# Neat Alternating Tilings of the Plane

Neat Alternating Tilings of the Plane

If adjacent tiles in a (polygonal) tiling always have a full side in common, the tiling is called neat.

Let a tiling be constructed from two sets of prototiles (e.g. polygons with and vertices, respectively).

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Then the tiling is called alternating if any two adjacent tiles T1 and T2 are not in the same prototile group (e.g. have different numbers of vertices).

In this Demonstration we will investigate neat alternating tilings of the plane with -gons and -gons.In other words, our tilings will have the following properties:• Each tile is a polygon.

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• There are only two prototiles used in the tiling.

• The tiles have either sides or sides where and are fixed.

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• Two adjacent tiles cannot have the same number of sides.

• The tiles are joined edge-to-edge.

Among all solutions for any given pair (, ) we will try to find the ones with the least number of prototiles.

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In his book New Mosaics, the author actually tried to find ALL known tilings with the smallest number of prototiles.

Some of these solutions are non-periodic.

In this Demonstration we restrict ourselves to giving only a few examples for each case {, }.

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