Sunday, July 8, 2018


The introduction to A Final Coloring Book of Tessellations noted that "as I stumble on additional patterns ... I can use the better discoveries to update and revise this and the previous books." In the past few weeks I have revised A Tessellating Coloring Book, More Tessellations Coloring Book, and A Final Coloring Book. New patterns (and thus deleted patterns) were six for the first book, eleven for the second, and seven for the last.

The three books contain over 300 tessellation designs and each is unique to one book. The first two of the books listed above were designed before the adult-coloring-book fad hit and were meant for children. The last book was intended for adults but it lacks the fussiness that many of the adult colorers seem to desire.

They all still have some weak designs and some geometric tilings that are in the public domain so if in the future I find additional interesting tessellating shapes, there may be more revisions.

Thursday, July 5, 2018

The ongoing journey

Exploring Tessellations: A Journey through Heesch Types And Beyond was published in October, 2015 and then for the next year was repeatedly updated as mistakes were corrected, new material added, and sections reorganized. The last of those updates was at the end of 2016. Since that last update I have designed an activity book and another maze book as well as updating two earlier maze books. The work on these books provided enough new material to justify going back and reviewing the content and organization of Exploring Tessellations.

This revision of June 2018 adds many new tilings as examples and deletes some old tilings that were substandard. It also reorganizes the section on Heesch types. The original organization of that section was based on efforts to find standing birds that tessellated. That whimsical organization was replaced with a more standard organization that stresses Heesch families. Heesch and his co-author showed that their list of 28 types consisted of nine main types from which the other 19 types could be derived by shrinking edges to zero. For example, type TTTT results when one of the TT pairs in type TTTTTT shrinks away.

These chess pieces were added as an example of isohedral class IH69. The tile has symmetry over a diagonal. It simultaneously fits Heesch types CCCC, G1G1G2G2, and CCGG. In this example two edges are straight, but that is not a requirement of class IH69.

This next image has been in the book from the start as an illustration of isohedral class IH91. It also has reflective symmetry. It is based on an isosceles triangle and one edge must be straight for it to have that symmetry. It fits both Heesch type CCC and CGG.
Finally, below is a new example of Heesch type CC3C3 (IH39), which is the hardest of the Heesch type to form into Escher-like tessellations. It could be a leaf on a twig or a rosebud.
The book has hundreds of other illustrations and is one of the few sources that explain both the Heesch classification and the isohedral classification of Grünbaum and Shepard.

Tuesday, June 5, 2018

Mazes Escher would like

Since Holiday Mazes was published in early 2015, I have continued to develop tessellations for several coloring books, an activities book, and Exploring Tessellations: A Journey through Heesch Types And Beyond. Recently I realized that I had more than enough new tessellation designs for another maze book focused on tessellations.

The result is Mazes Escher Would Like. The name is descriptive because the most of the mazes use Escher-like tessellations, that is, shapes that both tessellate and resemble real world objects. It is also an obvious attempt to have a title that might turn up in search results on Amazon, which is the only place that will be selling the book.

The book contains 83 mazes moderately difficult mazes suitable for older children and adults. Something new for my maze books is a table showing the Heesch types and isohedral classifications of each pattern used.

There are 36 IH groups represented and 24 of the 27 Heesch types. The Heesch types missing are C3C3C3C3C3C3, C3C3C3C3, and C3C3C6C6.

The book has a lot of bird patterns (32 of the 83) because for some reason birds pop out when I am looking for tessellations.

One of the odder bird designs in the book is the one below, shown in the sample maze that I use to keep track of the many maze typefaces I have. It makes an interesting maze.
It presented a challenge because my maze generation program only allows triangle-based mazes to have two orientations but this pattern has four. The solution was to print it using two typefaces. My maze generating programs were written in the over twenty years ago in a defunct computer language for an obsolete operating system that no longer runs on modern hardware. Sometimes it takes some effort and creativity to get the output I want.

In late 2015 I was playing with a group of tilings that I dubbed the "Fab Fours" and which found a home in the Delightful Designs: A Coloring Book of Magical Patterns. I tried to use a few shapes that I had found then to make mazes. An example is a three-edged shape that makes for a visually attractive maze.

It is not an Escher-like tessellation, but Escher also dabbled in various geometric shapes. (There are four mazes at the end that are not framed with tessellations. Three of them have over/under paths and I wanted to include a few just in case this is the last maze book I ever design.)

Tuesday, May 29, 2018

A couple of revisions to tessellating maze books

When I designed Tantalizing Tessellating Mazes in 2011 and 2012 I not only had enough tessellation patterns to produce 70 mazes for this book, but had enough left over to do another 70 mazes in More Tessellating Mazes. Many of the tessellations were common geometric shapes. At the time I did not see this as a shortcoming because I was more focused on mazes than on tessellations.

In the six years since the publication of these two books I have devoted considerable time to finding new tessellation patterns. Most of what I have found has been used in other maze books, but I early in 2018 I realized that I still had enough unused material to revise and upgrade both Tantalizing Tessellating Mazes and More Tessellating Mazes.

Both books had 70 pages of mazes and 35 pages of solutions, with two solutions per page. The revisions shrink the solutions so that four fit on a page. This change frees up space for an additional fourteen mazes with no change in page count. In addition I dropped patterns that I thought were the least interesting and replaced them with new patterns and mazes. As a result, both books now have a greater percentage of Escher-like mazes, that is, mazes based on shapes that resemble real world objects rather than geometric, abstract shapes. In Tantalizing Tessellating Mazes 35 of the mazes are new and 49 are carried over from the previous edition. The numbers for More Tessellating Mazes are 27 new and 57 holdovers.

