Sunday, March 13, 2016

A chevron tiling

I became interested in tessellating shapes based on the template of a chevron because of an oddity in Grunbaum and Shepard’s classification of tessellating isohedral types. In their 93 classes there are twenty four that have at least one edge that is shapeable and at least one edge that is an unshapeable straight line. Nineteen of them can be viewed as a bisection or trisection of other classes with symmetry that both tiling and tile share and four have symmetry that allows them to be bisected. The one that does not fit into the pattern is IH58. It lacks a parent, but it if had one, it would be a tile that has the template of a six-edged chevron.
(An example of IH58 is shown below on the top. It has four edges. Two opposite edges are straight lines. The other two opposite edges are formed with center-point rotation and are translated. The tiling has pmg symmetry. The bottom part of the figure shows the parent tile and tiling. It also has pmg symmetry and is a special case of IH15. In IH15 each pair of adjacent edges formed with center-point rotation is shaped identically, so two different shaped edges can be used in their construction. IH15 can have a template of a convex hexagon; the parent of IH58 must have one interior angle equal or greater than 180 degrees. Bisecting IH15 tilings on the axis of symmetry results in IH49 tilings.)
An earlier post mentioned how equilateral chevrons with six edges could tessellate and how the template of that equilateral template could be fitted with identical centro-symmetric edges. Tiles based on equilateral triangles, equilateral rhombuses, and regular hexagons with identical centro-symmetric edges can be placed so that only vertices meet and no edges, resulting in patterns of tiles and voids. (See the posts on Fab FoursFab Twos?, and An Unfinished Journey. ) There seems to be no way that equilateral chevrons can be arranged so all six vertices of the chevron will meet with no common edges, but in the pattern below five vertices meet with no common edges. The tiles are equilateral chevrons and the voids are equilateral triangles and hexagons.
If the chevrons are replaced with tiles with identical centro-symmetric sides, patterns such as the following result. The voids are shapes with identical centro-symmetric sides formed on templates of equilateral triangles and regular hexagons.
Below is another arrangement of chevrons with five corners touching. There are several others similar.
I have not found a use for the above, though perhaps I could include a few examples in Delightful Designs: A Coloring Book of Magical Patterns if I ever revise it. Here is what the previous figure would look like as a coloring page. How would you color it?
(Retrieved after being accidentally deleted.)