Dixon elliptic functions explained

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

\operatorname{cm} ^{3} z + \operatorname{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.^[1]

Definition

The functions sm and cm can be defined as the solutions to the initial value problem:^[2]

(d)/(dz) \operatorname{cm} z = - \operatorname{sm} ^{2} z, (d)/(dz) \operatorname{sm} z = \operatorname{cm} ^{2} z, \operatorname{cm}( 0 ) = 1 , \operatorname{sm}( 0 ) = 0

Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:^[3]

z = \int ^{\operatorname{sm} _{0} z} (dw)/(( 1- w ^{3)} ^{2/3}) = \int _{\operatorname{cm} z} ^{1} (dw)/(( 1- w ^{3)} ^{2/3})

which can also be expressed using the hypergeometric function:^[4]

\operatorname{sm} ^{-1} (z) = z {} _{2F} _{1l(\tfrac13,} \tfrac 23 ; \tfrac 43 ; z ^{3r)}

Parametrization of the cubic Fermat curve

Both sm and cm have a period along the real axis of

\pi _{3} = \Beta l ( \tfrac 13 , \tfrac 13 r ) = \tfrac{\sqrt{ 3 }}{ 2 \pi}\Gamma ^{3l(\tfrac{1}{3}r) ≈} 5 . 29991625 with \Beta the beta function and \Gamma the gamma function:^[5] \begin{aligned} \tfrac 13 \pi _{3} & = \int ^{0} _{-inf ty} (dx)/(( 1- x ^{3)} ^{2/3}) = \int ^{1} _{0} (dx)/(( 1- x ^{3)} ^{2/3}) = \int ^{inf ty} _{1} (dx)/(( 1- x ^{3)} ^{2/3}) \\[ 8 mu] & ≈ 1 . 76663875 \end{aligned}

They satisfy the identity

\operatorname{cm} ^{3} z + \operatorname{sm} ^{3} z = 1 . The parametric function t \mapsto (\operatorname{cm} t, \operatorname{sm} t), t \in l [{ - \tfrac 13 }\pi _{3,} \tfrac 23 \pi _{3r]} parametrizes the cubic Fermat curve x ^{3} + y ^{3} = 1 , with \tfrac 12 t representing the signed area lying between the segment from the origin to ( 1 , 0 ) , the segment from the origin to (\operatorname{cm} t, \operatorname{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.^[6] To see why, apply Green's theorem:A = \tfrac 12 \int ^{t} _{0} (x dy - y dx ) = \tfrac 12 \int ^{t} _{0} (\operatorname{cm} ^{3} t + \operatorname{sm} ^{3} t) dt = \tfrac 12 \int ^{t} _{0} dt = \tfrac 12 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 \tfrac 16 \pi _{3} :\begin{aligned} \tfrac 12 \pi _{3} & = \int ^{inf ty} _{-inf ty} l (( 1 - x ^{3)} ^{1/3} + x r ) dx \\[ 8 mu] \tfrac 16 \pi _{3} & = \int ^{0} _{-inf ty} l (( 1 - x ^{3)} ^{1/3} + x r ) dx = \int ^{1} _{0} ( 1 - x ^{3)} ^{1/3} dx . \end{aligned}

Symmetries

The function

\operatorname{sm} z has zeros at the complex-valued points z = \tfrac 1 \sqrt{ 3 }\pi _{3i(a} + b\omega) for any integers a and b , where \omega is a cube root of unity, \omega = \exp \tfrac 23 i \pi = - \tfrac 12 + \tfrac\sqrt{ 3 } 2 i (that is, a + b\omega is an Eisenstein integer). The function \operatorname{cm} z has zeros at the complex-valued points z = \tfrac 13 \pi _{3} + \tfrac 1 \sqrt{ 3 }\pi _{3i(a} + b\omega) . Both functions have poles at the complex-valued points z = - \tfrac 13 \pi _{3} + \tfrac 1 \sqrt{ 3 }\pi _{3i(a} + b\omega) .

