Resistance of infinite triangular and honeycomb lattices

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Numbers

$L$, $s$, $m$, $n$

$r_L(A_0,s_{m,n})$

triangular, A, 1, 0:

1/3

comment: Adjacent vertices: $t(1,0)=1/3$.

honeycomb, B, 0, 0:

2/3

comment: Adjacent vertices: $h_B(0,0)=2/3$.

triangular, A, 2, 0:

0.4613510849794984686213606734147889010771207452001040554692025855025454527895874873051465513739831098

triangular, A, 2, 1:

0.4359911241769174323559863299592722161281062940666146389320653739153939402718729230140933909796751117

honeycomb, A, 1, 0:

comment: Second-neighbor vertices: $h_A(1,0)=1$.

triangular, A, 3, 0:

0.5362130197539816234563280809774668129254489424012486656304310260305454334750498476617586164877973180

triangular, A, 3, 1:

0.5132889542179204951132649831296944139738648036664065279936602029103030346926979484038002882317088921

honeycomb, B, 0, 1:

1.102657790843584099022652996625938882794772960733281305598732040582060606938539589680760057646341778

honeycomb, B, 1, 1:

1.230675542489749234310680336707394450538560372600052027734601292751272726394793743652573275686991555

triangular, A, 4, 0:

0.5891518971297856069904942780704461507969043280145677656883619703563633905422482227205171923576353770

triangular, A, 4, 1:

0.5700986139893823374497917840600884988654694869300035583188505972518788440665337395686436893658738186

triangular, A, 4, 2:

0.5627909281898226136828580472368556408731785497340617216177514318511515028604457444693591929512151022

honeycomb, A, 2, 0:

1.384053254938495405864082020244366703231362235600312166407607756507636358368762461915439654121949330

honeycomb, A, 2, 1:

1.307973372530752297067958989877816648384318882199843916796196121746181820815618769042280172939025335

triangular, A, 5, 0:

0.6301777984807299896464781384744651644386001735102276309777286876606030756320617520824706691046200543

triangular, A, 5, 1:

0.6141616682826186769702879404252839648432189391579615904421554569716370730066551287207932449108236263

triangular, A, 5, 2:

0.6049852163621650084855173745207840097802013586057750785407434953912726298221055454356336012560625959

honeycomb, B, 1, 2:

1.410631163374336396090611986503755531179091842933125222394928162328242427754158358723040230585367114

honeycomb, B, 2, 1:

1.462569032612758422582516296218661044075835901399427694919385779736000009657268819821693967443092896

honeycomb, B, 2, 2:

1.510853058951400587190953737521950127976434491267759249093293814443393920957335283370705456093489320

triangular, A, 6, 0:

0.6636899372645523846754806232403457129088943339715741313379117540005444993345376861237735923622198297

triangular, A, 6, 1:

0.6499508099624022961361590243427155966196822503796503539778937438317182089511371731044733700431089333

triangular, A, 6, 2:

0.6406056737712468231132870430832043685584560375075451551493542473379862890258443394130209473376409787

triangular, A, 6, 3:

0.6372691975777545912113620577984875757606837796993033671756057435437094401510487750063509317147594536

honeycomb, A, 3, 0:

1.608639059261944870368984242932400438776346827203745996891293078091636300425149542985275849463391954

honeycomb, A, 3, 1:

1.539866862653761485339794949389083241921594410999219583980980608730909104078093845211400864695126676

triangular, A, 7, 0:

0.6920215369166247689934749618571506126446652017347105630195448274155728302952877019586131909968157900

triangular, A, 7, 1:

0.6800193341325774785703062952125136535653680639126029430372047606318273360888011582463054769928326337

triangular, A, 7, 2:

0.6710504478426884238411141056204807154035559542179906729230075563877109806189229740404762895505164770

triangular, A, 7, 3:

0.6662167025998519420352817557914743489433939429845898678367194879018694016051964401703982465505747863

honeycomb, B, 1, 3:

