Methods of Construction

The simplest way of creating such a pattern is to structure the polygon in question (such as a pentagon) within an implied regular grid of triangles, squares, or hexagons.

For example, three pentagons can be inscribed within a regular triangle as follows:

That is, if you start with a regular triangle and attach a pentagon to each base, then the resulting three pentagons can in turn be inscribed within a regular triangle. At that point, you're home free. We already know that regular triangles will tile the plane, so these sets of three pentagons will also repeat naturally within the plane.

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