On the real line,

\operatorname{sm}x =0 \leftrightarrow x\in\pi _{3Z} , which is analogous to \sin x =0 \leftrightarrow x\in\pi Z .

Fundamental reflections, rotations, and translations

Both and commute with complex conjugation,

\begin{align} \operatorname{cm} \bar{z} & = \overline{\operatorname{cm} z}, \\ \operatorname{sm} \bar{z} & = \overline{\operatorname{sm} z}. \end{align}

Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function is invariant under $\backslash tfrac13$ turn rotations of the complex plane, and $\backslash tfrac13$ turn rotations of the domain of cause

\tfrac 13 turn rotations of the codomain:\begin{align} \operatorname{cm} \omega z & = \operatorname{cm} z = \operatorname{cm} \omega ^{2} z, \\ \operatorname{sm} \omega z & = \omega \operatorname{sm} z = \omega ^{2} \operatorname{sm} \omega ^{2} z. \end{align}

Each Dixon elliptic function is invariant under translations by the Eisenstein integers

a + b\omega scaled by \pi _{3,} \begin{align} \operatorname{cm} l (z + \pi _{3(a} + b\omega) r ) = \operatorname{cm} z, \\ \operatorname{sm} l (z + \pi _{3(a} + b\omega) r ) = \operatorname{sm} z. \end{align}

Negation of each of and is equivalent to a

\tfrac 13 \pi _{3} translation of the other,\begin{align} \operatorname{cm}( - z) & = (1)/(\operatorname{cm) z} = \operatorname{sm} l (z + \tfrac 13 \pi _{3r),} \\ \operatorname{sm}( - z) & =- (\operatorname{sm)/(z}{\operatorname{cm}) z} = (1)/(\operatorname{sm) l (z - \tfrac 13 \pi _{3r)}} = \operatorname{cm} l (z + \tfrac 13 \pi _{3r). \end{align}}

For

n \in \{0, 1 , 2 \}, translations by \tfrac 13 \pi _{3\omega} give\begin{align} \operatorname{cm} l (z + \tfrac 13 \omega ^{n\pi} _{3 r )} & = \omega ^{2n} (- \operatorname{sm)/(z}{\operatorname{cm}) z}, \\ \operatorname{sm} l (z + \tfrac 13 \omega ^{n\pi} _{3 r )} & = \omega ^{n (1)/(\operatorname{cm)} z}. \end{align}

Specific values

!z !\operatorname{cm} z !\operatorname{sm} z

{ - \tfrac 13 }\pi _{3}	inf ty	inf ty
{ - \tfrac 16 }\pi _{3}	\sqrt[ 3 ]{ 2 }	-1
0	1	0
{\tfrac 16 }\pi _{3}	1 / \sqrt[ 3 ]{ 2 }	1 / \sqrt[ 3 ]{ 2 }
{\tfrac 13 }\pi _{3}	0	1
{\tfrac 12 }\pi _{3}	-1	\sqrt[ 3 ]{ 2 }
{\tfrac 23 }\pi _{3}	inf ty	inf ty