1.589368836625663603909388013496244468820908157066874777605071837671757572245841641276959769414632886

honeycomb, B, 2, 3:

1.619600587961284456019384848167249725764783232997658751942941826192727312234281535634202594085380029

honeycomb, B, 3, 2:

1.646178496397125446245914814426638553712512840330471807930262232013333381619677432441803170548797813

honeycomb, B, 3, 3:

1.695463530873149567896614143108001462587822757345819989637643593638787668083831809950919498211306514

triangular, A, 8, 0:

0.7165625589092653982389269493260198725794992134791068015438372847818311676290472244387654511268086719

triangular, A, 8, 1:

0.7059190285303879369529148040757553916244307675661882795805552022237033213152694923482645742531004854

triangular, A, 8, 2:

0.6974842442788090608226942521386338921909798756055036573843154799317141760176519219022318447512342869

triangular, A, 8, 3:

0.6920259223112429423689562631543424325434555549180520601525576186911504720249068114064278516277072429

triangular, A, 8, 4:

0.6901322714963269296416893093008566524508183885800580837830722735487898262052593009857152125222497793

honeycomb, A, 4, 0:

1.767455691389356820971482834211338452390712984043703297065085911069090171626744668161551577072906131

honeycomb, A, 4, 1:

1.710295841968147012349375352180265496596408460790010674956551791755636532199601218705931068097621456

honeycomb, A, 4, 2:

1.688372784569467841048574141710566922619535649202185164853254295553454508581337233408077578853645307

triangular, A, 9, 0:

0.7382088751562142960515759894087399012567028681482427280540638383303069038093497957150892021161083377

triangular, A, 9, 1:

0.7286534421609887252413405682693589691633828379296554795141523192488503145195534321311805825708634665

triangular, A, 9, 2:

0.7207730547840621898507457976508553496026894127355502329842765413324241033412755154124908990800480633

triangular, A, 9, 3:

0.7151176780718953933521282471178558104063795002826377556282912003234688427167355799594259770032008191

triangular, A, 9, 4:

0.7121541895778859045208299089567531758656056678375323233599397140533496049856746513803195393884673087

honeycomb, B, 2, 4:

1.737874758541369959618167205817728149518849395269840358477345524494302976749085029473636483573151517

honeycomb, B, 3, 4:

1.773412179401786621410574002555818614505592754102532914449368024579879307615437091009954126634332822

honeycomb, B, 4, 2:

1.772761360914152630653892796278423660433581266945611878699238422633696762504656835340626395463340294

honeycomb, B, 4, 3:

1.789245498634166022905597099006156473488889784693740227301749549614788546895294413725070535532760041

honeycomb, B, 4, 4:

1.833491363893134273607260356970195280078723440682756987108246114988603539180965103523781106373079058

triangular, A, 10, 0:

0.7575719724627282524622775312799456309810380781710696986457478530163540919103331982350656656043342952

triangular, A, 10, 1:

0.7489059176216573375627851596538779828264571206899844545532466268429777671388057436771714654118900630

triangular, A, 10, 2:

0.7415512179643451927910947095009053170905176449588603803689344219818586238835728211953019034372251779

triangular, A, 10, 3:

0.7359127176979468299443022048026227171404458500704558454294106242849493415948698449385405124646641438

triangular, A, 10, 4:

0.7323559397814254326052410560039272950951006405287324144592472236472253583360347347165452147070838691

triangular, A, 10, 5:

0.7311391362285388246361346692067836888322742548781813027765302540561135256454368298334834410820948333

honeycomb, A, 5, 0:

1.890533395442189968939434415423395493315800520530682892933186062981809226896185256247412007313860163

honeycomb, A, 5, 1:

1.842485004847856030910863821275851894529656817473884771326466370914911219019965386162379734732470879

honeycomb, A, 5, 2:

1.814955649086495025456552123562352029340604075817325235622230486173817889466316636306900803768187788

triangular, A, 11, 0:

