Dixon elliptic functions

The Dixon elliptic functions cm, sm applied to a real-valued argument x. Both functions are periodic with real period π₃ ≈ 5.29991625

In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity ${\mathrm{cm}}^{3}z+{\mathrm{sm}}^{3}z=1$, as real functions they parametrize the cubic Fermat curve $x^3+y^3=1$, just as the trigonometric functions sine and cosine parametrize the unit circle $x^2+y^2=1$.

They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873.

Definition

The functions sm and cm can be defined as the solutions to the initial value problem:

$\frac{d}{dz}\mathrm{cm}z=-{\mathrm{sm}}^{2}z,\frac{d}{dz}\mathrm{sm}z={\mathrm{cm}}^{2}z,\mathrm{cm}(0)=1,\mathrm{sm}(0)=0$

Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:

$z=\underset{0}{\overset{\mathrm{sm}z}{\int }}\frac{dw}{(1-w^3{)}^{2/3}}=\underset{\mathrm{cm}z}{\overset{1}{\int }}\frac{dw}{(1-w^3{)}^{2/3}}$

which can also be expressed using the hypergeometric function:

${\mathrm{sm}}^{-1}(z)=z{}_2{F}_1(\frac{1}{3},\frac{2}{3};\frac{4}{3};z^3)$

Parametrization of the cubic Fermat curve

The function t ↦ (cm t, sm t) parametrizes the cubic Fermat curve, with area of the sector equal to half the argument t.

Both sm and cm have a period along the real axis of ${\pi }_3=\mathrm{B}(\frac{1}{3},\frac{1}{3})=\frac{\sqrt{3}}{2\pi }{\mathrm{\Gamma }}^3\left(\frac{1}{3}\right)\approx 5.29991625$ with $\mathrm{B}$ the beta function and $\mathrm{\Gamma }$ the gamma function:

$\begin{array}{rl}\frac{1}{3}{\pi }_3& =\underset{-\mathrm{\infty }}{\overset{0}{\int }}\frac{dx}{(1-x^3{)}^{2/3}}=\underset{0}{\overset{1}{\int }}\frac{dx}{(1-x^3{)}^{2/3}}=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{dx}{(1-x^3{)}^{2/3}}\\ & \approx 1.76663875\end{array}$

They satisfy the identity ${\mathrm{cm}}^{3}z+{\mathrm{sm}}^{3}z=1$. The parametric function $t\mapsto (\mathrm{cm}t,\mathrm{sm}t),$ $t\in [-\frac{1}{3}{\pi }_3,\frac{2}{3}{\pi }_3]$ parametrizes the cubic Fermat curve $x^3+y^3=1,$ with $\frac{1}{2}t$ representing the signed area lying between the segment from the origin to $(1,0)$, the segment from the origin to $(\mathrm{cm}t,\mathrm{sm}t)$, and the Fermat curve, analogous to the relationship between the argument of the trigonometric functions and the area of a sector of the unit circle. To see why, apply Green's theorem:

$A=\frac{1}{2}\underset{0}{\overset{t}{\int }}(xdy-ydx)=\frac{1}{2}\underset{0}{\overset{t}{\int }}({\mathrm{cm}}^{3}t+{\mathrm{sm}}^{3}t)dt=\frac{1}{2}\underset{0}{\overset{t}{\int }}dt=\frac{1}{2}t.$

Notice that the area between the $x+y=0$ and $x^3+y^3=1$ can be broken into three pieces, each of area $\frac{1}{6}{\pi }_3$:

$\begin{array}{rl}\frac{1}{2}{\pi }_3& =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}((1-x^3{)}^{1/3}+x)dx\\ \frac{1}{6}{\pi }_3& =\underset{-\mathrm{\infty }}{\overset{0}{\int }}((1-x^3{)}^{1/3}+x)dx=\underset{0}{\overset{1}{\int }}(1-x^3{)}^{1/3}dx.\end{array}$

Symmetries

The Dixon elliptic function sm z in the complex plane, illustrating its double periodicity (ω = e^2πi/3).

