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Display properties: number-header: $p_c$-Numbers: []+Numbers:+- params:+ lattice: square+ d: '2'+ percolation: site+ number: 0.592746050788 +/- 6e-12+ comment: 'The published determinations disagree beyond their stated errors in the+ twelfth decimal: $0.59274605079210(2)$ CITE{Jacobsen15} from the eigenvalue formulation+ of critical polynomials, $0.592746050786(3)$ CITE{Mertens22} from the exact spanning+ probabilities of $n\times n$ squares with $n\leq 24$, and $0.5927460507896(1)$+ CITE{YangZhou24} and $0.59274605079016(1)$ CITE{Jacobsen24}, the last two as quoted+ in CITE{Wiki}. The entry is the interval $0.592746050788\pm 6\cdot 10^{-12}$,+ which contains all four with their error bars; the first ten decimals, $0.5927460507$,+ are OEIS A377420 CITE{OEISsquare}.'+- params:+ lattice: square+ d: '2'+ percolation: bond+ number: 1/2+ comment: '$p_c=\frac12$, exact: conjectured from series and self-duality by Sykes+ and Essam CITE{SykesEssam} and proved by Kesten CITE{Kesten}.'+ equals: HREF{Rational_numbers#1/2}+- params:+ lattice: triangular+ d: '2'+ percolation: site+ number: 1/2+ comment: '$p_c=\frac12$, exact CITE{SykesEssam}: the triangular lattice is self-matching,+ so its site threshold is $\frac12$ by the argument of Kesten CITE{Kesten}.'+ equals: HREF{Rational_numbers#1/2}+- params:+ lattice: triangular+ d: '2'+ percolation: bond+ number: '0.3472963553338606977034332535386295920007513543681387744724827562641316442780294708430332263147991480'+ comment: $p_c=2\sin\frac{\pi}{18}=2\cos\frac{4\pi}{9}$, exact, the root in $(0,1)$+ of $x^3-3x+1$, OEIS A130880 CITE{OEIStri}; found by Sykes and Essam CITE{SykesEssam}+ from the star–triangle transformation and proved by Wierman CITE{Wierman}; $\cos\frac{4\pi}{9}$+ is in the HREF{Cos_pi_times_x_at_rational_numbers#4/9}[table of $\cos(\pi x)$].+- params:+ lattice: honeycomb+ d: '2'+ percolation: site+ number: 0.697040230 +/- 5e-9+ comment: $0.697040230(5)$ CITE{Jacobsen14}, from critical polynomials on bases of+ up to $8$ unit cells; the honeycomb lattice is also called the hexagonal lattice.+- params:+ lattice: honeycomb+ d: '2'+ percolation: bond+ number: '0.6527036446661393022965667464613704079992486456318612255275172437358683557219705291569667736852008520'+ comment: $p_c=1-2\sin\frac{\pi}{18}$, exact, the root in $(0,1)$ of $x^3-3x^2+1$,+ OEIS A178959 CITE{OEISkagome}; found by Sykes and Essam CITE{SykesEssam} and proved+ by Wierman CITE{Wierman}; $1$ minus the bond threshold of the triangular lattice,+ its planar dual.+- params:+ lattice: kagome+ d: '2'+ percolation: site+ number: '0.6527036446661393022965667464613704079992486456318612255275172437358683557219705291569667736852008520'+ comment: $p_c=1-2\sin\frac{\pi}{18}=1-2\cos\frac{4\pi}{9}$, exact CITE{SykesEssam},+ the root in $(0,1)$ of $x^3-3x^2+1$, OEIS A178959 CITE{OEISkagome}; it equals+ the bond threshold of the honeycomb lattice, since the kagome lattice is the line+ graph of the honeycomb lattice; $\cos\frac{4\pi}{9}$ is in the HREF{Cos_pi_times_x_at_rational_numbers#4/9}[table+ of $\cos(\pi x)$].+- params:+ lattice: kagome+ d: '2'+ percolation: bond+ number: 0.52440499916744820 +/- 1e-17+ comment: '$0.52440499916744820(1)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; not known exactly: the conjectured polynomial $3p^2+6p^3-12p^4+6p^5-p^6=1$+ has its root at $0.52442971\ldots$, which differs from the estimate in the fifth+ decimal.'+- params:+ lattice: 3-12-12+ d: '2'+ percolation: site+ number: '0.8079007641202843312833520393286119147318350108627217209152260722915676700747830202460187405840713765'+ comment: '$p_c=\sqrt{1-2\sin\frac{\pi}{18}}$, exact CITE{SudingZiff}, the root in+ $(0,1)$ of $x^6-3x^4+1$, OEIS A174849 CITE{OEIStt}: contracting the edges of $(3,12^2)$+ that lie in no triangle gives the kagome lattice, and a contracted edge is open+ when both of its ends are, with probability $p^2$.'