0.7750879238483930448886933027673460339462584901257898125361949420307414097034322196659092425781014748

triangular, A, 11, 1:

0.7671616002642237493539127880979159425151764346862083710888675420255603518358028433375263090540079163

triangular, A, 11, 2:

0.7602883977214438094765093713664021359519248602458700696477137864549363168742223714488651296888011847

triangular, A, 11, 3:

0.7547737777999722646708851552623147478582057880816462000848066337270138998327100195635628314070841460

triangular, A, 11, 4:

0.7509046377859805063957182632798777827897821138353080890509077246977052214588903987839162491533427873

triangular, A, 11, 5:

0.7489072793263051367823330426596705955361164562682791705104038244677656136146011034035855691507333222

honeycomb, B, 2, 5:

1.847239630302084608182396806840055595321086496910853524257092734326254699795259865877618134226884645

honeycomb, B, 3, 5:

1.859752558416030508569092358029272343181876335271281824132253199700895991854605013569447793504527201

honeycomb, B, 4, 5:

1.894290276725750962752925103242464725901501363047839575397777888463958357589854053907737284058552614

honeycomb, B, 5, 3:

1.882860087711166422810166475402475954099341175812623600865703486272968358998998659855005480308463028

honeycomb, B, 5, 4:

1.904718152016267796219734007851203930021357227045157099569403448141341570983636641238287562291573538

honeycomb, B, 5, 5:

1.943818545707684670458117786057526473967176757861452116293534185492865783917736611310717631509948817

triangular, A, 12, 0:

0.7910786782975218380278566661022482314966678115484283966072329290389543752214906692816111575752680448

triangular, A, 12, 1:

0.7837768459008303400671000215130252280847460908314120188932268930390096437085818819018630070904953381

triangular, A, 12, 2:

0.7773385440302897116661113420068986433006339680531241722570751507693428816794416450962833439504251418

triangular, A, 12, 3:

0.7719993286481746008142670736765001821205582050053968395333523433828643768750011558233449248721746632

triangular, A, 12, 4:

0.7679863669819426486034193089475803520560060543739859764785209015597695183097035251914082688262969190

triangular, A, 12, 5:

0.7654917451282907257681288120031509890528178936887489273430571404995075970654231648898550983641943239

triangular, A, 12, 6:

0.7646449377072901945064424128046132219106073784104251189222767794382763655672203887941278424469507973

honeycomb, A, 6, 0:

1.991069811793657154026441869721037138726683001914722394013735262001633498003613058371320777086659489

honeycomb, A, 6, 1:

1.949852429887206888408477073028146789859046751138951061933681231495154626853411519313420110129326800

honeycomb, A, 6, 2:

1.921817021313740469339861129249613105675368112522635465448062742013958867077533018239062842012922936

honeycomb, A, 6, 3:

1.911807592733263773634086173395462727282051339097910101526817230631128320453146325019052795144278361

Definition

For network $L$ and destination vertex $s_{m,n}$, the normalized effective resistance is $r_L(A_0,s_{m,n})=R_{\mathrm{eff}}(A_0,s_{m,n})/R_0$, where every edge has resistance $R_0>0$ [1].

Parameters

$L$

— network ($L$ is triangular or honeycomb.)

$s$

— destination sublattice ($s=A$ for the triangular network; $s=A$ or $B$ for the honeycomb network.)

$m$

— first cell coordinate (The integers $m,n$ satisfy the representative conventions in (8); coincident endpoints are excluded.)

$n$

— second cell coordinate (The integers $m,n$ satisfy the representative conventions in (8).)

Formulas

(1)

$h_A(m,n)=3t(m,n)$ and $h_B(m,n)=t(m,n)+t(m+1,n)+t(m+1,n+1)$ [1], equations (77)-(78).

