038671166162228786195248747488629196083785742620075654950599544182824762761843337

ideal hyperbolic

0.3075099339099167894084781444906372839296260168755274529408395825566508414787374003073649255853659697

ideal hyperbolic

0.4569465810444636253749666225476833366117677300148315083943622472674843580708053855987331210088565639

ideal hyperbolic

0.5501870432773077729364615286577971688228682157313040393575663273602943178877886914465502688711933802

ideal hyperbolic

0.6150198678198335788169562889812745678592520337510549058816791651133016829574748006147298511707319394

ideal hyperbolic

0.6630600573937072666146993995962419521039116376389440360942224440270661096457804865052235279686702609

ideal hyperbolic

0.7002186781059915338894550821994734828740334120340562007714245520399301030862764283242842669008368871

ideal hyperbolic

0.7298766859369736445838719746526885999655402633539271382507445050013918352984867178647107502495980198

ideal hyperbolic

10:

0.7541264053760242060353440063990818146818579702658021893106529753065525105263991016427327716679487545

ideal hyperbolic

11:

0.7743392552574493613127432765031018104084279556006666262089549333684270521377501876270070118711987958

ideal hyperbolic

12:

0.7914546947139006595240509984575753873901499031703441206265571105763652251450307576859535548810736036

ideal hyperbolic

13:

0.8061393179618277504807854467268173713228305622305605378455327063888006194545028242289889199925610350

ideal hyperbolic

14:

0.8188799335574937299894550366873070199316815929643799787842854905632033530016159541764997385893145564

ideal hyperbolic

15:

0.8300407278820330507598234388631735710792581840235652010470183793271630743412629306341836035309486192

ideal hyperbolic

16:

0.8398997979331158783209192047799140438880708970017918783674930127392298387527969579611241733320506434

ideal hyperbolic

17:

0.8486732920121929219883275217661990736457319532171196684497339717169399432757261578241639064899854769

ideal hyperbolic

18:

0.8565318224301696435298641091843217381429621584904512138924818565256651898756251703837604504987922850

ideal hyperbolic

19:

0.8636119020896801206782137815796834378645339082529600829873164873683778264587410832949851475797094545

ideal hyperbolic

20:

0.8700240868988696381736590661057728043028076914593310363975976879986610401546448900283714135348901507

ideal hyperbolic

21:

0.8758588839614320974330244748952451406106160198376991764461720605256791483509347111790289569691220444

ideal hyperbolic

22:

0.8811911116389275634943515257564056016838959501134633620485298645354189666781546673655230733226180096

ideal hyperbolic

23:

0.8860831662358807174654452781114220449190562039890338662639866043641058031327193981671911263199043457

ideal hyperbolic

24:

0.8905875031526581586761650440436075453403999630094345739805227712249365324832141172220321902784922455

ideal hyperbolic

25:

0.8947485448915946719908866417919345498152949285213696861293775084495505300685049492243852549662372703

ideal hyperbolic

26:

0.8986041649746615508380802130746613652137021049441949451846172057734769719552517104851865319452822548

ideal hyperbolic

27:

0.9021868540265383795339117792268379220317449755340737776911590604473196271670097404881115653037176217

ideal hyperbolic

28:

0.9055246448479184828185571642691864248153502191755916849015793642698688176592416129105235626142125188

ideal hyperbolic

29:

0.9086418527538440039858232829370575975164074650389177851269647660024595535739455539449500245312387621

ideal hyperbolic

30:

0.9115596728958880296592092287529472483648417740100598432906574031799419656964041138930293286763560911

Definition

For the domain argument $D=P_n$ or $H_n$, the interior conformal radius is $r(D,0)=f'(0)>0$, where $f:\mathbb D\to D$ is the conformal bijection satisfying $f(0)=0$ [1].

Parameters

geometry

— polygon geometry

$n$

— number of sides ($n\geq3$)

Formulas

(1)

$\displaystyle r(P_n,0)=\frac{\Gamma(1-1/n)}{\Gamma(1+1/n)\Gamma(1-2/n)}=\frac{n}{B(1/n,1-2/n)}$, where $\Gamma$ is the Gamma function and $B(a,b)=\int_0^1t^{a-1}(1-t)^{b-1}\,dt$ is Euler's beta integral [3]. The Gamma expression follows from [1], equation (1.5), by the reflection and duplication formulas [4].