More specific values

!z !\operatorname{cm} z !\operatorname{sm} z

{ - \tfrac 14 }\pi _{3}	(1+ \sqrt{ 3)/(+) \sqrt{ 2 \sqrt{ 3 }}} { 2 }	(-1- \sqrt{ 3+2 \sqrt{ 3)/() }}
- {\tfrac 29 }\pi _{3}	(\sqrt[ 6 ] 3)/(2 \sin\left( (19 \pi\right))/())	- (2 \cos\left( (1 { 18)/())/(\pi\right)}{\sqrt[ 6 ] 3 })
- {\tfrac 19 }\pi _{3}	(2 \sin\left( (29 \pi\right))/())/(\sqrt[ 6 ] 3)	- (\sqrt[ 6 ] 3)/(2 \cos\left( (1 { 18)/()) \pi\right)}
- {\tfrac{ 1 }{ 12 }}\pi _{3}	(-1+ \sqrt{ 3)/(+) \sqrt{ 2 \sqrt{ 3 }}} { 2 \sqrt[ 3 ]{ 2 }}	(-1+ \sqrt{ 3)/(-) \sqrt{ 2 \sqrt{ 3 }}} { 2 \sqrt[ 3 ]{ 2 }}
{\tfrac{ 1 }{ 12 }}\pi _{3}	(-1+ \sqrt{ 3+2 \sqrt{ 3)/() }}	(1+ \sqrt{ 3)/(-) \sqrt{ 2 \sqrt{ 3 }}} { 2 }
{\tfrac 19 }\pi _{3}	(\sqrt[ 6 ] 3)/(2 \sin\left( (29 \pi\right))/())	(2 \sin\left( (19 \pi\right))/())/(\sqrt[ 6 ] 3)
{\tfrac 29 }\pi _{3}	(2 \sin\left( (19 \pi\right))/())/(\sqrt[ 6 ] 3)	(\sqrt[ 6 ] 3)/(2 \sin\left( (29 \pi\right))/())
{\tfrac 14 }\pi _{3}	(1+ \sqrt{ 3)/(-) \sqrt{ 2 \sqrt{ 3 }}} { 2 }	(-1+ \sqrt{ 3+2 \sqrt{ 3)/() }}
{\tfrac{ 5 }{ 12 }}\pi _{3}	(-1+ \sqrt{ 3)/(-) \sqrt{ 2 \sqrt{ 3 }}} { 2 \sqrt[ 3 ]{ 2 }}	(-1+ \sqrt{ 3)/(+) \sqrt{ 2 \sqrt{ 3 }}} { 2 \sqrt[ 3 ]{ 2 }}
{\tfrac 49 }\pi _{3}	- (\sqrt[ 6 ] 3)/(2 \cos\left( (1 { 18)/()) \pi\right)}	(2 \sin\left( (29 \pi\right))/())/(\sqrt[ 6 ] 3)
{\tfrac 59 }\pi _{3}	- (2 \cos\left( (1 { 18)/())/(\pi\right)}{\sqrt[ 6 ] 3 })	(\sqrt[ 6 ] 3)/(2 \sin\left( (19 \pi\right))/())
{\tfrac{ 7 }{ 12 }}\pi _{3}	(-1- \sqrt{ 3+2 \sqrt{ 3)/() }}	(1+ \sqrt{ 3)/(+) \sqrt{ 2 \sqrt{ 3 }}} { 2 }

Sum and difference identities

The Dixon elliptic functions satisfy the argument sum and difference identities:^[7]

\begin{aligned} \operatorname{cm}( u + v ) & = (\operatorname{sm)/(u) \operatorname{cm} u - \operatorname{sm} v \operatorname{cm} v } { \operatorname{sm} u \operatorname{cm} ^{2} v - \operatorname{cm} ^{2} u \operatorname{sm} v } \\[ 8 mu] \operatorname{cm}( u - v ) & = (\operatorname{cm)/(^{2}) u \operatorname{cm} v - \operatorname{sm} u \operatorname{sm} ^{2} v } { \operatorname{cm} u \operatorname{cm} ^{2} v - \operatorname{sm} ^{2} u \operatorname{sm} v } \\[ 8 mu] \operatorname{sm}( u + v ) & = (\operatorname{sm)/(^{2}) u \operatorname{cm} v - \operatorname{cm} u \operatorname{sm} ^{2} v } { \operatorname{sm} u \operatorname{cm} ^{2} v - \operatorname{cm} ^{2} u \operatorname{sm} v } \\[ 8 mu] \operatorname{sm}( u - v ) & = (\operatorname{sm)/(u) \operatorname{cm} u - \operatorname{sm} v \operatorname{cm} v } { \operatorname{cm} u \operatorname{cm} ^{2} v - \operatorname{sm} ^{2} u \operatorname{sm} v } \end{aligned}

