A Voronoi partition read on a grid, square cells taking the tone of their nearest seed

Mosaic

Throw a handful of seeds, lay a grid over them, and give each square the tone of the seed nearest its middle. The territories of a Voronoi partition come back, but their borders now follow the grid instead of cutting across it: no diagonal, no vertex where three cells meet, only steps. Read it as tesserae set by hand, or as a map whose regions were sampled rather than traced.

Example

<?php

declare(strict_types=1);

use Atelier\Pattern\Pattern;
use Atelier\Pattern\PatternRegistry;
use Atelier\Svg\Document;
use Atelier\Svg\Dumper\CompactXmlDumper;
use Atelier\Svg\Element\Shape\RectElement;

require __DIR__.'/vendor/autoload.php';

$mosaic = Pattern::mosaic(size: 150, cells: 24, sites: 14, tones: 7, seed: 1);

$document = Document::create(360, 200);

(new PatternRegistry($mosaic))->attachTo($document);

$surface = new RectElement();
$surface->setX('0')->setY('0')->setWidth('360')->setHeight('200');
$surface->setAttribute('fill', $mosaic->fill());

$document->getRootElement()?->appendChild($surface);

echo (new CompactXmlDumper())->dump($document);

Options

Setting	Default	Effect
`size`	`150`	side of the repeating tile, so how far the eye travels before the map comes round again
`cells`	`24`	squares across the tile, at least 2; more cells means finer steps along a border
`sites`	`14`	seeds sharing the tile, at least 2; more seeds means smaller territories
`tones`	`7`	distinct tones in the tiling, at least 2 and at most `sites`
`seed`	`1`	where the seeds fall and which tone each one takes
`grid`	`true`	draws the rules between the squares
`seeds`	`true`	marks each seed with a dot, which reads as the centre of its territory
`withColor(string)`	`currentColor`	the one ink every tone is an opacity of
`withOpacity(?float)`	none	opacity of the whole tile at once
`withAngle(float)`	`0`	turns the whole paving through `patternTransform`
`withId(string)`	derived	fixes the identifier instead of deriving it from the tile content

cells and size are read together: a square is size / cells wide, and a square below about three units reads as noise rather than as masonry.

tones is the number of opacities the squares are painted at, spread evenly between 0.15 and 0.65, the bounds a filled voronoi uses. A square takes the index of its seed modulo tones, not a value derived from how far that seed is, so the surface reads as a few flat tones rather than as a continuum. Two neighbouring territories can land on the same tone and merge into one shape, which is what keeps the tiling from looking mechanically striped.

How the tile closes

The distance from a square to a seed is measured on the torus, not in the plane: along each axis the shorter of the two ways is kept, straight across the tile or out through one edge and back in through the other. A square on the right edge can therefore belong to a seed living past the left one.

A territory reaching over an edge is therefore continued by the squares of the opposite edge, which measure themselves against the same seed and take the same tone. Nothing is drawn twice and no square is clipped, since the grid divides the tile exactly. The map closes because of where the distance was measured, not because a correction was applied afterwards.

The same seed draws the same map, run after run. It takes part in the derived identifier, so two seeds are two definitions in the <defs> rather than one.