# Greek Wave Frieze

> The running wave of Greek pottery, drawn by one formula and wound into a spiral mosaic; the band slides along itself without end.

The running wave, or Vitruvian scroll, is the band of curling waves that Greek potters painted around the rims of their vases. This document draws the band with one formula and winds it into a spiral.

Every point of the plane is given a place on a spiral band. $a$ is its angle as a fraction of a turn, $p$ its distance from the centre, and $u = a + \lfloor p - a \rfloor$ counts the turns of the band up to that point, so the band winds outward by one unit per turn. Along the band, $s = \pi(u^2 + u) - T$ measures the position in cells; the factor $\pi(u^2 + u)$ is chosen so that every cell is one unit square, however far it is from the centre. Inside a cell, $h$ and $v$ are the coordinates measured from the cell's centre, between $-1/2$ and $1/2$.

Without a twist, the lower half of every cell is dark and the upper half light: plain stripes. The twist angle $w$ turns each point around its cell's centre by an angle that is largest at the centre and fades to zero at the edge of the cell, and $d$ is the height of the turned point. Where $d \le 0$ the field is dark. The turn curls the straight boundary between the halves into a spiral, the crest of the wave. The twist $k + 1.5u$ grows slowly along the band, so the outer waves are wound a little tighter than the inner ones; edit $k$ to loosen or tighten them all.

$T$ slides the whole band along itself by one cell per loop, so the waves travel outward around the spiral without end.

After a Desmos graph by Fabrice Neyret.

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