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

\begin{aligned} \operatorname{cm}(x + \omega y) & = (\operatorname{sm)/(x) \operatorname{cm} x - \omega\operatorname{sm} y \operatorname{cm} y } { \operatorname{sm} x \operatorname{cm} ^{2} y - \omega\operatorname{cm} ^{2} x \operatorname{sm} y }\\[ 4 mu] & = (\operatorname{cm)/(x) (\operatorname{sm} ^{2} x \operatorname{cm} ^{2} y + \operatorname{cm} x \operatorname{sm} ^{2} y \operatorname{cm} y + \operatorname{sm} x \operatorname{cm} ^{2} x \operatorname{sm} y) }{\operatorname{sm} ^{2} x \operatorname{cm} ^{4} y + \operatorname{sm} x \operatorname{cm} ^{2} x \operatorname{sm} y \operatorname{cm} ^{2} y + \operatorname{cm} ^{4} x \operatorname{sm} ^{2} y} \\[ 4 mu] & + \omega (\operatorname{sm)/(x) \operatorname{sm} y (\operatorname{cm} ^{3} x - \operatorname{cm} ^{3} y )}{\operatorname{sm} ^{2} x \operatorname{cm} ^{4} y + \operatorname{sm} x \operatorname{cm} ^{2} x \operatorname{sm} y \operatorname{cm} ^{2} y + \operatorname{cm} ^{4} x \operatorname{sm} ^{2} y} \\[ 8 mu] \operatorname{sm}(x + \omega y) & = (\operatorname{sm)/(^{2}) x \operatorname{cm} y - \omega ^{2\operatorname{cm}} x \operatorname{sm} ^{2} y } { \operatorname{sm} x \operatorname{cm} ^{2} y - \omega\operatorname{cm} ^{2} x \operatorname{sm} y } \\[ 4 mu] & = (\operatorname{sm)/(x) (\operatorname{sm} x \operatorname{cm} x \operatorname{cm} ^{2} y + \operatorname{sm} y\operatorname{cm} ^{3} x + \operatorname{sm} y\operatorname{cm} ^{3} y ) }{\operatorname{sm} ^{2} x \operatorname{cm} ^{4} y + \operatorname{sm} x \operatorname{cm} ^{2} x \operatorname{sm} y \operatorname{cm} ^{2} y + \operatorname{cm} ^{4} x \operatorname{sm} ^{2} y} \\[ 4 mu] & + \omega (\operatorname{sm)/(y) (\operatorname{sm} x \operatorname{cm} ^{3} x + \operatorname{sm} x \operatorname{cm} ^{3} y + \operatorname{cm} ^{2} x \operatorname{sm} y \operatorname{cm} y) }{\operatorname{sm} ^{2} x \operatorname{cm} ^{4} y + \operatorname{sm} x \operatorname{cm} ^{2} x \operatorname{sm} y \operatorname{cm} ^{2} y + \operatorname{cm} ^{4} x \operatorname{sm} ^{2} y} \end{aligned}

Multiple-argument identities

Argument duplication and triplication identities can be derived from the sum identity:^[8]

