Geometry: Plane Tiling Mathematica Package

By Xah Lee. Date: . Last updated: .

PlaneTiling is a Mathematica package that draws any of the 17 wallpaper patterns. This page shows some of examples of what this package can do.

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

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

wallpaper16 1001
Random kaleidoscope style wallpaper. This one has symmetry 632 (p6)
In this example, it has symmetry 3*3 (p3lm) and using pentacle as motif.

WallpaperPlot can also plot the symmetry diagram of wallpapers. Shown here are wallpaper groups with 3-fold symmetries. Any symmetry element graphics can be user specified. For example, the last image displays the rotation elements of group *632 (p6m), using numbers instead of traditional symbols.

wallpaperGroupElements13 wallpaperGroupElements14 wallpaperGroupElements15 wallpaperGroupElements16 wallpaperGroupElements17 wallpaperGroup17rot

Assorted graphics manipulation functions lets you transform or cut graphics.

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

Assorted functions generates lattices or networks colored regularly. Here are some examples of triangular network with regularly directed edges.

directedNetwork1 directedNetwork2 directedNetwork3 directedNetwork4
lattice networks

An example using the package to generate wallpaper designs. In this example, three separate motifs are mapped to a triangular lattice — a design with Islamic flavor. The motifs are generated by a PolygonMotif and two StarMotif with different parameters, then the motifs are mapped to a triangular lattice generated by LatticeCoordinates.
In this example, LineTransform is recursively applied to a triangular network, then CookieStamp2DGraphics cuts out a nice hexagon boundary.

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