Tessellation of hexagons, squares and triangles.

Grid generating the tessellation.

There is a nice semiregular tessellation with sixfold rotational symmetry made of hexagons, squares and triangles. Its grid is easy to find. From two of these grids we get a rather complicated tiling with twelvefold rotational symmetry:

Quasiperiodic tiling resulting from doubling the grid.

Note that we have now some single hexagons. At the lower right we have a large nearly symmetric structure. Especially if one thinks of units of one rhomb together with a triangle or two hexagons together with two rhombs. In these units there are always two possible positions for the triangle or the hexagon. If these are not distinguished, then there would be much more symmetry.

An interesting structure arises in the lower left, up to a ring with connected squares. Alternating, there are single squares and four squares making a larger square. As the double grid becomes very complicated I do not further look at this.

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I came across this image while doing a google search for tessellation images of regular convex polygons for my quilting projects. I had been trying to figure out how to create tessellations with hexagons so I could turn a corner in a section of a “Grandmother’s Flower Garden” quilt, which is simply hexagons sewn together, and have the hexagons oriented 90 degrees to how they had been. Now I see this image, I realize I was SO close, with the introduction of these 30 degree skinny diamonds! The more I look at this image above, the more I see ways to use all these shapes to create perfectly symmetrical designs. I love that you’re using your math skills to alter photographic images, and create beauty. Have you ever considered creating millefiori quilt patterns?

Thanks for your comment. I have looked up millefiori quilt patterns. They are very interesting and beautiful. Some resemble quasiperiodic tilings with “defects” that make them more symmetric and flowerlike. Currently, I am trying to create same new quasiperiodic tilings with more rosette-like symmetric centers.