+- params:+ lattice: 3-12-12+ d: '2'+ percolation: bond+ number: 0.740420798850811610 +/- 2e-18+ comment: $0.740420798850811610(2)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the three-twelve or truncated+ hexagonal lattice.+- params:+ lattice: 4-6-12+ d: '2'+ percolation: site+ number: 0.7478008 +/- 2e-7+ comment: $0.7478008(2)$ CITE{Jacobsen14}, from critical polynomials computed by+ transfer matrices; the lattice is also called the cross or truncated trihexagonal+ lattice.+- params:+ lattice: 4-6-12+ d: '2'+ percolation: bond+ number: 0.693733124922 +/- 2e-12+ comment: $0.693733124922(2)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the cross or truncated trihexagonal+ lattice.+- params:+ lattice: 4-8-8+ d: '2'+ percolation: site+ number: 0.7297232 +/- 5e-7+ comment: $0.7297232(5)$ CITE{Jacobsen14}, from critical polynomials computed by+ transfer matrices; the lattice is also called the four-eight, bathroom-tile or+ truncated square lattice.+- params:+ lattice: 4-8-8+ d: '2'+ percolation: bond+ number: 0.6768031243900113 +/- 3e-16+ comment: $0.6768031243900113(3)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the four-eight, bathroom-tile+ or truncated square lattice.+- params:+ lattice: 3-4-6-4+ d: '2'+ percolation: site+ number: 0.62181207 +/- 7e-8+ comment: $0.62181207(7)$ CITE{Jacobsen14}, from critical polynomials computed by+ transfer matrices; the lattice is also called the ruby or rhombitrihexagonal lattice.+- params:+ lattice: 3-4-6-4+ d: '2'+ percolation: bond+ number: 0.524831461573 +/- 1e-12+ comment: $0.524831461573(1)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the ruby or rhombitrihexagonal+ lattice.+- params:+ lattice: 3-3-3-3-6+ d: '2'+ percolation: site+ number: 0.579498 +/- 3e-6+ comment: $0.579498(3)$ CITE{SudingZiff}, from hull-walk gradient percolation, with+ the uncertainty as listed in CITE{Wiki}; the paper prints $(2)$; the lattice is+ also called the snub hexagonal or maple-leaf lattice.+- params:+ lattice: 3-3-3-3-6+ d: '2'+ percolation: bond+ number: 0.4343283172240 +/- 6e-13+ comment: $0.4343283172240(6)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the snub hexagonal or maple-leaf+ lattice.+- params:+ lattice: 3-3-4-3-4+ d: '2'+ percolation: site+ number: 0.550806 +/- 3e-6+ comment: $0.550806(3)$ CITE{SudingZiff}, from hull-walk gradient percolation, with+ the uncertainty as listed in CITE{Wiki}; the paper prints $(2)$; the lattice is+ also called the snub square, puzzle or Shastry–Sutherland lattice.+- params:+ lattice: 3-3-4-3-4+ d: '2'+ percolation: bond+ number: 0.4141378565917 +/- 1e-13+ comment: $0.4141378565917(1)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the snub square, puzzle or+ Shastry–Sutherland lattice.+- params:+ lattice: 3-3-3-4-4+ d: '2'+ percolation: site+ number: 0.550213 +/- 3e-6+ comment: $0.550213(3)$ CITE{SudingZiff}, from hull-walk gradient percolation, with+ the uncertainty as listed in CITE{Wiki}; the paper prints $(2)$; the lattice is+ also called the frieze, trellis or elongated triangular lattice.+- params:+ lattice: 3-3-3-4-4+ d: '2'+ percolation: bond+ number: 0.41964035886369 +/- 2e-14+ comment: $0.41964035886369(2)$ CITE{ScullardJacobsen}, from the eigenvalue formulation+ of critical polynomials; the lattice is also called the frieze, trellis or elongated+ triangular lattice.+- params:+ lattice: D-3-3-4-3-4+ d: '2'+ percolation: site+ number: 0.6501834 +/- 2e-7+ comment: $0.6501834(2)$ CITE{Jacobsen14}, from critical polynomials computed by+ transfer matrices; the lattice is also called the Cairo pentagonal lattice.