Tilings That Are Not Edge-to-edge
| With star polygons | |
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Octagrams and squares |
Dodecagrams and equilateral triangles |
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Two tilings by regular polygons of two kinds. Two elements of the same kind are congruent. |
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| Tilings by convex polygons | |
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Six triangles surround every hexagon. No pair of triangles has a common boundary, if their sides have a length lower than the side length of hexagons. |
Squares of two different sizes. Several proofs of the Pythagorean identity are based on such a tiling, for all right triangles that are not isosceles. |
Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive; there are eight families of such uniform tilings, 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 about this topic: Tiling By Regular Polygons