hbeLabs.com

## Regular polygons (3 to 130 sides) - Lowest Dirichlet eigenvalue of the Laplacian

By: Robert S. Jones
rsjones7 at yahoo dot com
June, 2015

The lowest Dirichlet eigenvalues of the Laplacian for the regular polygons, with number of sides $S$ and area $\pi$, are presented in the following table. These are eigenvalue solutions of $$\nabla^2\Phi(k;\mathbf{r})+k^2\Phi(k;\mathbf{r})=0$$ subject to the boundary condition $$\Phi(k;\mathbf{r})=0\mbox{ for } \mathbf{r}\in\partial\Omega$$ where $\partial\Omega$ is the boundary of the regular polygon $\Omega$. The eigenvalue is $\lambda=k^2$.

The calculation becomes more and more difficult for increasing values of $S$. At $S=130$, the calculation takes about one day of CPU time (i7, quad-core, 8 threads, 8GB RAM). For low values of $S$, the convergence rate is quite good allowing me, for example, to calculate the the regular pentagon lowest Dirichlet eigenvalue to 1000 digits (which took about 2.7 days CPU time, after increasing the RAM to 16 GB).

Upper and lower bounds are presented such that all of the eigenvalues have a relative error of $\epsilon<10^{-30}$. The notation is such that if $\lambda=1.23_{34}^{43}$ then $1.2334<\lambda<1.2343$.

The last two columns list the approximate number of correct digits in the estimate $\tilde\lambda_a$ using the asymptotic formulas (see below, here $a=4,5,6$ is the highest order of the expansion). That number is $\widetilde{D}_a=-\log_{10}(|\tilde\lambda_a-\lambda|/\lambda)$ where $\lambda$ is the exact value.

The first two, the equilateral triangle and the square, of course, have closed-form results, $\lambda(3)=4\,\pi/\sqrt{3}$ and $\lambda(4)=2\,\pi$

The last row is the unit-radius circle eigenvalue, $j_{0,1}^2$.

Two text files with these 30-digit results for the pentagon to the 130-sided, $\pi$-area regular polygon are provided here:

