WolframLang: Plane Tiling Package 📦

By Xah Lee. Date: 1997-10-30. Last updated: 2024-03-12.

PlaneTiling is a WolframLang / Mathematica package that draws any possible wallpaper patterns, and useful for creating any periodic tiling illustrations.

Download

https://github.com/xahlee/WolframLanguage-PlaneTiling

Notebook Example

plane tiling mathematica 2016-05-04 35001 — plane tiling package

plane tiling mathematica 2016-05-04 — plane tiling package

Features Gallery

Map a Motif to Lattice

generate patterns by mapping a motif to lattice.

xunre tarci — Red Stars relief. It is made by mapping stars to a triangular grid, then raise the center point of the stars, then turn on a non-colored light source.

perturb the lattice to get a slight random effect:

starry night — Starry Night, generated by putting perturbed star polygon on a perturbed triangular grid.

Wallpaper Design by Specifying Symmetry

The function WallpaperPlot allows plotting of any periodic wallpaper design. User specifies the symmetry type and motif. This means that it'll plot any periodic tilings, Islamic patterns, or Celtic knots designs

starSky 2024-03-12 gyds — In this example, it has symmetry 3*3 (p3lm) and using pentacle as motif.

wallpaper16 1001 — Random kaleidoscope style wallpaper. This one has symmetry 632 (p6)

Plot the Symmetry Diagram of Wallpapers

plot the symmetry diagram of wallpapers.

Any symmetry element graphics can be user specified.

plane tiling 2024-03-12 233934 — Rotation elements of group *632 (p6m), using numbers instead of traditional symbols:

plane tiling 2024-03-12 233952 — showing only glide-reflection elements of the group *632 (p6m), using red line.

Dissect Graphics

cutStar — This one is done by cutting a `StarMotif` with parallel lines in four directidions using `CutPolygon`, then scaling the polygons apart.

Lattice and Network

Function to generate lattice colored regularly. This allows you to map motif to dots, accordin to color code.

Function to generates directed graph laid out in a regular lattice. This allows you to apply line transform to the edges according to the arrow direction.

Line Tranform on Edges of Tiling

One technique of generating patterns is to apply Line Transform on the edges of regular tilings.

plane tiling 2024-03-12 234143 — a fractal tiling created by the technique of line transform on edges of a hexagonal tiling.

Line Transform is recursively applied to a triangular network.

a pattern created by line tranform. plane tiling 2024-03-12 234234