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Why are there only 3 regular polygons that tile the plane?

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This is because of the interior angles of the regular polygons. For the regular polygon to tessellate the plane, the interior angle needs to be a factor of 360 degrees. Consider, for example, the pentagon. read more

For the regular polygon to tessellate the plane, the interior angle needs to be a factor of 360 degrees. Consider, for example, the pentagon. The interior angles of a regular pentagon are 108 degrees. read more

Well, if you tile the plane by congruent regular polygons, there must be $n$ polygons meeting at each vertex. Thus the interior angles of each polygon must be $2\pi/n$, for some positive integer $n$. For $n=3$, we get polygons with angles of $2\pi/3$, which are regular hexagons. read more

Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges. There are seven families of isogonal each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. read more

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