$S$$\lambda \epsilon \widetilde{D}_4 \widetilde{D}_5 \widetilde{D}_6 37.25519745693687140237631303056862 1.215$$ 1.226$$1.988 46.28318530717958647692528676655901 1.982$$ 1.999$$4.068 56.02213793204263387829800871005_{402}^{592}$$3.13\times 10^{-31}$$2.562$$ 2.585$$3.786 65.9174178316136612156885745768_{3851}^{4127}$$4.66\times 10^{-31}$$3.032$$ 3.059$$4.155 75.8664493126559858577124749417_{5839}^{6058}$$3.71\times 10^{-31}$$3.427$$ 3.459$$4.532 85.83849143359244285051664037956_{027}^{492}$$7.94\times 10^{-31}$$3.768$$ 3.804$$4.882 95.82182680227026573173554644371_{507}^{762}$$4.36\times 10^{-31}$$4.069$$ 4.109$$5.202 105.81126035921911602278881646881_{013}^{374}$$6.20\times 10^{-31}$$4.337$$ 4.381$$5.494 115.80423063671740072187839445285_{306}^{754}$$7.70\times 10^{-31}$$4.579$$ 4.628$$5.762 125.7993698043565000793150253110_{0613}^{1122}$$8.77\times 10^{-31}$$4.800$$ 4.852$$6.009 135.79590026685601470979077106333_{349}^{895}$$9.41\times 10^{-31}$$5.003$$ 5.059$$6.239 145.79335700527119455327322707868_{188}^{752}$$9.71\times 10^{-31}$$5.191$$ 5.251$$6.452 155.7914500106515799756938484981_{4604}^{5169}$$9.74\times 10^{-31}$$5.365$$ 5.429$$6.652 165.7899918999902085343497522138_{2606}^{3161}$$9.57\times 10^{-31}$$5.528$$ 5.596$$6.840 175.7888578719811046986171966351_{8657}^{9195}$$9.27\times 10^{-31}$$5.682$$ 5.753$$7.017 185.78796259185784686421256838015_{194}^{509}$$5.42\times 10^{-31}$$5.826$$ 5.901$$7.184 195.78724635138196124300803664483_{173}^{663}$$8.45\times 10^{-31}$$5.962$$ 6.041$$7.343 205.78666651414037221353091296202_{392}^{856}$$8.00\times 10^{-31}$$6.091$$ 6.174$$7.494 215.7861920775968442730282037573_{2687}^{3124}$$7.54\times 10^{-31}$$6.214$$ 6.300$$7.637 225.78580012942836502757458604434_{176}^{587}$$7.08\times 10^{-31}$$6.331$$ 6.420$$7.775 235.785473486454901632048264070_{19447}^{20012}$$9.76\times 10^{-31}$$6.442$$ 6.536$$7.906 245.7851990897900240918344636129_{7661}^{8182}$$8.99\times 10^{-31}$$6.549$$ 6.646$$8.032 255.78496689413042350141867068388_{314}^{794}$$8.29\times 10^{-31}$$6.651$$ 6.752$$8.153 265.78476908631484297799227471769_{006}^{450}$$7.65\times 10^{-31}$$6.749$$ 6.853$$8.270 275.78459952723648464059322282715_{248}^{815}$$9.79\times 10^{-31}$$6.844$$ 6.951$$8.382 285.784453347751719951196794842_{39684}^{40203}$$8.95\times 10^{-31}$$6.935$$ 7.045$$8.490 295.78432665236541120738029338594_{357}^{832}$$8.19\times 10^{-31}$$7.022$$ 7.136$$8.594 305.78421629939226404411903673407_{022}^{458}$$7.52\times 10^{-31}$$7.107$$ 7.224$$8.695 315.784119736080032703344528384_{59653}^{60185}$$9.18\times 10^{-31}$$7.189$$ 7.309$$8.793 325.7840348737024443183305074869_{2753}^{3239}$$8.38\times 10^{-31}$$7.268$$ 7.392$$8.888 335.7839599920405088123350325226_{2669}^{3114}$$7.66\times 10^{-31}$$7.344$$ 7.471$$8.979 345.78389366569480925203347656871_{102}^{629}$$9.10\times 10^{-31}$$7.418$$ 7.549$$9.069 355.78383470677098820284070000522_{066}^{547}$$8.30\times 10^{-31}$$7.490$$ 7.624$$9.155 365.7837821199558806276999199658_{4812}^{5374}$$9.70\times 10^{-31}$$7.560$$ 7.697$$9.239 375.78373506704984629196244063652_{200}^{712}$$8.83\times 10^{-31}$$7.628$$ 7.768$$9.321 385.7836928387732922677065179222_{5965}^{6432}$$8.06\times 10^{-31}$$7.694$$ 7.837$$9.401 395.78365483221087114338620791091_{180}^{716}$$9.24\times 10^{-31}$$7.759$$ 7.905$$9.479 405.7836205326559735769513685588_{8691}^{9180}$$8.43\times 10^{-31}$$7.821$$ 7.970$$9.555 415.7835894989127288572435410875_{3914}^{4468}$$9.56\times 10^{-31}$$7.882$$ 8.035$$9.629 425.7835613513319606799637449504_{6674}^{7179}$$8.72\times 10^{-31}$$7.942$$ 8.097$$9.701 435.78353576202197197127744676206_{295}^{862}$$9.78\times 10^{-31}$$8.000$$ 8.158$$9.771 445.7835124467992680334748033579_{8534}^{9051}$$8.92\times 10^{-31}$$8.056$$ 8.218$$9.840 455.78349115853885630291385887880_{169}^{744}$$9.91\times 10^{-31}$$8.112$$ 8.276$$9.908 465.78347168165616811010513722523_{471}^{995}$$9.05\times 