The function $\mathrm{sm}z$ has zeros at the complex-valued points $z=\frac{1}{\sqrt{3}}{\pi }_{3}i(a+b\omega )$ for any integers $a$ and $b$, where $\omega$ is a cube root of unity, $\omega =\exp \frac{2}{3}i\pi =-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ (that is, $a+b\omega$ is an Eisenstein integer). The function $\mathrm{cm}z$ has zeros at the complex-valued points $z=\frac{1}{3}{\pi }_3+\frac{1}{\sqrt{3}}{\pi }_{3}i(a+b\omega )$. Both functions have poles at the complex-valued points $z=-\frac{1}{3}{\pi }_3+\frac{1}{\sqrt{3}}{\pi }_{3}i(a+b\omega )$.

On the real line, $\mathrm{sm}x=0\leftrightarrow x\in {\pi }_3\mathbb{\mathbb{Z}}$, which is analogous to $\sin x=0\leftrightarrow x\in \pi \mathbb{\mathbb{Z}}$.

Fundamental reflections, rotations, and translations

Both cm and sm commute with complex conjugation,

$\begin{array}{rl}\mathrm{cm}\overline{z}& =\overline{\mathrm{cm}z},\\ \mathrm{sm}\overline{z}& =\overline{\mathrm{sm}z}.\end{array}$

Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function cm is invariant under $\frac{1}{3}$ turn rotations of the complex plane, and $\frac{1}{3}$ turn rotations of the domain of sm cause $\frac{1}{3}$ turn rotations of the codomain:

$\begin{array}{rl}\mathrm{cm}\omega z& =\mathrm{cm}z=\mathrm{cm}{\omega }^{2}z,\\ \mathrm{sm}\omega z& =\omega \mathrm{sm}z={\omega }^2\mathrm{sm}{\omega }^{2}z.\end{array}$

Each Dixon elliptic function is invariant under translations by the Eisenstein integers $a+b\omega$ scaled by ${\pi }_3,$

$\begin{array}{r}\mathrm{cm}(z+{\pi }_3(a+b\omega ))=\mathrm{cm}z,\\ \mathrm{sm}(z+{\pi }_3(a+b\omega ))=\mathrm{sm}z.\end{array}$

Negation of each of cm and sm is equivalent to a $\frac{1}{3}{\pi }_3$ translation of the other,

$\begin{array}{rl}\mathrm{cm}(-z)& =\frac{1}{\mathrm{cm}z}=\mathrm{sm}(z+\frac{1}{3}{\pi }_3),\\ \mathrm{sm}(-z)& =-\frac{\mathrm{sm}z}{\mathrm{cm}z}=\frac{1}{\mathrm{sm}(z-\frac{1}{3}{\pi }_3)}=\mathrm{cm}(z+\frac{1}{3}{\pi }_3).\end{array}$

For $n\in \mathbb{\{}0,1,2\},$ translations by $\frac{1}{3}{\pi }_3\omega$ give

$\begin{array}{rl}\mathrm{cm}(z+\frac{1}{3}{\omega }^n{\pi }_3)& ={\omega }^{2n}\frac{-\mathrm{sm}z}{\mathrm{cm}z},\\ \mathrm{sm}(z+\frac{1}{3}{\omega }^n{\pi }_3)& ={\omega }^n\frac{1}{\mathrm{cm}z}.\end{array}$

Specific values

$z$	$\mathrm{cm}z$	$\mathrm{sm}z$
$-\frac{1}{3}{\pi }_3$	$\mathrm{\infty }$	$\mathrm{\infty }$
$-\frac{1}{6}{\pi }_3$	$\sqrt[3]{2}$	$-1$
$0$	$1$	$0$
$\frac{1}{6}{\pi }_3$	$1/\sqrt[3]{2}$	$1/\sqrt[3]{2}$
$\frac{1}{3}{\pi }_3$	$0$	$1$
$\frac{1}{2}{\pi }_3$	$-1$	$\sqrt[3]{2}$
$\frac{2}{3}{\pi }_3$	$\mathrm{\infty }$	$\mathrm{\infty }$