(2)

$\displaystyle r(H_n,0)=\frac{\Gamma(1-1/n)\Gamma(1/2+1/n)}{\Gamma(1+1/n)\Gamma(1/2-1/n)}=\frac{B(1/2+1/n,1/2)}{B(1/2-1/n,1/2)}$ [1], equation (1.6).

(3)

$\displaystyle \frac{r(P_n,0)^2}{A_n}=\frac{2^{4/n}}{\pi}r(H_n,0),\qquad A_n=\frac n2\sin\frac{2\pi}{n}$, where $A_n$ is the area of the regular polygon of circumradius one.

(4)

$\displaystyle r(P_4,0)=\frac{\sqrt2}{K},\qquad r(H_4,0)=\frac{\pi}{2K^2}$, where $K=$ $K(1/2)$ uses the elliptic parameter convention. Equivalently, $K=\pi/[2\operatorname{AGM}(1,1/\sqrt2)]$, where AGM denotes the arithmetic-geometric mean [2].

Comments

(5)

Write $\mathbb D=\{z:|z|<1\}$ and take vertices $e^{2\pi i j/n}$ for $0\leq j<n$. The Euclidean domain $P_n$ is the interior of their convex hull, with centre $0$ and circumradius $1$. The ideal hyperbolic domain $H_n$ contains $0$ and has sides formed by arcs orthogonal to $|z|=1$ joining adjacent vertices. The two geometries specify domains of the same conformal-radius function.

(6)

The conformal h-radius used in [1], equation (1.3), equals the ordinary conformal radius at the centre $0$. These conformal radii are not hyperbolic distances.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
R = RealBallField(461)
def radius(geometry, n):
    a = QQ(1) / n
    if geometry == "euclidean":
        return R(1-a).gamma() / (R(1+a).gamma()*R(1-2*a).gamma())
    if geometry == "ideal-hyperbolic":
        h = QQ(1) / 2
        return R(1-a).gamma()*R(h+a).gamma() / (R(1+a).gamma()*R(h-a).gamma())
    raise ValueError("Unknown polygon geometry")
values = {(geometry, n): radius(geometry, n)
          for geometry in ("euclidean", "ideal-hyperbolic")
          for n in range(3, 31)}

References

[1]

R. W. Barnard, P. Hadjicostas and A. Yu. Solynin, The Poincaré metric and isoperimetric inequalities for hyperbolic polygons, Transactions of the American Mathematical Society 357 (2005), 3905–3932, Theorems 1.2 and 1.3. (doi)

Links

[2]

DLMF 19.8: arithmetic-geometric mean and elliptic integrals

[3]

DLMF 5.12: Euler beta integral

[4]

DLMF 5.5: Gamma reflection and duplication

Similar tables

Area of the regular n-gon — Its circumradius-one branch gives the areas of the same Euclidean polygons.

Complete elliptic integral of the first kind — The lemniscatic value determines both square radii.

Values of the Gamma function at rational numbers — Gamma values at rational arguments determine every radius in these families.

Dirichlet eigenvalues of the classical planar domains — Its regular Euclidean polygons use the same circumradius-one normalization.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds both the Euclidean and ideal hyperbolic families for every side count $3\leq n\leq30$, with centre $0$ and vertices on the unit circle. The discrete index is the order of polygonal symmetry. This contiguous range includes familiar small symmetries and nonconstructible polygons; $30$ is a noncanonical editorial cutoff limiting depth as the domains approach the disk, while the two geometries provide breadth)

How they were obtained:

Computed with Arb real balls at 461 working bits, starting from exact rational Gamma arguments. The 128-bit guard left at least 455 bits of relative accuracy in every entry. Exact rational endpoint comparisons confirm that each serialized 100-significant-digit decimal contains the full computed ball within one unit of its last place.

Every value was independently compared with smooth beta-integral quadrature, without Gamma or beta-function evaluations, at 130 and 170 decimal digits. Three known beta-integral controls were checked first at each precision; the largest relative discrepancy from an Arb midpoint was less than 2.7e-131. Both square values were also checked with a separately iterated AGM and a rigorous bracket. The Euclidean area formula in [1], equation (1.5), was checked for every side count at 600 bits. These checks test the implementation; the error guarantee comes from Arb. The attached generator, checks and argument-selection note describe the calculation and selection.