10^{-31}$$8.166$$ 8.333$$9.974 475.78345382750846329737964591391_{342}^{920}$$9.98\times 10^{-31}$$8.219$$ 8.389$$10.038 485.78343743054686541925063606475_{348}^{877}$$9.12\times 10^{-31}$$8.271$$ 8.444$$10.101 495.78342234508394017728694551658_{365}^{944}$$9.99\times 10^{-31}$$8.321$$ 8.497$$10.163 505.78340844256821299478011619632_{077}^{606}$$9.14\times 10^{-31}$$8.371$$ 8.550$$10.224 515.78339560927790344216639813994_{232}^{718}$$8.38\times 10^{-31}$$8.419$$ 8.601$$10.284 525.7833837443627027000195590153_{1956}^{2483}$$9.11\times 10^{-31}$$8.467$$ 8.652$$10.342 535.7833727581755982440480496612_{8572}^{9056}$$8.36\times 10^{-31}$$8.514$$ 8.701$$10.399 545.78336257084729274289731414400_{023}^{546}$$9.04\times 10^{-31}$$8.559$$ 8.750$$10.455 555.78335311106423638564481084216_{098}^{579}$$8.31\times 10^{-31}$$8.604$$ 8.797$$10.511 565.7833443150181293546384478158_{8517}^{9035}$$8.94\times 10^{-31}$$8.648$$ 8.844$$10.565 575.78333612550029210309036935466_{229}^{706}$$8.22\times 10^{-31}$$8.691$$ 8.890$$10.618 585.7833284911188091539136722748_{1900}^{2410}$$8.82\times 10^{-31}$$8.734$$ 8.935$$10.670 595.78332136562003385373793961146_{241}^{711}$$8.12\times 10^{-31}$$8.776$$ 8.980$$10.722 605.78331470729905942821079780018_{166}^{669}$$8.67\times 10^{-31}$$8.817$$ 9.024$$10.772 615.7833084784862442505710587362_{1718}^{2253}$$9.23\times 10^{-31}$$8.857$$ 9.066$$10.822 625.78330264509892841017038059784_{138}^{705}$$9.80\times 10^{-31}$$8.896$$ 9.109$$10.871 635.7832971762491756994220505872_{0528}^{1051}$$9.02\times 10^{-31}$$8.935$$ 9.150$$10.919 645.7832920438997850274611258286_{1636}^{2189}$$9.55\times 10^{-31}$$8.974$$ 9.191$$10.967 655.7832872225619902161013793189_{5601}^{6111}$$8.81\times 10^{-31}$$9.011$$ 9.232$$11.014 665.7832826890292492111478490456_{5885}^{6424}$$9.30\times 10^{-31}$$9.048$$ 9.271$$11.060 675.7832784221423469802979701702_{2974}^{3472}$$8.59\times 10^{-31}$$9.085$$ 9.310$$11.105 685.7832744025817283876486673365_{4675}^{5199}$$9.04\times 10^{-31}$$9.121$$ 9.349$$11.150 695.78327061268356062546650679310_{308}^{858}$$9.50\times 10^{-31}$$9.156$$ 9.387$$11.194 705.7832670362765177201049535052_{8863}^{9439}$$9.95\times 10^{-31}$$9.191$$ 9.424$$11.237 715.7832636585366972675053570571_{5508}^{6041}$$9.20\times 10^{-31}$$9.225$$ 9.461$$11.280 725.78326046585843427039079456440_{041}^{599}$$9.63\times 10^{-31}$$9.259$$ 9.497$$11.322 735.78325744573907894604087036089_{114}^{630}$$8.92\times 10^{-31}$$9.293$$ 9.533$$11.363 745.7832545866760630843215845592_{6763}^{7303}$$9.31\times 10^{-31}$$9.325$$ 9.569$$11.404 755.78325187807479995190728441292_{112}^{674}$$9.70\times 10^{-31}$$9.358$$ 9.604$$11.445 765.7832493101661516716346291929_{1870}^{2391}$$9.00\times 10^{-31}$$9.390$$ 9.638$$11.485 775.78324687393236029853061852313_{162}^{704}$$9.36\times 10^{-31}$$9.421$$ 9.672$$11.524 785.7832445610404785123055631187_{1525}^{2088}$$9.72\times 10^{-31}$$9.453$$ 9.705$$11.563 795.7832423637824563386561208792_{0871}^{1395}$$9.03\times 10^{-31}$$9.483$$ 9.739$$11.602 805.78324027502114444346026329033_{075}^{617}$$9.36\times 10^{-31}$$9.514$$ 9.771$$11.640 815.78323828814156470878881891374_{106}^{668}$$9.70\times 10^{-31}$$9.543$$ 9.804$$11.677 825.7832363970068770166311668301_{4963}^{5486}$$9.02\times 10^{-31}$$9.573$$ 9.836$$11.714 835.78323459591853914193516915928_{038}^{579}$$9.33\times 10^{-31}$$9.602$$ 9.867$$11.751 845.78323287958021583661957400684_{188}^{746}$$9.64\times 10^{-31}$$9.631$$ 9.898$$11.787 855.7832312430650447977618691596_{8600}^{9176}$$9.95\times 10^{-31}$$9.659$$ 9.929$$11.823 865.78322968178591229998729418953_{020}^{557}$$9.26\times 10^{-31}$$9.687$$ 9.959$$11.858 875.78322819146843072380099753098_{290}^{843}$$9.55\times 10^{-31}$$9.715$$ 9.989$$11.893 885.7832267681263447879041212711_{1715}^{2285}$$9.83\times 10^{-31}$$9.743$$10.019$$11.927 895.78322540803912364438644198743_{300}^{831}$$9.17\times 