\begin{align} \operatorname{cm} 2 u & = (\operatorname{cm)/(^{3}) u - \operatorname{sm} ^{3} u} { \operatorname{cm} u ( 1 + \operatorname{sm} ^{3} u) } = (2 \operatorname{cm)/(^{3}) u - 1 } { 2 \operatorname{cm} u - \operatorname{cm} ^{4} u }, \\[ 5 mu] \operatorname{sm} 2 u & = (\operatorname{sm)/(u) ( 1 + \operatorname{cm} ^{3} u)} { \operatorname{cm} u ( 1 + \operatorname{sm} ^{3} u) } = (2 \operatorname{sm)/(u) - \operatorname{sm} ^{4} u} { 2 \operatorname{cm} u - \operatorname{cm} ^{4} u }, \\[ 5 mu] \operatorname{cm} 3 u & = (\operatorname{cm)/(^{9}) u - 6 \operatorname{cm} ^{6} u + 3 \operatorname{cm} ^{3} u + 1 } { \operatorname{cm} ^{9} u + 3 \operatorname{cm} ^{6} u - 6 \operatorname{cm} ^{3} u + 1 }, \\[ 5 mu] \operatorname{sm} 3 u & = (3 \operatorname{sm)/(u) \operatorname{cm}u (\operatorname{sm} ^{3u} \operatorname{cm} ^{3u} - 1 )} { \operatorname{cm} ^{9} u + 3 \operatorname{cm} ^{6} u - 6 \operatorname{cm} ^{3} u + 1 }. \end{align} From these formulas it can be deduced that expressions in form \operatorname{cm}( (k \pi _{3})/(2 ^{n3} ^{m}) ) and \operatorname{sm}( (k \pi _{3})/(2 ^{n3} ^{m}) ) are either signless infinities, or origami-constructibles for any n, m, k \in N (In this paragraph, M = set of all origami-constructibles \cup \{ inf ty}\ ). Because by finding \operatorname{cm}( (x)/(2) ) , quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if \operatorname{cm} x \in M , then \operatorname{cm}( (x)/(2) ) \in 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 \operatorname{cm} x \in M then \operatorname{cm}( (x)/(3) ) \in M is true. Statement \operatorname{cm} x \in M ⇒ \operatorname{cm}(nx) \in M is true, because any multiple argument formula is a rational function. If \operatorname{cm} x \in M , then \operatorname{sm} x \in M because \operatorname{sm} x = \omega ^{p\sqrt[3]{1-\operatorname{cm}} ^{3} x} where p \in \{ 0 , 1 , 2 \} .

Specific value identities

The

\operatorname{cm} function satisfies the identities

where

\operatorname{cl} is lemniscate cosine and \varpi is Lemniscate constant.

Power series

The and functions can be approximated for

|z| < \tfrac 13 \pi _{3} by the Taylor series\begin{aligned} \operatorname{cm} z & = c _{0} + c ^{3} _{1 z} + c ^{6} _{2 z} + c ^{9} _{3 z} + … + c ^{3 n} _{nz} + … \\[ 4 mu] \operatorname{sm} z & = s _{0z} + s ^{4} _{1 z} + s ^{7} _{2 z} + s ^{10} _{3 z} + … + s ^{3 n +1} _{nz} + … \end{aligned}

whose coefficients satisfy the recurrence

c _{0} = s _{0} = 1 , ^[9] \begin{aligned} c _{n} & =- (1)/(3 n) \sum ^{n -1} _{k =0} s _{ks} _{n-1-k} \\[ 4 mu] s _{n} & = (1)/(3 n +1) \sum ^{n} _{k =0} c _{kc} _{n-k} \end{aligned}

These recurrences result in:^[10]

\begin{aligned} \operatorname{cm} z & = 1 - (1)/(3) z ^{3} + (1)/(18) z ^{6} - (23)/(2268) z ^{9} + (25)/(13608) z ^{12} - (619)/(1857492) z ^{15} + … \\[ 8 mu] \operatorname{sm} z & = z - (1)/(6) z ^{4} + (2)/(63) z ^{7} - (13)/(2268) z ^{10} + (23)/(22113) z ^{13} - (2803)/(14859936) z ^{16} + … \end{aligned}

Relation to other elliptic functions

Weierstrass elliptic function

\wp(z) = \wp l (z; 0 , \tfrac 1 { 27 } r ), with lattice Λ = \pi _{3Z} ⊕ \pi _{3\omegaZ} a scaling of the Eisenstein integers, can be defined as:^[11] \wp(z) = (1)/(z ^{2}) + \sum _{λ\inΛ\setminus\{0\} }\!\left(\frac 1 - \frac 1 \right)