More specific values

$z$	$\mathrm{cm}z$	$\mathrm{sm}z$
$-\frac{1}{4}{\pi }_3$	$\frac{1+\sqrt{3}+\sqrt{2\sqrt{3}}}{2}$	$\frac{-1-\sqrt{3+2\sqrt{3}}}{\sqrt[3]{4}}$
$-\frac{1}{12}{\pi }_3$	$\frac{-1+\sqrt{3}+\sqrt{2\sqrt{3}}}{2\sqrt[3]{2}}$	$\frac{-1+\sqrt{3}-\sqrt{2\sqrt{3}}}{2\sqrt[3]{2}}$
$\frac{1}{12}{\pi }_3$	$\frac{-1+\sqrt{3+2\sqrt{3}}}{\sqrt[3]{4}}$	$\frac{1+\sqrt{3}-\sqrt{2\sqrt{3}}}{2}$
$\frac{1}{4}{\pi }_3$	$\frac{1+\sqrt{3}-\sqrt{2\sqrt{3}}}{2}$	$\frac{-1+\sqrt{3+2\sqrt{3}}}{\sqrt[3]{4}}$
$\frac{5}{12}{\pi }_3$	$\frac{-1+\sqrt{3}-\sqrt{2\sqrt{3}}}{2\sqrt[3]{2}}$	$\frac{-1+\sqrt{3}+\sqrt{2\sqrt{3}}}{2\sqrt[3]{2}}$
$\frac{7}{12}{\pi }_3$	$\frac{-1-\sqrt{3+2\sqrt{3}}}{\sqrt[3]{4}}$	$\frac{1+\sqrt{3}+\sqrt{2\sqrt{3}}}{2}$

Sum and difference identities

The Dixon elliptic functions satisfy the argument sum and difference identities:

$\begin{array}{rl}\mathrm{cm}(u+v)& =\frac{\mathrm{sm}u\mathrm{cm}u-\mathrm{sm}v\mathrm{cm}v}{\mathrm{sm}u{\mathrm{cm}}^{2}v-{\mathrm{cm}}^{2}u\mathrm{sm}v}\\ \mathrm{cm}(u-v)& =\frac{{\mathrm{cm}}^{2}u\mathrm{cm}v-\mathrm{sm}u{\mathrm{sm}}^{2}v}{\mathrm{cm}u{\mathrm{cm}}^{2}v-{\mathrm{sm}}^{2}u\mathrm{sm}v}\\ \mathrm{sm}(u+v)& =\frac{{\mathrm{sm}}^{2}u\mathrm{cm}v-\mathrm{cm}u{\mathrm{sm}}^{2}v}{\mathrm{sm}u{\mathrm{cm}}^{2}v-{\mathrm{cm}}^{2}u\mathrm{sm}v}\\ \mathrm{sm}(u-v)& =\frac{\mathrm{sm}u\mathrm{cm}u-\mathrm{sm}v\mathrm{cm}v}{\mathrm{cm}u{\mathrm{cm}}^{2}v-{\mathrm{sm}}^{2}u\mathrm{sm}v}\end{array}$

These formulas can be used to compute the complex-valued functions in real components:

$\begin{array}{rl}\mathrm{cm}(x+\omega y)& =\frac{\mathrm{sm}x\mathrm{cm}x-\omega \mathrm{sm}y\mathrm{cm}y}{\mathrm{sm}x{\mathrm{cm}}^{2}y-\omega {\mathrm{cm}}^{2}x\mathrm{sm}y}\\ & =\frac{\mathrm{cm}x({\mathrm{sm}}^{2}x{\mathrm{cm}}^{2}y+\mathrm{cm}x{\mathrm{sm}}^{2}y\mathrm{cm}y+\mathrm{sm}x{\mathrm{cm}}^{2}x\mathrm{sm}y)}{{\mathrm{sm}}^{2}x{\mathrm{cm}}^{4}y+\mathrm{sm}x{\mathrm{cm}}^{2}x\mathrm{sm}y{\mathrm{cm}}^{2}y+{\mathrm{cm}}^{4}x{\mathrm{sm}}^{2}y}\\ & +\omega \frac{\mathrm{sm}x\mathrm{sm}y({\mathrm{cm}}^{3}x-{\mathrm{cm}}^{3}y)}{{\mathrm{sm}}^{2}x{\mathrm{cm}}^{4}y+\mathrm{sm}x{\mathrm{cm}}^{2}x\mathrm{sm}y{\mathrm{cm}}^{2}y+{\mathrm{cm}}^{4}x{\mathrm{sm}}^{2}y}\\ \mathrm{sm}(x+\omega y)& =\frac{{\mathrm{sm}}^{2}x\mathrm{cm}y-{\omega }^2\mathrm{cm}x{\mathrm{sm}}^{2}y}{\mathrm{sm}x{\mathrm{cm}}^{2}y-\omega {\mathrm{cm}}^{2}x\mathrm{sm}y}\\ & =\frac{\mathrm{sm}x(\mathrm{sm}x\mathrm{cm}x{\mathrm{cm}}^{2}y+\mathrm{sm}y{\mathrm{cm}}^{3}x+\mathrm{sm}y{\mathrm{cm}}^{3}y)}{{\mathrm{sm}}^{2}x{\mathrm{cm}}^{4}y+\mathrm{sm}x{\mathrm{cm}}^{2}x\mathrm{sm}y{\mathrm{cm}}^{2}y+{\mathrm{cm}}^{4}x{\mathrm{sm}}^{2}y}\\ & +\omega \frac{\mathrm{sm}y(\mathrm{sm}x{\mathrm{cm}}^{3}x+\mathrm{sm}x{\mathrm{cm}}^{3}y+{\mathrm{cm}}^{2}x\mathrm{sm}y\mathrm{cm}y)}{{\mathrm{sm}}^{2}x{\mathrm{cm}}^{4}y+\mathrm{sm}x{\mathrm{cm}}^{2}x\mathrm{sm}y{\mathrm{cm}}^{2}y+{\mathrm{cm}}^{4}x{\mathrm{sm}}^{2}y}\end{array}$

