# Wallpaper Group Pattern Maker (17 Symmetries)

> Tile one motif with any of the 17 wallpaper symmetry groups into a seamless SVG and CSS. Choose a plane symmetry group from p1 to p6m, a motif or your own SVG path, colors and cell size. Get a seamless repeat tile as SVG, a CSS background snippet and a larger preview built from the International Tables symmetry operations.

Part of usefulweb.app (https://usefulweb.app/). Part of [Lumina](https://lumina.usefulweb.app/) (ownerApp: lumina). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://luminawallpaperpattern.usefulweb.app/ (interactive tool at https://luminawallpaperpattern.usefulweb.app/tool)
- JSON API: POST https://luminawallpaperpattern.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://luminawallpaperpattern.usefulweb.app/api/v1/tool)
- OpenAPI: https://luminawallpaperpattern.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://luminawallpaperpattern.usefulweb.app/mcp - tool `lumina_wallpaper_pattern`

## Inputs
- group (select): Wallpaper group
- motif (select): Motif
- path (text): Custom SVG path (0–100 box; used with Custom SVG path)
- cell (integer): Cell width (px)
- stretch (integer): Cell height for oblique and rectangular lattices (% of width)
- motifSize (number): Motif size (% of cell width)
- offsetX (number): Motif position along the first cell edge (%)
- offsetY (number): Motif position along the second cell edge (%)
- rotation (number): Motif rotation (degrees)
- colors (text): Motif colors (1–6 six-digit HEX, separated by spaces)
- background (text): Background (six-digit HEX or none)
- coloring (select): Coloring
- width (integer): Preview width (px)
- height (integer): Preview height (px)

## Method
- Each of the 17 wallpaper groups is listed by its coordinate triplets from the International Tables for Crystallography (Vol. A, plane groups 1–17) in fractional coordinates of the conventional cell. Centred groups (cm, cmm) include the (½, ½) centring translation.
- Fractional operations become pixel transforms through the lattice basis: oblique (a, 0) and (a/3, b), rectangular and centred a × b, square a × a, and hexagonal (a, 0) and (−a/2, a√3/2). Cell stretch sets b = a × stretch for oblique, rectangular and centred lattices.
- The motif (a 0–100 box) is scaled to a share of the cell width, rotated and centred at a fractional position in the cell. Every symmetry copy is translated by all lattice vectors whose copy touches the rectangular repeat tile, so shapes crossing an edge appear on the opposite edge and the tile repeats seamlessly. The repeat tile is a × b, a × 3b for oblique lattices and a × a√3 for hexagonal ones.
- Coloring by symmetry operation cycles the motif colors over the operations of one cell; this is a coloring choice and can make the color pattern less symmetric than the shape pattern. Custom paths accept M, L, H, V, C, S, Q, T and Z commands only.