The function

\wp(z) solves the differential equation:\wp'(z) ^{2} = 4 \wp(z) ^{3} - \tfrac 1 { 27 }

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

z = \int ^{\wp(z)} _{inf ty} (dw)/(\sqrt{ 4 w ^{3} - \tfrac 1 { 27) }}

In terms of

\wp(z) , the Dixon elliptic functions can be written:^[12] \operatorname{cm} z = (3 \wp'(z) +1)/(3 \wp'(z) -1) , \operatorname{sm} z = (-6 \wp(z))/(3 \wp'(z) -1)

Likewise, the Weierstrass elliptic function

\wp(z) = \wp l (z; 0 , \tfrac 1 { 27 } r ) can be written in terms of Dixon elliptic functions:\wp'(z) = (\operatorname{cm)/(z) + 1 }{ 3 (\operatorname{cm} z - 1 )}, \wp(z) = (- \operatorname{sm)/(z}{ 3 (\operatorname{cm}) z - 1 )}

Jacobi elliptic functions

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.^[13] Let

k = e ^{5} , \theta = 3 ^{(1)/(4)} e ^{5} , s = \operatorname {sn}(u,k) , c = \operatorname {cn}(u,k) , and d = \operatorname {dn}(u,k) . Then, let\xi(u) = (-1+ \theta scd)/(1+ \theta scd) , η (u) = (2 ^{1/3} \left( 1+ \theta ^{2} s ^{2\right)})/(1+ \theta scd) .

Finally, the Dixon elliptic functions are as so:

\operatorname {sm}(z) = \xi \left( (z + \pi _{3/6})/(2 ^{1/3} \theta) \right) , \operatorname {cm}(z) = η \left( (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.^[14]

For example, defining

\pi _{n} = \Beta l (\tfrac 1 n, \tfrac 1 n r ) and \sin _{n} z,\cos _{n} z the inverses of an integral:z = \int ^{\sin _{n} z} _{0} (dw)/(( 1- w ^{n)} ^{(n-1)/n}) = \int ^{1} _{\cos _{n} z} (dw)/(( 1- w ^{n)} ^{(n-1)/n})

The area in the positive quadrant under the curve

x ^{n} + y ^{n} = 1 is\int ^{1} _{0} ( 1 - x ^{n)} ^{1/n} dx = (\pi _{n})/(2 n) .

The quartic

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

Applications

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.^[15]