+- params:+ lattice: D-3-3-4-3-4+ d: '2'+ percolation: bond+ number: 0.5858621434083 +/- 1e-13+ comment: $1-p_c^{\mathrm{bond}}$ of the snub square lattice $(3^2,4,3,4)$, its planar+ dual, whose bond threshold $0.4141378565917(1)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-3-3-3-4-4+ d: '2'+ percolation: site+ number: 0.6470471 +/- 2e-7+ comment: $0.6470471(2)$ CITE{Jacobsen14}, from critical polynomials computed by+ transfer matrices; the lattice is also called the prismatic pentagonal lattice.+- params:+ lattice: D-3-3-3-4-4+ d: '2'+ percolation: bond+ number: 0.58035964113631 +/- 2e-14+ comment: $1-p_c^{\mathrm{bond}}$ of the elongated triangular lattice $(3^3,4^2)$,+ its planar dual, whose bond threshold $0.41964035886369(2)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-3-3-3-3-6+ d: '2'+ percolation: site+ number: 0.639447 +/- 5e-6+ comment: $0.639447(5)$ CITE{Parviainen}, by simulation, the standard error of the+ estimate being about $5\cdot 10^{-6}$; the lattice is also called the floret pentagonal+ lattice.+- params:+ lattice: D-3-3-3-3-6+ d: '2'+ percolation: bond+ number: 0.5656716827760 +/- 6e-13+ comment: $1-p_c^{\mathrm{bond}}$ of the snub hexagonal lattice $(3^4,6)$, its planar+ dual, whose bond threshold $0.4343283172240(6)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-3-6-3-6+ d: '2'+ percolation: site+ number: 0.585040 +/- 5e-6+ comment: $0.585040(5)$ CITE{Parviainen}, by simulation, the standard error of the+ estimate being about $5\cdot 10^{-6}$; the lattice is also called the rhombille+ or dice lattice.+- params:+ lattice: D-3-6-3-6+ d: '2'+ percolation: bond+ number: 0.47559500083255180 +/- 1e-17+ comment: $1-p_c^{\mathrm{bond}}$ of the kagome lattice $(3,6,3,6)$, its planar dual,+ whose bond threshold $0.52440499916744820(1)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-3-4-6-4+ d: '2'+ percolation: site+ number: 0.582410 +/- 5e-6+ comment: $0.582410(5)$ CITE{Parviainen}, by simulation, the standard error of the+ estimate being about $5\cdot 10^{-6}$; the lattice is also called the deltoidal+ trihexagonal or ruby-dual lattice.+- params:+ lattice: D-3-4-6-4+ d: '2'+ percolation: bond+ number: 0.475168538427 +/- 1e-12+ comment: $1-p_c^{\mathrm{bond}}$ of the rhombitrihexagonal lattice $(3,4,6,4)$,+ its planar dual, whose bond threshold $0.524831461573(1)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-4-8-8+ d: '2'+ percolation: site+ number: 1/2+ comment: '$p_c=\frac12$, exact CITE{SykesEssam}: every face of the tetrakis square+ lattice is a triangle, so the lattice is self-matching.'+ equals: HREF{Rational_numbers#1/2}+- params:+ lattice: D-4-8-8+ d: '2'+ percolation: bond+ number: 0.3231968756099887 +/- 3e-16+ comment: $1-p_c^{\mathrm{bond}}$ of the truncated square lattice $(4,8^2)$, its+ planar dual, whose bond threshold $0.6768031243900113(3)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-4-6-12+ d: '2'+ percolation: site+ number: 1/2+ comment: '$p_c=\frac12$, exact CITE{SykesEssam}: every face of the kisrhombille+ lattice is a triangle, so the lattice is self-matching.'+ equals: HREF{Rational_numbers#1/2}+- params:+ lattice: D-4-6-12+ d: '2'+ percolation: bond+ number: 0.306266875078 +/- 2e-12+ comment: $1-p_c^{\mathrm{bond}}$ of the truncated trihexagonal lattice $(4,6,12)$,+ its planar dual, whose bond threshold $0.693733124922(2)$ is CITE{ScullardJacobsen}.+- params:+ lattice: D-3-12-12+ d: '2'+ percolation: site+ number: 1/2+ comment: '$p_c=\frac12$, exact CITE{SykesEssam}: every face of the triakis triangular+ lattice is a triangle, so the lattice is self-matching.'+ equals: HREF{Rational_numbers#1/2}+- params:+ lattice: D-3-12-12+ d: '2'+ percolation: bond+ number: 0.259579201149188390 +/- 2e-18+ comment: $1-p_c^{\mathrm{bond}}$ of the truncated hexagonal lattice $(3,12^2)$,+ its planar dual, whose bond threshold $0.740420798850811610(2)$ is CITE{ScullardJacobsen}.