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GenerateTiling (1.0.0) current version: 2.0.0 »

Source Notebook

Generate a tiling from a set of rectangular array tiles

Contributed by: Wolfram Research

ResourceFunction["GenerateTiling"][tiles,{},size]

covers a size array with tiles.

ResourceFunction["GenerateTiling"][tiles,seed,size]

covers a size array with tiles and starting seed array.

ResourceFunction["GenerateTiling"][tiles,{},size,"Boundary" → "Periodic"]

covers a size toroidal array with tiles.

Details and Options

A pattern is a rectangular array of positive integer values suitable for use in the resource function TilingPatternPlot.

A tile mask is a subset of positions within a rectangular array, such as the a values in {{a,a,a},{_,a,_}}. This particular mask is also known as the Tetris T shape.

A tile is an array that can contain integers or blanks.

All tiles in a tileset fit in an array of the same size, say {a,b}. If all subarrays of that size in a larger pattern matches a tile in the tileset, then that tileset can be used to make the given pattern.

An all-zero tile

leads to an all white or all zero pattern

An all-one tile

leads to an all black or all one pattern

The two tiles

lead to a checkboard pattern

. No subset of the tiles will make a larger pattern, so these two tiles produce a minimal tiling.

The tileset

will produce the all-white pattern, but the second tile is not necessary. Therefore, this is not a minimal tileset.

The above patterns have a size of 4×4.

The input tiles should be an array of integer or Blank (_) values.

Consider

and {{_,0,0},{0,1,_}}. These both represent a 4-cell tile within a 6-cell array, where the integers indicate the coloring of the tile and the Blank (_) values are wildcards.

A tiling with a periodic boundary can be divided into identical pieces. Basically, this gives a tiling on a torus that may be used for an infinite tiling.

Examples

Basic Examples (3)

Generate a tiling for a set of tiles:

In[1]:=

Out[2]=

The tiles used:

In[3]:=

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The generated tiling:

In[4]:=

Out[4]=

Scope (2)

Generate a tiling from a complicated set of tiles:

In[5]:=

tiles = {{{0, 1, 1}, {1, 1, 1}}, {{1, 0, 1}, {1, 1, 1}}, {{1, 1, 0}, {1, 1, 1}}, {{1, 1, 1}, {1, 0, 1}}, {{0, 1, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 0, 1}}, {{0, 0, 0}, {0, 1, 0}}, {{0, 0, 0}, {1, 0, 0}}, {{1, 0, 1}, {0, 0, 0}}, {{1, 1, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1, 1}}, {{0, 1, 0}, {1, 1, 0}}, {{1, 0, 0}, {1, 0, 1}}};
tiling = ResourceFunction["GenerateTiling"][tiles, {}, 30];

The tiles used:

In[6]:=

Out[6]=

The generated tiling:

In[7]:=

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Generate a tiling around a {{1,1}} initial condition:

In[8]:=

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A tiling around a {{0,0}} initial condition isn't possible for this set of tiles:

In[10]:=

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Options (3)

Boundary (3)

Generate a tiling on a size 30 torus (i.e. a tiling with a size 30 periodic boundary):

In[12]:=

tiles = {{{0, 1, 1}, {1, 1, 1}}, {{1, 0, 1}, {1, 1, 1}}, {{1, 1, 0}, {1, 1, 1}}, {{1, 1, 1}, {1, 0, 1}}, {{0, 1, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 0, 1}}, {{0, 0, 0}, {0, 1, 0}}, {{0, 0, 0}, {1, 0, 0}}, {{1, 0, 1}, {0, 0, 0}}, {{1, 1, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1, 1}}, {{0, 1, 0}, {1, 1, 0}}, {{1, 0, 0}, {1, 0, 1}}};
ResourceFunction["TilingPatternPlot"][
ResourceFunction["GenerateTiling"][tiles, {}, 30, "Boundary" -> "Periodic"], {Frame -> False, Mesh -> None, PixelConstrained -> 4}]

Out[12]=

The tiles cover a size 12 section, but the toroidal wrapping between the top and bottom edge does not correspond to the tile set:

In[13]:=

Out[13]=

These tiles cannot cover a size 12 torus:

In[14]:=

Out[14]=

Neat Examples (3)

Generate a tiling from a complicated set of tiles:

In[15]:=

$tiles = {{{0, 0, _}, {0, _, 0}, {1, _, _}}, {{0, 0, _}, {0, _, 1}, {0, _, _}}, {{0, 0, _}, {1, _, 0}, {0, _, _}}, {{0, 0, _}, {1, _, 0}, {1, _, _}}, {{0, 1, _}, {0, _, 0}, {1, _, _}}, {{0, 1, _}, {0, _, 1}, {0, _, _}}, {{0, 1, _}, {1, _, 1}, {1, _, _}}, {{1, 0, _}, {0, _, 0}, {1, _, _}}, {{1, 0, _}, {1, _, 0}, {0, _, _}}, {{1, 0, _}, {1, _, 1}, {1, _, _}}, {{1, 1, _}, {0, _, 0}, {0, _, _}}, {{1, 1, _}, {0, _, 0}, {1, _, _}}, {{1, 1, _}, {1, _, 0}, {0, _, _}}, {{1, 1, _}, {1, _, 1}, {0, _, _}}}; tiling = ResourceFunction["GenerateTiling"][tiles, {}, 79, "Boundary" -> "Periodic"];$