Thursday, March 4, 2010

Extrinsic Vertices from mixed n-gon tilings.

Plots of the extrinsic vertices (my term) for tilings from mixes of 4-, 5-, and 6-gon regular polygons with 3-gon tiles are generally more interesting than those with just triangles and one of the 4-, 5-, or 6-sided regular polygons. Less variegated plots come from tile sets that include only polygons with angles that are multiples of 18 degrees (pentagons and triangles that subdivide them), or multiples of 30 degrees (hexagons and their triangles), or multiples of 45 degrees (rectangles and their triangles).

These images show that it is not the complexity of the tiling, but the tile set itself that determines the patterning of the vertices and extrinsic vertices. The first image is from an animation. The second image is the tiling that is the underlying structure for the first and last images. The last image of the extrinsic vertices reveals its regularity. The tile set includes a pentagon and two triangles whose angles are multiples of 36 degrees. The tiling is highly irregular, including gaps that can't be filled with the tile set. It is an edge-to-edge tiling though.

Compare these images to those from my last post where the tile sets are generally mixed 4-, 5-, and 6-gon regular polygons with 3-gon tiles.