---
tag: PPW3GYXH0P
kind: concept
name: Dixon elliptic functions
msc: ["33E05"]
specializes: [FF9ZPCZ59K]
sources:
  - url: https://en.wikipedia.org/wiki/Dixon_elliptic_functions
same_as:
  wikidata: Q28402589
  wikipedia: Dixon elliptic functions
---

## Summary

The Dixon elliptic functions $\operatorname{sm}$ and $\operatorname{cm}$ are two [[elliptic functions|FF9ZPCZ59K]] that parametrise the Fermat cubic $x^3 + y^3 = 1$, as $\sin$ and $\cos$ parametrise the circle $x^2 + y^2 = 1$.
They satisfy $\operatorname{sm}' = \operatorname{cm}^2$ and $\operatorname{cm}' = -\operatorname{sm}^2$.
Their period lattice has a threefold symmetry: rotation by a primitive cube root of unity $\omega$ maps it to itself.
They are named after Alfred Cardew Dixon.

## Explanations

### L1 human

The point $(\cos t, \sin t)$ runs around the circle $x^2 + y^2 = 1$.
The point $(\operatorname{cm} t, \operatorname{sm} t)$ runs along the cubic curve $x^3 + y^3 = 1$ in the same way.
Near $t = 0$, $\operatorname{sm} t \approx t$ and $\operatorname{cm} t \approx 1$, just as $\sin t \approx t$ and $\cos t \approx 1$.
Unlike sine and cosine, they have poles, because the real cubic curve goes off to infinity.
In the complex plane they repeat along a lattice of triangles.