Tiling with 12-fold symmetry (n=6) using the generalized method
The tiling for n=6 showing 12-fold symmetry is shown below.
There is no triangle in this tiling, making this different than the older tiling for n=6.
The following pictures show at least the first 3 inflations of each of the prototiles.
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Tile 1, first 5 inflations PDF |
Tile 1, 6th inflation PDF |
Tile 1, 7th inflation PDF |
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Tile 2, first 5 inflations PDF |
Tile 2, 6th inflation PDF |
Tile 3, first 5 inflations PDF |
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Tile 4, first 4 inflations PDF |
Tile 5, first 3 inflations PDF |
Tile 6, first 4 inflations PDF |
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Tile 7, first 3 inflations PDF |
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Periodic tiles and fixed points
In looking at periodic tiles, there are two pairs, with cycle 2. One pair is the bottom middle and right crosses, each having the other at its center. The other pair is the double-crosses at top-right and bottom-left, again having each other at their center.
There is one fixed point vertex pattern in the tiling, emerging at every vertex, where the mandala below emerges with higher iterations. This mandala was created using the fifth iteration of the thin rhomb.
Other comments
This was my first time to try inkscape, and to try using a single color with shading. I originally liked the effect, and the set of prototiles has a certain beauty. I say originally, because when I saw the beautiful rendering by Dieter K. Steemann on Pinterest, I realized I will never do a single color again.
Dieter Steemann’s additional pictures from the tiling made me realize the fact that 4 of the prototiles are period 2, pairwise. Here is where you can check out more on Dieter K. Steemann’s pinterest site.
Back to Filling the Gaps n-fold tiling.
Copyright 2020 by Jim Millar