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

• O. S. Adams (1925). Elliptic functions applied to conformal world maps (No. 297). US Government Printing Office. ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1121925.pdf
• 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 $\backslash int\; dx\; /\; (1\; -\; x^3)^$ to elliptic integrals”. Messenger of Mathematics 11, 142–143. https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify=
• 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 ”, 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
• J. F. Cox (1935) “Répresentation de la surface entière de la terre dans une triangle équilatéral”, Bulletin de la Classe des Sciences, Académie Royale de Belgique 5e, 21, 66–71.
• G. Dillner (1873) “Traité de calcul géométrique supérieur”, Chapter 16, Nova acta Regiae Societatis Scientiarum Upsaliensis, Ser. III 8, 94–102. https://archive.org/details/novaactaregiaeso38kung/page/94/
• A. C. . Dixon . Alfred Cardew Dixon . On the doubly periodic functions arising out of the curve . Quarterly Journal of Pure and Applied Mathematics . 1890 . XXIV . 167–233 .
• A. Dixon (1894) The elementary properties of the elliptic functions. MacMillian. https://archive.org/details/elempropellipt00dixorich/
• Van Fossen Conrad . Eric . Flajolet . Philippe . Philippe Flajolet . math/0507268 . 2005 . . 2223029 . Art. B54g, 44 . The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion . 54. 2005math......7268V.
• A. Gambini, G. Nicoletti, & D. Ritelli (2021) “Keplerian trigonometry”. Monatshefte für Mathematik 195(1), 55–72. https://doi.org/10.1007/s00605-021-01512-0
• R. Grammel (1948) “Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen”. Archiv der Mathematik 1(1), 47–51. https://doi.org/10.1007/BF02038206
• J. C. Langer & D. A. Singer (2014) “The Trefoil”. Milan Journal of Mathematics 82(1), 161–182. https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf
• M. Laurent (1949) “Tables de la fonction elliptique de Dixon pour l’intervalle 0-0, 1030”. Bulletin de l’Académie Royale des Sciences de Belgique Classe des Sciences, 35, 439–450.
• L. P. Lee (1973) “The Conformal Tetrahedric Projection with some Practical Applications”. The Cartographic Journal, 10(1), 22–28. https://doi.org/10.1179/caj.1973.10.1.22
• L. P. Lee (1976) Conformal Projections Based on Elliptic Functions. Toronto: B. V. Gutsell, York University. Cartographica Monographs No. 16. . Supplement No. 1 to The Canadian Cartographer 13.
• E. Lundberg (1879) “Om hypergoniometriska funktioner af komplexa variabla”. Manuscript, 1879. Translation by Jaak Peetre “On hypergoniometric functions of complex variables”. https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps
• J. Magis (1938) “Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral”. Bulletin Géodésique 59(1), 247–256. http://doi.org/10.1007/BF03029866
• M. D. McIlroy (2011) “Wallpaper maps”. Dependable and Historic Computing. Springer. 358–375. https://link.springer.com/chapter/10.1007/978-3-642-24541-1_27
• W. P. Reinhardt & P. L. Walker (2010) “Weierstrass Elliptic and Modular Functions”, NIST Digital Library of Mathematical Functions, §23.5(v). https://dlmf.nist.gov/23.5#v
• P. L. Robinson (2019) “The Dixonian elliptic functions”. https://arxiv.org/abs/1901.04296
• H. A. Schwarz (1869) “Ueber einige Abbildungsaufgaben”. Crelles Journal 1869(70), 105–120. http://doi.org/10.1515/crll.1869.70.105
• B. R. Seth & F. P. White (1934) “Torsion of beams whose cross-section is a regular polygon of n sides”. Mathematical Proceedings of the Cambridge Philosophical Society, 30(2), 139. http://doi.org/10.1017/s0305004100016558 
• D. Shelupsky (1959) “A generalization of the trigonometric functions”. The American Mathematical Monthly 66(10), 879–884. https://www.jstor.org/stable/2309789

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/

Notes and References

1. Dixon (1890), Dillner (1873). Dillner uses the symbols W = \operatorname{sm}, W _{1} = \operatorname{cm}.
2. Dixon (1890), Van Fossen Conrad & Flajolet (2005), Robinson (2019).
3. The mapping for a general regular polygon is described in Schwarz (1869).
4. van Fossen Conrad & Flajolet (2005) p. 6.
5. Dillner (1873) calls the period 3 w . Dixon (1890) calls it 3λ ; Adams (1925) and Robinson (2019) each call it 3 K . Van Fossen Conrad & Flajolet (2005) call it \pi _{3} . Also see OEIS A197374.
6. Dixon (1890), Van Fossen Conrad & Flajolet (2005)
7. Dixon (1890), Adams (1925)
8. Dixon (1890), p. 185–186. Robinson (2019).
9. Adams (1925)
10. van Fossen Conrad & Flajolet (2005). Also see OEIS A104133, A104134.
11. Reinhardt & Walker (2010)
12. Chapling (2018), Robinson (2019). Adams (1925) instead expresses the Dixon elliptic functions in terms of the Weierstrass elliptic function \wp(z; 0 , -1 ).
13. van Fossen Conrad & Flajolet (2005), p.38
14. Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).
15. Adams (1925), Cox (1935), Magis (1938), Lee (1973), Lee (1976), McIlroy (2011), Chapling (2016).