Multiple-argument identities

Argument duplication and triplication identities can be derived from the sum identity:

$\begin{array}{rl}\mathrm{cm}2u& =\frac{{\mathrm{cm}}^{3}u-{\mathrm{sm}}^{3}u}{\mathrm{cm}u(1+{\mathrm{sm}}^{3}u)}=\frac{2{\mathrm{cm}}^{3}u-1}{2\mathrm{cm}u-{\mathrm{cm}}^{4}u},\\ \mathrm{sm}2u& =\frac{\mathrm{sm}u(1+{\mathrm{cm}}^{3}u)}{\mathrm{cm}u(1+{\mathrm{sm}}^{3}u)}=\frac{2\mathrm{sm}u-{\mathrm{sm}}^{4}u}{2\mathrm{cm}u-{\mathrm{cm}}^{4}u},\\ \mathrm{cm}3u& =\frac{{\mathrm{cm}}^{9}u-6{\mathrm{cm}}^{6}u+3{\mathrm{cm}}^{3}u+1}{{\mathrm{cm}}^{9}u+3{\mathrm{cm}}^{6}u-6{\mathrm{cm}}^{3}u+1},\\ \mathrm{sm}3u& =\frac{3\mathrm{sm}u\mathrm{cm}u({\mathrm{sm}}^{3}u{\mathrm{cm}}^{3}u-1)}{{\mathrm{cm}}^{9}u+3{\mathrm{cm}}^{6}u-6{\mathrm{cm}}^{3}u+1}.\end{array}$

From these formulas it can be deduced that expressions in form $\mathrm{cm}(\frac{k{\pi }_3}{2^n{3}^m})$ and $\mathrm{sm}(\frac{k{\pi }_3}{2^n{3}^m})$ are either signless infinities, or origami-constructibles for any $n,m,k\in \mathbb{\mathbb{N}}$ (In this paragraph, $\mathbb{𝕄}=$ set of all origami-constructibles $\cup \{\mathrm{\infty }\}$). Because by finding $\mathrm{cm}(\frac{x}{2})$, quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if $\mathrm{cm}x\in \mathbb{𝕄}$, then $\mathrm{cm}(\frac{x}{2})\in \mathbb{𝕄}$. To find one-third of argument value of cm, equation which is reductible to cubic or lesser degree in some cases by variable exchange $t=x^3$ has to be solved as seen from triplication formula from that follows: if $\mathrm{cm}x\in \mathbb{𝕄}$ then $\mathrm{cm}(\frac{x}{3})\in \mathbb{𝕄}$ is true. Statement $\mathrm{cm}x\in \mathbb{𝕄}$ $\Rightarrow$ $\mathrm{cm}(nx)\in \mathbb{𝕄}$ is true, because any multiple argument formula is a rational function. If $\mathrm{cm}x\in \mathbb{𝕄}$, then $\mathrm{sm}x\in \mathbb{𝕄}$ because $\mathrm{sm}x={\omega }^p\sqrt[3]{1-{\mathrm{cm}}^{3}x}$ where $p\in \{0,1,2\}$.

