Available from Amazon.

From the Introduction:

This coloring book is the result of an unexpected and serendipitous discovery made while I was studying tessellations, attempting to learn how many distinct, tessellating shapes could be derived from a square template when given four identical edges. My results showed that 15 distinct shapes were possible when the edge was asymmetric, only two when the edge had mirror symmetry, and four when the edge had central-point rotation.

What, you may wonder, is center-point rotation? It means that you form an edge by beginning with a line segment (1) that is shaped in some way (2), making sure that the line does not cross itself. A copy of this shaped line (3) is then rotated 180 degrees (4) and one end is connected to an end of the original shaped line (5). This connected line is then used to from a closed figure by rotation and/or flips. Not all lines will work; some will cross when they are arranged to make the four shapes.

The four distinct shapes that can be formed in this way all tessellate, that is, each shape can fit together with copies of itself to fill the plane with no gaps of overlap. So far none of this was too exciting, but then I rotated the shapes 45 degrees so that instead of fitting together edge to edge, they would fit together vertex to vertex, forming a checkerboard-like pattern of tiles and voids. The results amazed me because the voids took the same four shapes that the tiles had. (When I thought about it, I realized that this must happen because the voids share their four edges from the tiles and thus are confined to the same four shapes.)

Three of the four distinct shapes have either mirror or rotational symmetry and one is asymmetric. Two of the symmetric shapes have only two orientations, one has four orientations, and the asymmetric shape has eight orientations, giving a total of 16 possible tiles to use in creating patterns. Using just these four shapes in different orientations results in a huge number of possible patterns that can be altered by simply changing a tile.

I have searched for previous use of the magical possibilities of these sets of tessellating tiles and have found some that are close. David Bailey at http://www.tess-elation.co.uk has three of the four shapes and has done research on the family. "Rep-Tiling the Plane" in the May 2000 issue of *Scientific American* put all four shapes in a pattern. The article started with a completely different approach to tiling and never mentioned the importance of the edges having central-point rotation. If you know of previous use of patterns of the sort this book uses, I would like to hear about it.

The patterns in this book can all be considered tessellation patterns, though most are not patterns of a single shape. They are two-, three-, or four-tile tessellations, a type of tessellation that previously had not interested me.

In addition to using sets of edges to form tiles based on square templates, I have used the same edges to form tiles based on triangular and hexagonal templates. Some of these patterns that are visually interesting are included at the end of the book.

The story of how this set of tiles was found and some technical details about it is contained in *Exploring Tessellations: A Journey through Heesch Types and Beyond* (CreateSpace, 2015). If you would like to make your own patterns with these shapes, the typefaces that I used to design this book are available at myfonts.com. Search for FabFours.

(If you do not understand what this Introduction is saying, play with the patterns on the following pages for a while and then come back and read it again.)

I apologize in advance for errors that remain in the book.

Robert Schenk

November 2015