Doubling the tesselation of dodecagons and triangles

Semiregular tessellation of dodecagons (in blue) and triangles.

Grid for the tesselation.

It is still too hot for my brain to do something really new and thus I continue with the semiregular tessellations. Today I am looking at the tessellation of dodecagons and triangles. Dodecagons are regular polygons with twelve sides. The grid combines the grid of rhombs for the tessellation of hexagons and triangles with the grid of triangles for the tessellation of hexagons, see the last two posts.

The doubled grid gives this tiling of twelve-fold symmetry:

Quasiperiodic tiling resulting from the tessellation of dodecagons and triangles.

Ring of dodecagons.

Part of the double grid making a ring of dodecagons.

Many things are similar to the tilings with hexagons. Dodecagons come in pairs. Below the center we see a characteristic ring of dodecagons. There are parts of rings higher up at the left and the right. It is difficult to find out which part of the double grid results in rings.

Clearly the points of the single grid with twelve lines meeting are important. They are the corners of hexagons and at their centers. It seems that rings arise if two of them form a twelve-pointed star. The length of the side of these hexagon equals the distance between the centers in exactly the same way as for the Socolar tiling, see “Doubling the tessellation of hexagons“. Thus this tiling is related to the Socolar and the shield tilings.

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