# Gyroid

> Schoen's gyroid, the minimal surface that divides space into two intertwined labyrinths, approximated by the level set of sin x cos y + sin y cos z + sin z cos x; the level animates.

In 1970 Alan Schoen, working at NASA on lightweight structures, found a minimal surface that nobody had seen before and named it the gyroid. Like a soap film, it has zero mean curvature at every point. It repeats in all three directions, and unlike the triply periodic minimal surfaces known before it, it contains no straight lines and has no mirror planes. It divides space into two intertwined labyrinths of passages that never meet.

The exact gyroid has no simple equation. This document draws the standard approximation, the level set $G(x, y, z) = c$ with $G = \sin x\cos y + \sin y\cos z + \sin z\cos x$, over one and a half periods in each direction. At $c = 0$ the two labyrinths are congruent: $G(-x, -y, -z) = -G(x, y, z)$, so a point reflection swaps them. As $c$ animates between $-1$ and $1$, one labyrinth widens and the other narrows. Since $|G| \le 3/2$, at $c = \pm 3/2$ the surface would shrink to isolated points and vanish.

Gyroids appear wherever two materials must fill space in equal measure and both stay connected: in block copolymers, in the wing scales of some butterflies, where the structure produces their green colour, and in lattices printed for strength at low weight. Rotate the surface to look down its passages.

After a Desmos graph by Fabrice Neyret.

- [Open the interactive version](https://graph-paper.io/showcase/en/gyroid)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Gyroid)