Friday, August 3, 2018

Butterflies in Mazes Escher Would Like

Here are some further notes on the mazes in Mazes Escher Would Like.

One surprise I found was a hexagonal pattern that looked like it fit into a rectangular grid but when I made a maze with it, I found that it drifted. Below is a moth pattern that does not drift. The edges are labeled clockwise, with 1 being the horizontal edge on the right. Notice that to go down the column, one follows the pattern of 35263526 etc.

Below is a different butterfly pattern with its six edges labeled in the same way. Notice following the 35263526 route takes down and to the right.

One of my maze-making programs allows a correction to be made so that the maze will line up correctly.

Perhaps the most interesting pattern that I have found in the past few years is of butterflies such as the ones below.

The shape of the tile has mirror symmetry but it does not fit into any of the symmetrical types of Gr├╝nbaum and Shepard's classification. The tiling above is type CG1CG2G1G2. Because of the symmetry of the the shape, the translation block appears to be two rather than four. 

There is another tiling possible, a TCCTCC type, shown below. To see the difference between the two, follow a diagonal from southwest to northeast. Below all the shapes have the same orientation along this diagonal. Above there are two oriented the same, then two different, etc. 

Some of the other shapes that are included in the book have appeared in posts on this blog during the past few years and many have appeared in coloring books or in Exploring Tessellations, but not in other maze books.

Seventy-nine of the 83 mazes use tessellations. Ten are geometric or abstract shapes, one is a symbol, 13 are mammals including people, 7 are other vertebrates, 11 are invertebrates, 6 are non-living objects, and 32 are bird shapes.