Dixon elliptic functions
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 formula_1, as real functions they parametrize the cubic Fermat curve formula_2, just as the trigonometric functions sine and cosine parametrize the unit circle formula_3. 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: Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral: which can also be expressed using the hypergeometric function: Parametrization of the cubic Fermat curve. Both sm and cm have a period along the real axis of formula_7 with formula_8 the beta function and formula_9 the gamma function: They satisfy the identity formula_11. The parametric function formula_12 formula_13 parametrizes the cubic Fermat curve formula_14 with formula_15 representing the  signed area lying between the segment from the origin to formula_16, the segment from the origin to formula_17, 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: Notice that the area between the formula_19 and formula_2 can be broken into three pieces, each of area formula_21: Symmetries. The function formula_23 has zeros at the complex-valued points formula_24 for any integers formula_25 and formula_26, where formula_27 is a cube root of unity, formula_28 (that is, formula_29 is  an Eisenstein integer). The function formula_30 has zeros at the complex-valued points formula_31. Both functions have poles at the complex-valued points formula_32. Sum and difference identities. The Dixon elliptic functions satisfy the argument sum and difference identities: Other expressions. Power series expansion. The coefficients formula_49 and formula_50 of the power series expansions satisfy the recurrence formula_52 These recurrences result in: Weierstrass elliptic function. The equianharmonic Weierstrass elliptic function formula_55 with lattice formula_56 a scaling of the Eisenstein integers, can be defined as: The function formula_58 solves the differential equation: We can also write it as the inverse of  the integral: In terms of formula_58, the Dixon elliptic functions can be written: Likewise, the Weierstrass elliptic function formula_63 can be written in terms of Dixon elliptic functions: Jacobi elliptic functions. The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley. Let formula_65, formula_66, formula_67, formula_68, and formula_69. Then, let Finally, the Dixon elliptic functions are as so: Generalized trigonometry. Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an formula_74 case, and the functions sm and cm as an formula_75 case. For example, defining formula_76 and  formula_77 the inverses of an integral: The area in the positive quadrant under the curve formula_79 is 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.