(2)

$t(m,n)=\dfrac1\pi\int_0^{\pi/2}\dfrac{1-e^{-|m-n|\sigma(y)}\cos((m+n)y)}{\sinh\sigma(y)\cos y}\,dy$, where $\cosh\sigma(y)=2/\cos y-\cos y$ and $\sigma(y)\geq0$ [3], equation (62). The integrand has limiting value $|m-n|$ at $y=0$.

(3)

$\begin{aligned}6t(m,n)&=t(m+1,n)+t(m-1,n)+t(m,n+1)\\&\quad +t(m,n-1)+t(m+1,n+1)+t(m-1,n-1)\end{aligned}$ for $(m,n)\ne(0,0)$. On honeycomb, $3h_A(m,n)=h_B(m,n)+h_B(m-1,n)+h_B(m-1,n-1)$ for $(m,n)\ne(0,0)$, and $3h_B(m,n)=h_A(m,n)+h_A(m+1,n)+h_A(m+1,n+1)$ for all integers $m,n$. These are the harmonic equations of the declared networks.

(4)

Let $D(m,n)=\max(|m|,|n|,|m-n|)$. The triangular graph distance is $D(m,n)$; the honeycomb distances from $A_0$ are $d_A(m,n)=2D(m,n)$ and $d_B(m,n)=1+2\min\{D(m,n),D(m+1,n),D(m+1,n+1)\}$. Every two-edge path between honeycomb $A$ vertices makes one triangular step, and the three $A$ neighbors of $B_{m,n}$ determine $d_B$.

(5)

$t(1,0)=1/3$, $t(2,0)=8/3-4\sqrt3/\pi$, $t(2,1)=-2/3+2\sqrt3/\pi$, and $t(3,1)=-5+10\sqrt3/\pi$ [4], equations (3)-(6). In particular, $h_B(0,0)=2/3$ and $h_A(1,0)=1$.

Comments

(6)

Put $a_1=(1,0)$, $a_2=(-1/2,\sqrt3/2)$ and $u=ma_1+na_2$. The triangular vertices are $A_u=u$, with edges having coordinate differences $\pm(1,0)$, $\pm(0,1)$ and $\pm(1,1)$. The honeycomb vertices are $A_u=u$ and $B_u=u+\delta$, where $\delta=(2a_1+a_2)/3$; join $A_u$ to $B_u$, $B_{u-a_1}$ and $B_{u-a_1-a_2}$ [1], equation (64). The coordinates label vertices; the geometric edge lengths do not change their common resistance.

(7)

The infinite-network value is the limit of periodic quotients with the two endpoints held fixed [2]. The source is $A_0$. Triangular vertices have only the label $A$; the honeycomb label $s=A$ or $B$ specifies the destination basis vertex. Write $t(m,n)$ for triangular resistance and $h_A(m,n)$ or $h_B(m,n)$ for honeycomb resistance. In units of $R_0$, the positive combinatorial Laplacian is $Lf(v)=\deg(v)f(v)-\sum_{w\sim v}f(w)$, with degrees six and three respectively. For unit resistors, the entries are the numerical resistances in ohms.

(8)

For triangular and honeycomb $A$ destinations, choose the unique representative with $m\geq2n\geq0$ and $m>0$. Triangular rotations and reflections give this wedge [4], equations (49)-(51); for honeycomb $A$ destinations, translation and endpoint exchange supply inversion in addition to the symmetries fixing $A_0$. For honeycomb $B$ destinations, choose the lexicographically greatest pair among $C^j(m,n)$ and $SC^j(m,n)$ for $j=0,1,2$, where $C(m,n)=(-n-1,m-n)$ and $S(m,n)=(m,m-n)$. These are the 120-degree rotation about $A_0$ and reflection through the line $A_0B_0$.

(9)

Entries are ordered by increasing graph distance, then triangular before honeycomb, then $A$ before $B$, then increasing $m,n$. The seed $t(0,0)=h_A(0,0)=0$ is excluded because the endpoints coincide. The vertex $B_0$ is distinct from $A_0$, so $h_B(0,0)$ is included.