Specific value identities

The $\mathrm{cm}$ function satisfies the identities $$\begin{array}{rl}\mathrm{cm}\frac{2}{9}{\pi }_3& =-\mathrm{cm}\frac{1}{9}{\pi }_3\mathrm{cm}\frac{4}{9}{\pi }_3,\\ \mathrm{cm}\frac{1}{4}{\pi }_3& =\mathrm{cl}\frac{1}{3}\varpi ,\end{array}$$

where $\mathrm{cl}$ is lemniscate cosine and $\varpi$ is Lemniscate constant.

Power series

The cm and sm functions can be approximated for $|z|<\frac{1}{3}{\pi }_3$ by the Taylor series

$\begin{array}{rl}\mathrm{cm}z& =c_0+c_1{z}^3+c_2{z}^6+c_3{z}^9+\cdots +c_n{z}^{3n}+\cdots \\ \mathrm{sm}z& =s_{0}z+s_1{z}^4+s_2{z}^7+s_3{z}^{10}+\cdots +s_n{z}^{3n+1}+\cdots \end{array}$

whose coefficients satisfy the recurrence $c_0=s_0=1,$

$\begin{array}{rl}{c}_n& =-\frac{1}{3n}\sum _{k=0}^{n-1}{s}_k{s}_{n-1-k}\\ s_n& =\frac{1}{3n+1}\sum _{k=0}^n{c}_k{c}_{n-k}\end{array}$

These recurrences result in:

$\begin{array}{rl}\mathrm{cm}z& =1-\frac{1}{3}{z}^3+\frac{1}{18}{z}^6-\frac{23}{2268}{z}^9+\frac{25}{13608}{z}^{12}-\frac{619}{1857492}{z}^{15}+\cdots \\ \mathrm{sm}z& =z-\frac{1}{6}{z}^4+\frac{2}{63}{z}^7-\frac{13}{2268}{z}^{10}+\frac{23}{22113}{z}^{13}-\frac{2803}{14859936}{z}^{16}+\cdots \end{array}$

Relation to other elliptic functions

Weierstrass elliptic function

Elliptic curve $y^2=4x^3-\frac{1}{27}$ for the Weierstrass ℘-function $z\mapsto \mathrm{\wp }(z;0,\frac{1}{27})$ related to the Dixon elliptic functions.

The equianharmonic Weierstrass elliptic function $\mathrm{\wp }(z)=\mathrm{\wp }(z;0,\frac{1}{27}),$ with lattice $\mathrm{\Lambda }={\pi }_3\mathbb{\mathbb{Z}}\oplus {\pi }_3\omega \mathbb{\mathbb{Z}}$ a scaling of the Eisenstein integers, can be defined as:

$\mathrm{\wp }(z)=\frac{1}{z^2}+\sum _{\lambda \in \mathrm{\Lambda }\setminus \{0\}}(\frac{1}{(z-\lambda {)}^2}-\frac{1}{{\lambda }^2})$

The function $\mathrm{\wp }(z)$ solves the differential equation:

${\mathrm{\wp }}^{\prime }(z{)}^2=4\mathrm{\wp }(z{)}^3-\frac{1}{27}$

We can also write it as the inverse of the integral:

$z=\underset{\mathrm{\infty }}{\overset{\mathrm{\wp }(z)}{\int }}\frac{dw}{\sqrt{4w^3-\frac{1}{27}}}$

In terms of $\mathrm{\wp }(z)$, the Dixon elliptic functions can be written:

$\mathrm{cm}z=\frac{3{\mathrm{\wp }}^{\prime }(z)+1}{3{\mathrm{\wp }}^{\prime }(z)-1},\mathrm{sm}z=\frac{-6\mathrm{\wp }(z)}{3{\mathrm{\wp }}^{\prime }(z)-1}$

Likewise, the Weierstrass elliptic function $\mathrm{\wp }(z)=\mathrm{\wp }(z;0,\frac{1}{27})$ can be written in terms of Dixon elliptic functions:

${\mathrm{\wp }}^{\prime }(z)=\frac{\mathrm{cm}z+1}{3(\mathrm{cm}z-1)},\mathrm{\wp }(z)=\frac{-\mathrm{sm}z}{3(\mathrm{cm}z-1)}$

Jacobi elliptic functions

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley. Let $k=e^{5i\pi /6}$, $\theta =3^{\frac{1}{4}}{e}^{5i\pi /12}$, $s=\mathrm{sn}(u,k)$, $c=\mathrm{cn}(u,k)$, and $d=\mathrm{dn}(u,k)$. Then, let

$\xi (u)=\frac{-1+\theta scd}{1+\theta scd}$, $\eta (u)=\frac{2^{1/3}(1+{\theta }^2{s}^2)}{1+\theta scd}$.

