A journey through Heesch types and beyond

Months ago I began trying to sort the various tessellations I had developed for mazes into their appropriate types. (See past posts on this blog for background information.) The impetus for this project came from playing with the program TesselManiac! to see how many of the program's templates (or Heesch types) could be used to tessellate standing birds. The project seemed quite limited when I began and I was unsure how I would put together the results, but the more I learned the more I realized how much more there was to learn. Starting with the Heesch classification of 28 types, I quickly learned that there was another classification, Grünbaum and Shepard's 93 isohedral classes. Trying to make sense of the 93 classes took months. An unexpected and short e-mail exchange with another person doing tessellations made me realize that I needed to become comfortable with the 17 symmetry groups. Sorting my letter tessellations, I found some that I could not classify and was introduced to anisohedral tiles and tilings. So far I have not ventured into the topic of aperiodic tessellations and I lack the tools to display them.

Several times I have thought that the project was finished only to have some further insight or find a new way of seeing a topic that required additions or reorganization. I have once again reached a point where the book looks finished but I will not be surprised if further revisions are needed.

In the process of writing the book, which has the title Exploring Tessellations: A Journey through Heesch Types And Beyond, I created many new tessellations. I wanted at least one bird tessellation to illustrate each of the 28 Heesch types, and though a few are weak, I was able to accomplish this. I had no examples for some of the isohedral classes so I needed to develop those. Most of these are abstract, geometric tilings.

The book begins with illustrations and explanations of the 17 wallpaper groups of symmetry and then examines the 28 Heesch types, avoiding as much as possible mathematical jargon. Next is a look at the isohedral classes, followed by themed examples. The book ends with a few mazes to illustrate how tessellations fit maze making. The book is more advanced than the short introductions to tessellations aimed at children but it is simpler than those written for people who want the mathematics of the topic explained. It contains hundreds of illustrations with comments.

Past posts have illustrated many of the bird tessellations done for the book. Below is a shark tessellation that is featured on the book's cover.

 The 93 isohedral classes include the 28 Heesch types, 21 classes with unshapeable edges, 24 with both shapeable and unshapeable edges, and 20 that restrict the edges of Heesch types, often resulting in tilings that satisfy more than one Heesch type. The book ignores the 21 groups that have no shapeable edges. They are grids of polygons that often cannot be distinguished if they lack internal markings. In trying to make sense of the rest, I eventually realized that if a class has mirror symmetry of both tile and tiling, a bisection (or in one case a trisection) resulted in a tiling that formed another isohedral class. Below on the top is an example of IH12 with its bisection, IH22, on the bottom.

 Exploring different ways to make puzzle pieces, I found that there were only two tiles possible if the template shape was a square and if the edges were identically shaped with central symmetry. However, one of them fit together in several ways.

Playing with different ways to form square tiles, I found that there were only four possible tiles if the sides had identical central rotation. Three of them had symmetry and fit various special isohedral classes. The fourth was asymmetrical. Below is an example.

One final thing I found as I worked on this is only a few people are seriously interested in tessellations. World wide the number is probably only in the hundreds.

Exploring Tessellations is available from CreateSpace and Amazon.

Tessellations are all around us

The picture is of the metal platform at one end of a caboose that is at the Whistle Stop Restaurant and Monon Connection Museum north of Monon, Indiana. Note the cmm symmetry. It is an example of isohedral class IH67, which is a bisection of isohedral class IH17. IH17 is hexagonal, with two opposite edges straight and unshapable. The other four edges are identically shaped with central rotation. One half is the mirror image of the other. It fits six of the seven hexagonal Heesch types, all but C3C3C3C3C3C3. There is also a second way of bisecting IH17, by cutting the example show below horizontally rather than vertically. That will yield an IH26 tiling. Finally, IH17 can be quartered, resulting in an IH54 tiling.

This is a tiny bit of what I learned during my Journey through Heesch Types and Beyond.

Restricted tesselation types part 2

(This post is a continuation of a previous post. It will not make much sense except as a continuation of that post.)