10^{-31}$$9.770$$10.049$$11.962 905.78322410773152267822276798164_{137}^{683}$$9.43\times 10^{-31}$$9.796$$10.078$$11.995 915.783222863954922345132559440_{09855}^{10417}$$9.69\times 10^{-31}$$9.823$$10.106$$12.029 925.7832216736702720961404775421_{0778}^{1354}$$9.95\times 10^{-31}$$9.849$$10.135$$12.062 935.7832205340324857277389860275_{8692}^{9230}$$9.29\times 10^{-31}$$9.875$$10.163$$12.094 945.783219442376150669518408021_{29845}^{30398}$$9.53\times 10^{-31}$$9.901$$10.191$$12.127 955.78321839620242804116261639329_{040}^{606}$$9.77\times 10^{-31}$$9.926$$10.218$$12.159 965.783217393167033007010852241_{39585}^{40114}$$9.13\times 10^{-31}$$9.951$$10.245$$12.190 975.7832164310691962277585187415_{2773}^{3315}$$9.36\times 10^{-31}$$9.976$$10.272$$12.222 985.78321550784151722801068379624_{253}^{808}$$9.57\times 10^{-31}$$10.000$$10.299$$12.253 995.7832146215406294155617881961_{5703}^{6270}$$9.79\times 10^{-31}$$10.024$$10.326$$12.283 1005.7832137703386044343671448760_{6733}^{7265}$$9.17\times 10^{-31}$$10.048$$10.352$$12.314 1015.7832129525150306224265566440_{5557}^{6099}$$9.37\times 10^{-31}$$10.072$$10.378$$12.344 1025.7832121664497066780323274791_{2494}^{3048}$$9.57\times 10^{-31}$$10.096$$10.403$$12.374 1035.78321141061589730038223792298_{352}^{917}$$9.76\times 10^{-31}$$10.119$$10.429$$12.403 1045.7832106835741026399496287805_{6681}^{7258}$$9.96\times 10^{-31}$$10.142$$10.454$$12.432 1055.7832099839662979373801765921_{2700}^{3241}$$9.34\times 10^{-31}$$10.165$$10.479$$12.461 1065.7832093105106038060342151456_{8977}^{9528}$$9.52\times 10^{-31}$$10.187$$10.503$$12.490 1075.7832086619963512745011282308_{8693}^{9255}$$9.70\times 10^{-31}$$10.209$$10.528$$12.518 1085.78320803727950899717733132378_{083}^{655}$$9.87\times 10^{-31}$$10.232$$10.552$$12.546 1095.7832074352784430036203824301_{5931}^{6468}$$9.27\times 10^{-31}$$10.254$$10.576$$12.574 1105.78320685496998202644834606904_{166}^{713}$$9.43\times 10^{-31}$$10.275$$10.600$$12.602 1115.7832062953857638544958683846_{8888}^{9402}$$8.87\times 10^{-31}$$10.297$$10.623$$12.629 1125.783205755608840330598899278_{69493}^{70058}$$9.75\times 10^{-31}$$10.318$$10.646$$12.656 1135.7832052347705205764174624901_{3556}^{4130}$$9.91\times 10^{-31}$$10.339$$10.670$$12.683 1145.7832047320474338019919255227_{7639}^{8179}$$9.32\times 10^{-31}$$10.360$$10.693$$12.710 1155.7832042466587946646835703161_{8765}^{9314}$$9.46\times 10^{-31}$$10.381$$10.715$$12.736 1165.78320377786385559803797727272_{276}^{833}$$9.61\times 10^{-31}$$10.401$$10.738$$12.763 1175.7832033249595318512911938361_{7560}^{8084}$$9.05\times 10^{-31}$$10.422$$10.760$$12.788 1185.7832028872781861783972732175_{4920}^{5452}$$9.18\times 10^{-31}$$10.442$$10.782$$12.814 1195.7832024641855612037927177369_{4957}^{5497}$$9.31\times 10^{-31}$$10.462$$10.804$$12.840 1205.7832020550788484815190743450_{7680}^{8227}$$9.44\times 10^{-31}$$10.482$$10.826$$12.865 1215.78320165938488416452948767025_{078}^{632}$$9.57\times 10^{-31}$$10.501$$10.848$$12.890 1225.78320127655846202071020955606_{248}^{810}$$9.69\times 10^{-31}$$10.521$$10.869$$12.915 1235.78320090608075527914348181390_{019}^{588}$$9.81\times 10^{-31}$$10.540$$10.890$$12.940 1245.7832005474578394714033936689_{3655}^{4230}$$9.93\times 10^{-31}$$10.559$$10.911$$12.964 1255.78320020021930905447076146118_{363}^{906}$$9.37\times 10^{-31}$$10.578$$10.932$$12.989 1265.7831998639169811697955997275_{4738}^{5287}$$9.48\times 10^{-31}$$10.597$$10.953$$13.013 1275.7831995381236804121745520138_{6591}^{7147}$$9.59\times 10^{-31}$$10.615$$10.973$$13.037 1285.78319922243209895698523832013_{030}^{555}$$9.06\times 10^{-31}$$10.634$$10.994$$13.060 1295.78319891645372682901545245421_{112}^{643}$$9.16\times 10^{-31}$$10.652$$11.014$$13.084 1305.78319861981784749432269771828_{013}^{587}$$9.91\times 10^{-31}$$10.671$$11.034$$13.107 \infty$$5.783185962946784521175995758455...$------------

