Tilings

This is a TikZ library for working with tilings. It can be used to define tiles which can be placed alongside each other to produce a tiling.

To use this library, load the TikZ package and then use:

\usetikzlibrary{tilings}

There are also libraries with pre-defined tiles:

• Penrose tiles

  \usetikzlibrary{tilings.penrose}
  

  This defines sets of Penrose tiles: kite/dart, the two rhombi, and the pentagonal tile sets.

• polykite tiles

  \usetikzlibrary{tilings.polykite}
  

  These are the tiles defined in the aperiodic monotile. Currently defined are the hat, the turtle, the meta, super, and sub clusters.

Each of these loads the main tilings library and then defines certain sets of tiles.

Defining tiles

The command to define a tile is:

\DefineTile{<name>}{<edges>}{<vertices>}

• The name is its label.
• The edges is a list of labels for each edge in terms of its matching pair. An edge labelled a will match against an edge labelled A. Edges labelled with numbers will match against themselves. Edges a to e (and A to E) and 1, 2, 3 are available by default, more edges can be defined if needed.
• The vertices is a list of vertices of the tile in the same order as the edges. The expectation is that these will go in the same direction around each tile (though this can be swapped when the tile is placed). The first edge goes between the first two vertices and so on. Each vertex is specified by giving its coordinates, these can be in cartesian form, { x, y}, or polar form, { θ: r} (θ in degrees), and can be prepended by one or two + signs to make them relative. These will be put through the LaTeX3 floating point parser.

When a tile has been defined it must still be baked. This is because there is the facility to replace the default straight edges by more complicated paths. To bake a tile is simple:

\BakeTile{<name>}

Using tiles

The simplest way to use the tiles is with the pic syntax of TikZ. When a tile is defined then it in turn defines a pic of the same name. The tiles can be positioned alongside existing tiles using a key:

align with=<tile> [back] along <edge> [using <index>]

• <tile> is the name of an existing tile.
• <edge> is the label of an edge of that tile; if there are multiple edges with the same label then they are further modified with an index, so the first listed edge of type a is a1 and so on (but if there is only one of a type it doesn't get an index).
• using <index> is for if the tile to be placed has multiple edges with the placing label; this is just the index of the edge.
• back this modifier reverses the alignment so that it matches against the edge in the "wrong" direction; this is useful if the previously placed tile was flipped.

As hinted above, a tile can be "flipped" using the key flip tile.

Various styles are used to render the tiles:

• <name> when used on the pic command (rather than directly as the pic type) this installs the <name> as the pic type and tries the style every tile pic.
• A tile is clipped against itself, and the clipping path tries the styles every tile clip and every <name> clip.
• The tile itself is rendered with every tile, every <name>, this tile, and the pic actions.

Tiles can be further customised using the keys:

• every tile before path
• every <name> before path
• this tile before path
• every tile after path
• every <name> after path
• this tile after path

These are simply executed and not applied to a path, so are provided as hooks at the corresponding stages of rendering to allow other code to be run; the arcs on the Penrose tiles are inserted using these styles.