IH34 is a restricted rhombus with 60 and 120 degree angles of Heesch type C3C3C3C3. All side are identical. It can be created using Tesselmaniac with template type CC6C6 and ignoring the C line. The example below is the vaguely S shaped figure. (In all of these examples, some lines have to be ignored or eliminated to get the desired tile.)

IH57 is a restricted parallelogram of Heesch type CCCC or TTTT. All sides have central-point rotation with opposite sides identical. It can be created using the Tesselmaniac template for TCTC and ignoring one of the C lines between tiles, the yellow line in the example below.

IH59 is a restricted rhombus of Heesch type G1G1G2G2 with all sides identical but without mirroring. The easiest way to get this tile in Tesselmaniac is with the CGG template, dropping the C sides. 

IH61 is a restricted square of Heesch type C4C4C4C4 with all sides identical. It can be created in Tesselmaniac using the CC4C4 and dropping the C section to unite the two segments.

IH62 is a square of Heesch type CCCC and also C4C4C4C4 with all sides identical. It can be created in Tesselmaniac with the template for C4C4C4C4 using only one set of the C4C4 pairs.

IH73 is a variant of IH61 with the further restriction that all sides are symmetrical around their middles. This cannot be exactly created in Tesselmaniac because Tesselmaniac does not have the drawing tools to line up points or segments exactly. Below is an approximation of the type.

These various special cases of Heesch types are mostly useful for generating abstract, geometrical shapes.

More information and more examples of these isohedral classes are available in Exploring Tessellations: A Journey through Heesch Types And Beyond. It is available from Amazon.

More Tessellations: A Coloring Book

Before I continue with posts about Exploring Tessellations: A Journey through Heesch Types and Beyond, I would like to mention a spinoff of that effort. For some of the isohedral classes that have shapeable sides, I had no examples in the collection of designs I used for mazes. For others I had only weak examples. I needed to develop examples that would clearly illustrate each class and show contrasts with other classes. As I finished the book (if in fact I have finished it —I expect I will be making revisions as I discover mistakes and as I learn more—there is so much more to know), I decided to reuse material in the format of a coloring book. The creation of More Tessellations: A Coloring Book was quick and easy compared to the time I have spent on Exploring Tessellations.

The book has the same format as an earlier coloring book of tessellations, A Tessellating Coloring Book. The cover is more attractive and an added feature is that the last page of the book identifies the Heesch type, isohedral class, and symmetry group of the designs used on each page. Three years ago I did not know enough about tessellations and symmetry to do a similar page.

The pages have many pictures of standing birds, though not those previously used in A Tessellating Coloring Book. (For some reason I find birds the easiest motif to tessellate.)

 I was very pleased with finding a bird design for Heesch type CC3C3C6C6.

These birds form a C3C3C3C3C3C3 type tessellation.

I like the abstract appearance of this pattern that on closer inspection is composed of mites (though they do not have enough legs).

IH90 is type CCC with all edges identically formed. It makes a visually appealing pattern

 IH14 is special case of type TTG1G2TG2G1 in which the TT edges are unshapeable straight lines and the four G sides are shaped identically. The right side of the tile is a mirror image of the left side. The illustration shows two ways it can be formed with the same sides. It has cm symmetry.

In a previous post I showed an illustration of class IH9. So far I have not converted it in Fontographer. The coloring book has a different illustration of IH9, shown below.

Some pages have more than one tessellation pattern. In the picture below are
examples of IH71 and IH61. Both patterns have all edges identically shaped, but the way the edges are arranged differs from one pattern to the other.

Here is the finished version of IH18, illustrated earlier while still in Tesselmaniac. It has p31m symmetry.

There is a very limited audience for all the books I have designed. I hope that the few people who eventually buy this one will enjoy it. It is available from CreateSpace and Amazon.

Restricted tessellation types Part 1

A couple of previous posts (here and here) looked at Heesch classification of tessellation types. As I dug deeper into the geometry of tessellations, I found that there was another classification scheme that is popular. Branko Grünbaum and Geoffrey Shepard in their book Tilings and Patterns (Freeman, 1987) found 93 isohedral classes of tilings. Tiles in isohedral tilings must have the same shape and size, must fit together to fill the plane with no overlaps or gaps, and the tile must play the same role throughout the pattern, which means that it must fit with its neighbors in only one way. (So a pattern in which some of the tiles have five neighbors and some have six is not isohedral.)  Twenty eight are the familiar Heesch types and another 20 fit Heesch categories but have additional restrictions on how some edges are formed. Some of these restricted cases result in tilings that fit into more than one Heesch type and about half result in tiles that have mirror symmetry. The other 45 have one or more lines that must be straight. For a complete list and a much more detailed explanation of these 93 types, see freespace.virgin.net/tom.mclean/index.html.