Estimates of these eigenvalues can be obtained using the asymptotic formula: $$\tilde\lambda_4=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\mathcal{O}\left(\frac{1}{S^5}\right)\right]$$ $$\tilde\lambda_5=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\frac{\left(12-2j_{0,1}^2\right)\zeta(5)}{S^5} +\mathcal{O}\left(\frac{1}{S^6}\right)\right]$$ $$\tilde\lambda_6=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\frac{\left(12-2\,j_{0,1}^2\right)\zeta(5)}{S^5} +\frac{\left(8+4\,j_{0,1}^2\right)\zeta^2(3)}{S^6} +\mathcal{O}\left(\frac{1}{S^7}\right) \right]$$ where $j_{0,1}$ is the first root of the Bessel function $J_0(x)$, and $\zeta(x)$ is the Riemann-Zeta function. This formula, which was not used to calculate my results (second column), does yield increasingly better results, achieving 30-digit precision for $S>20,000$. ($S>10^{30/7}$)

Thank you Mark Boady for offering asymptotic terms beyond the fourth order.

As an aside, the over 30-digit result for the lowest Dirichlet eigenvalue of the famous but somewhat obscure 65537-sided regular polygon (widipedia 65537-gon), with area $\pi$, is very close to $$\frac{\lambda_1}{j_{01}^2} = 1.000000000000017081474070818627484...$$ $$\lambda_1 = 5.783185962946883306517068556211370...$$ where the above formula for $\tilde\lambda_6$ above simply used, and there is some uncertainty in the final one or two digits.