Finally, the Dixon elliptic functions are as so:

$\mathrm{sm}(z)=\xi \left(\frac{z+{\pi }_3/6}{2^{1/3}\theta }\right)$, $\mathrm{cm}(z)=\eta \left(\frac{z+{\pi }_3/6}{2^{1/3}\theta }\right)$.

Generalized trigonometry

Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an $n=2$ case, and the functions sm and cm as an $n=3$ case.

For example, defining ${\pi }_n=\mathrm{B}(\frac{1}{n},\frac{1}{n})$ and ${\sin }_{n}z,{\cos }_{n}z$ the inverses of an integral:

$z=\underset{0}{\overset{{\sin }_{n}z}{\int }}\frac{dw}{(1-w^n{)}^{(n-1)/n}}=\underset{{\cos }_{n}z}{\overset{1}{\int }}\frac{dw}{(1-w^n{)}^{(n-1)/n}}$

The area in the positive quadrant under the curve $x^n+y^n=1$ is

$\underset{0}{\overset{1}{\int }}(1-x^n{)}^{1/n}dx=\frac{{\pi }_n}{2n}$.

The quartic $n=4$ case results in a square lattice in the complex plane, related to the lemniscate elliptic functions.

Applications

A conformal map projection of the globe onto an octahedron. Because the octahedron has equilateral triangle faces, this projection can be described in terms of sm and cm functions.

The Dixon elliptic functions are conformal maps from an equilateral triangle to a disk, and are therefore helpful for constructing polyhedral conformal map projections involving equilateral triangles, for example projecting the sphere onto a triangle, hexagon, tetrahedron, octahedron, or icosahedron.

See also

• Eisenstein integer
• Elliptic function
  • Abel elliptic functions
  • Jacobi elliptic functions
  • Lemniscate elliptic functions
  • Weierstrass elliptic function
• Lee conformal world in a tetrahedron
• Schwarz–Christoffel mapping

References

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• R. Bacher & P. Flajolet (2010) “Pseudo-factorials, elliptic functions, and continued fractions” The Ramanujan journal 21(1), 71–97. https://arxiv.org/pdf/0901.1379.pdf
• A. Cayley (1882) “Reduction of $\int dx/(1-x^3){}^{2/3}$ to elliptic integrals”. Messenger of Mathematics 11, 142–143. https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify={%22pages%22:%5b146%5d}
• F. D. Burgoyne (1964) “Generalized trigonometric functions”. Mathematics of Computation 18(86), 314–316. https://www.jstor.org/stable/2003310
• A. Cayley (1883) “On the elliptic function solution of the equation x³ + y³ − 1 = 0”, Proceedings of the Cambridge Philosophical Society 4, 106–109. https://archive.org/details/proceedingsofcam4188083camb/page/106/
• R. Chapling (2016) “Invariant Meromorphic Functions on the Wallpaper Groups”. https://arxiv.org/pdf/1608.05677
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External links

• Desmos plots:
  • Real-valued Dixon elliptic functions https://www.desmos.com/calculator/5s4gdcnxh2.
  • Parametrizing the cubic Fermat curve, https://www.desmos.com/calculator/elqqf4nwas
• On-Line Encyclopedia of Integer Sequences pages:
  • “Coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the Dixon elliptic function sm(x,0).” https://oeis.org/A104133
  • “Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).” https://oeis.org/A104134
  • “Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).” https://oeis.org/A197374
• Mathematics Stack Exchange discussions:
  • “On $x^3+y^3=z^3$, the Dixonian elliptic functions, and the Borwein cubic theta functions”, https://math.stackexchange.com/q/2090523/
  • “doubly periodic functions as tessellations (other than parallelograms)”, https://math.stackexchange.com/q/35671/