(10)

The square-network table uses the same normalization and periodic limit. In these planar networks, the separate origin Green-function integrals diverge, while their resistance differences converge. The three-dimensional Watson integrals instead converge at the origin. Kirchhoff indices sum pair resistances on finite connected graphs.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 120

def triangular(m, n):
    # Numerical quadrature; the attached generator supplies certified bounds.
    d, s = abs(m-n), m+n
    def integrand(y):
        if not y:
            return mp.mpf(d)
        if y == mp.pi/2:
            return (1-mp.cos(s*y))/2 if d == 0 else mp.mpf(1)/2
        c = mp.cos(y)
        x = 2*mp.asinh(mp.sin(y/2)*mp.sqrt((c+2)/c))
        decay = mp.exp(-d*x)
        return (-mp.expm1(-d*x)+2*decay*mp.sin(s*y/2)**2)/(mp.sin(y)*mp.sqrt(3+mp.sin(y)**2))
    return mp.quad(integrand, [0, mp.pi/4, mp.pi/2])/mp.pi

def resistance(lattice, sublattice, m, n):
    if lattice == 'triangular' and sublattice == 'A':
        return triangular(m, n)
    if lattice == 'honeycomb' and sublattice == 'A':
        return 3*triangular(m, n)
    if lattice == 'honeycomb' and sublattice == 'B':
        return triangular(m, n)+triangular(m+1, n)+triangular(m+1, n+1)
    raise ValueError('Unknown network or destination sublattice')

print(mp.nstr(resistance('honeycomb', 'B', 0, 1), 100))

References

[1]

József Cserti, Application of the lattice Green’s function for calculating the resistance of infinite networks of resistors, American Journal of Physics 68 (2000), 896–906. (arXiv) (doi)

[2]

József Cserti, Gábor Széchenyi and Gyula Dávid, Uniform tiling with electrical resistors (2011), section 2. (arXiv)

Links

[3]

Cserti: triangular and honeycomb resistance formulas, version 4, sections IV–V

[4]

Richard J. Mathar, Recurrence for the Atkinson-Steenwijk integrals for resistors in the infinite triangular lattice (2022)

Similar tables

Resistance of the infinite square lattice — the remaining regular Euclidean tiling graph, with the same edge normalization and periodic limit

Kirchhoff indices of connected graphs — sums effective resistances over vertex pairs in finite connected graphs

Watson integrals of the cubic lattices — convergent origin Green-function integrals for three-dimensional networks

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all 84 noncoincident endpoint-pair symmetry representatives through graph distance 12 in each network: 48 triangular, 15 honeycomb with destination $A$, and 21 honeycomb with destination $B$. This noncanonical finite reference bound covers short-separation axial, symmetry-line and asymmetric displacements, including six even and six odd honeycomb shells, before extending individual directions further)

How they were obtained:

The generator computes $a+b\sqrt3/\pi$ with $a,b\in\mathbb Q$. Exact integration of polynomial moments supplies the triangular axis seeds; exact harmonic recurrences supply the other displacements. The honeycomb reductions are (1). The three rows with $b=0$ are returned as exact rationals. The derivation is attached as derivation.md; the conjectured recurrences in the source are not used. Arb evaluates the remaining expressions at 461 working bits for 100 significant digits. Maximum measured cancellation is 39.10 bits, and every nonrational output retains at least 418 bits of relative accuracy. Every written decimal interval contains the full computed ball.

Independent mpmath quadrature at 125 and 150 decimal digits checks all 84 entries, after controls at the origin and four known triangular displacements. The largest difference between those precisions is less than $3\cdot10^{-126}$. Exact harmonicity holds at all selected endpoints. Breadth-first graph enumeration verifies the distances, symmetry representatives and all three group counts. Separate Fourier solves on periodic networks of side lengths 64, 128 and 256 check both networks, including exact finite-torus adjacent-edge controls; their errors decrease on every row. These are checks, while the certified digits follow from the exact arithmetic and Arb enclosures.