+- params:+ lattice: sc+ d: '3'+ percolation: site+ number: 0.31160768 +/- 15e-8+ comment: $0.31160768(15)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: sc+ d: '3'+ percolation: bond+ number: 0.24881185 +/- 10e-8+ comment: $0.24881185(10)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations; the earlier $0.24881182(10)$ of Wang, Zhou, Zhang, Garoni and+ Deng CITE{WangEtAl} agrees within the errors.+- params:+ lattice: bcc+ d: '3'+ percolation: site+ number: 0.2459615 +/- 2e-7+ comment: $0.2459615(2)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: bcc+ d: '3'+ percolation: bond+ number: 0.18028762 +/- 20e-8+ comment: $0.18028762(20)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: fcc+ d: '3'+ percolation: site+ number: 0.19923517 +/- 20e-8+ comment: $0.19923517(20)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: fcc+ d: '3'+ percolation: bond+ number: 0.12016377 +/- 15e-8+ comment: $0.12016377(15)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: diamond+ d: '3'+ percolation: site+ number: 0.4299870 +/- 4e-7+ comment: $0.4299870(4)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: diamond+ d: '3'+ percolation: bond+ number: 0.3895892 +/- 5e-7+ comment: $0.3895892(5)$ CITE{XuWangLvDeng}, from wrapping probabilities in Monte+ Carlo simulations.+- params:+ lattice: hypercubic+ d: '4'+ percolation: site+ number: 0.19688561 +/- 3e-8+ comment: $0.19688561(3)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '4'+ percolation: bond+ number: 0.16013122 +/- 6e-8+ comment: $0.16013122(6)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '5'+ percolation: site+ number: 0.14079633 +/- 4e-8+ comment: $0.14079633(4)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '5'+ percolation: bond+ number: 0.11817145 +/- 3e-8+ comment: $0.11817145(3)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '6'+ percolation: site+ number: 0.109016661 +/- 8e-9+ comment: $0.109016661(8)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '6'+ percolation: bond+ number: 0.09420165 +/- 2e-8+ comment: $0.09420165(2)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '7'+ percolation: site+ number: 0.088951121 +/- 1e-9+ comment: $0.088951121(1)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '7'+ percolation: bond+ number: 0.078675230 +/- 2e-9+ comment: $0.078675230(2)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '8'+ percolation: site+ number: 0.075210128 +/- 1e-9+ comment: $0.075210128(1)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '8'+ percolation: bond+ number: 0.0677084181 +/- 3e-10+ comment: $0.0677084181(3)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '9'+ percolation: site+ number: 0.0652095348 +/- 6e-10+ comment: $0.0652095348(6)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '9'+ percolation: bond+ number: 0.0594960034 +/- 1e-10+ comment: $0.0594960034(1)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '10'+ percolation: site+ number: 0.0575929488 +/- 4e-10+ comment: $0.0575929488(4)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '10'+ percolation: bond+ number: 0.0530925842 +/- 2e-10+ comment: $0.0530925842(2)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '11'+ percolation: site+ number: 0.0515896843 +/- 2e-10+ comment: $0.0515896843(2)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '11'+ percolation: bond+ number: 0.04794968373 +/- 8e-11+ comment: $0.04794968373(8)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '12'+ percolation: site+ number: 0.0467309755 +/- 1e-10+ comment: $0.0467309755(1)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '12'+ percolation: bond+ number: 0.04372385825 +/- 10e-11+ comment: $0.04372385825(10)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '13'+ percolation: site+ number: 0.04271507960 +/- 10e-11+ comment: $0.04271507960(10)$ CITE{MertensMoore}, from invasion percolation.+- params:+ lattice: hypercubic+ d: '13'+ percolation: bond+ number: 0.04018761703 +/- 6e-11+ comment: $0.04018761703(6)$ CITE{MertensMoore}, from invasion percolation.
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