WolframLang: Plane Tiling Mathematica Package

By Xah Lee. Date: . Last updated: .

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


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.

plane tiling 2024-03-12 234639
plane tiling 2024-03-12 234639
plane tiling 2024-03-12 234300
plane tiling 2024-03-12 234300
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 233921
plane tiling 2024-03-12 233921
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

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.

plane tiling 2024-03-12 234055
plane tiling 2024-03-12 234055

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.

directedNetwork1 directedNetwork2 directedNetwork3 directedNetwork4
lattice networks

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.

plane tiling 2024-03-12 234712
plane tiling 2024-03-12 234712
plane tiling 2024-03-12 234234
a pattern created by line tranform. plane tiling 2024-03-12 234234