The order of mazes has been changed. The new mazes are slightly more difficult than the mazes they replace, but they still seem best described as fairly easy and appropriate for ages 8 and older. Both books have many bird tessellations because I seem prone to find bird shapes as I toy with Tesselmaniac!. (Tantalizing Tessellating Mazes has 29 bird tessellations and More Tessellating Mazes has 21.)

Some of the shapes used in the books have previously appeared on this blog. Here is a bird tessellation illustrated with a small sample maze that I used to keep track of maze typefaces.

The pattern below comes from the quilting world. This maze allows not only horizontal and vertical passages but also passages through the corners. The shape is nothing special but the corner passages make even small mazes a challenge. (This is not a maze from the book but a sample maze to illustrate the pattern.)

Both books are available on Amazon and links to them are in the side margin.

Saturday, May 5, 2018

Tessellating activities

It has been over a year since I last published a book via CreateSpace, and since then CreateSpace has ceased to sell books; all sales are now through Amazon. If you click on any CreateSpace link in a previous post, it will take you to an Amazon page.

I have not spent much time or effort with tessellations (or mazes) since Tessellating Animals Activity Book was published in January 2017. A post at the time tried to indicate what kind of activities were in the book, but the illustrations were of partial puzzles that could not be solved. The sample pages that Amazon shows are even less revealing of what is in the book.

Below are examples of some of the types of puzzles in Tessellating Animals Activity Book. First is a mini-Sudoku. Normal Sudoku puzzles are 9-by-9 grids, but simpler versions can be constructed with 4-by-4 or 6-by-6 grids. The puzzle below uses numbers 1 through 6. All rows and all columns must have each number only once. In addition, there are six boxes indicated by the shading and each of those boxes must also have each of the six numbers only once.

The book contains 14 mini-Sudoku puzzles. (The bird-head shape was in preliminary versions of the book but was dropped before it was published because it is a low-quality design.)

A second type of puzzle is a word search, which also use grids. The grid here is provided by a weird shape that was never in any version of the book but which works well for this type of puzzle. Words can be vertical, horizontal, or diagonal and they may be reversed. Find the following words: reflect, valence, hexagon, rotate, escher, glide, flips, edge. The letters not used in any of these words will spell another word that is related to tessellations.

There are ten word-search puzzles in the book and all are larger than this one.

There are eleven pages of decoder puzzles in the book and most of those pages have more than one puzzle. In the puzzle below the letters have been replaced by their position in the alphabet: A=1, B=2, C=3, etc. 
This shape was a first attempt to find a standing bird that fit Heesch type TGTG. It is not very good and I have found better ways to illustrate the type.

The book contains a number of mazes, some of the traditional variety but others that are coded. They are simpler than the coded mazes in the book Hidden Path Mazes: Decode to Solve.

The various puzzles and activities are supplemented with explanations of basic characteristics of tessellations.

Friday, April 20, 2018

Tessellating wine glasses

Goblets, wine glasses, or chalices are easy to tessellate and if you search the Internet for images you should be able to find at least two ways in which the shape fits into a tiling pattern. Below is one of these ways.
The five-edged goblets are an example of isohedral class IH26. IH26 is the result of bisecting isohedral class IH17. If the straight bottom is considered an edge of central rotation, the pattern fits Heesch type TCTCC. If it is an edge of reflection, it is not a Heesch type.

If we slide alternate rows, each glass has six neighbors rather than the five in the pattern above.
When the six-edged goblets are centered over each other as in the image above, we have an example of isohedral class IH15, which simultaneously fits Heesch types TCCTCC, TCCTGG, and TG1G1TG2G2.

The other easy way to tessellate this shape is to stack them in the pattern below.

This arrangement is an example of isohedral class IH12 that simultaneously fits Heesch types TTTTTT and TG1G1TG2G2. Four of the edges are identically shaped.

All six edges can be identically shaped and the image still resembles a chalice or wine glass.

However, this shape can also be fit together in a different way, one that I have not found on the Internet.

It is an example of Heesch type TG1G1TG2G2 but not of TTTTTT. Can you find the TT pair?

Wednesday, October 4, 2017

Isohedral classification

I have finally begun to understand the main thrust of Grünbaum and Shepard's classification of isohedral tilings, though there are details that still elude me. They begin with the eleven Laves tilings, tilings in which the angles at each vertex are equal. In the tiling below there are vertices with three line converging, so all the angles at these vertices are 120 degrees. There are also vertices with six lines converging, so the angles here must be 60 degrees.

Grünbaum and Shepard then consider the various rotational and reflectional symmetries that each of the tiles can have. There are ten for the regular hexagon, with twofold, threefold, and sixfold rotational symmetry, plus reflection over one side, three sides, one diagonal, three diagonals, reflection over one diagonal and one edge, reflection over three sides and three diagonals, and no symmetry at all. They then see how each of these possibilities can fit together consistently. Each of those ten yield at least one isohedral class; the one with no symmetry yields the seven hexagonal Heesch types.

Can there be a class formed from reflectional symmetry in the above figure? The answer is, "No". If we bisect the tile, we get the result on the bottom in the figure. Notice that some of the tiles have five neighbors and some have only four. The symmetry line cuts the base of the tile but the base of the tile never abuts another base. It would need to do so for it to make fit into its own isohedral class.  The only isohedral class that the tiling above yields is IH21, which is Heesch type CC3C3C6C6.