TesselManiac! not only gives the Heesch type for each of its templates but also the isohedral class (IH) number. In addition to the 28 Heesch types,  TesselManiac! has templates for eight isohedral classes that have mirroring. (Twelve other restricted Heesch types do not have templates in TesselManiac! but most can, with only a little work, be formed using TesselManiac!, though to really see them properly the design must be taken out of TesselManiac! and redone in a different drawing program, eliminating unwanted lines. (I do this in Fontagrapher if I need to do it.) This and a future post illustrate these twelve restricted types and explain how Tesselmaniac can be made to produce them. (Kevin Lee seems not to have omitted these twelve from TesselManiac! because his focus was on Escher-like tessellations, not ornamental or abstract tessellations. )

IH8 is a restricted TTTTTT/TCCTCC hexagon. Each pair of opposite sides is equal and parallel. The sides have a central point rotation and are translated to opposite side. It can be constructed with the template for the pentagon TCCTC type with the final C side eliminated. It is a challenge to find visually interesting patterns for this type as you can see in this example from TesselManiac!. (Note that the straight line needs to be eliminated, which means that the design must be copied in another drawing program. The same note holds for all of the types discussed in this and the next post.)

IH9 is a restricted hexagon of TG1G2TG2G1 type. Opposite sides are parallel, with one pair equal at one length and with a central rotation shape and the other four sides equal at a second length with identical glide sides. It can be constructed using the CGCG template and ignoring one of the C lines as shown below. (The green line and all of its corresponding copies need to be removed.)

IH10 is a  restricted TTTTTT/C3C3C3C3C3C3 hexagonal tiling with identical T sides. It can be created in TesselManiac! with the C3C3C3C3 template. (All the straight lines need to be removed.)

IH11 is a restricted hexagon of that fits several Heesch types and is a special case of IH8 with more symmetry and thus more visual interest. Each of the six sides is identical with a central point rotation. It can be created in TesselManiac! in at least two ways. One way is with the CC6C6 template using only the C sides.

It can also be done with the C3C3C6C6 template.

The IH18 type is a more restricted variant of IH 10. It is a regular hexagon with all sides the same shape, but the sides also have central symmetry. It takes a little artistic talent to get it just right in TesselManiac! (but is easy in Fontographer because of its line manipulation tools). Below is an attempt to form it in TesselManiac!. (The straight lines forming the regular hexagons need to be eliminated.)

IH90 is the only restricted triangle in the IH classification system. It is a Heesch CCC type with all sides identical. It can be done in at least two ways in TesselManiac!.  One way is with the  CC3C3 template using only the C or central rotation sides.

It can also be done with C3C3C6C6 template using only the C6 sides. (Of course the straight lines need to be eliminated to get the final result.)

Most of what I term "restricted Heesch" classes change the symmetry of the tessellation. Tessellations that fit into most of these classes have additional symmetry, usually with with reflection. These tessellations belong to symmetry groups that have an "m" in their name such as cmm, pmg, and p31m.

Playing with this aspect of tessellations was an early step in what eventually has become a book unlike any of the others that I have published via CreateSpace. The title is Exploring Tessellations: A Journey through Heesch Types and Beyond. It is available from CreateSpace and Amazon.  More information about it will be included in future posts.

Exploring Symmetry Coloring Book

I recently published a new book and it is not the one that I have been working on for the past five months. Rather it is an offshoot from that book.

In the process of working on the still uncompleted book, I realized I needed to get much more comfortable with symmetry. Mathematicians have shown that there are only 17 groups of two-dimensional symmetry in patterns that periodically repeat in more than one direction. My way of getting familiar with these 17 groups was to review the patterns that I have used to make mazes and classify them by group. Making a book ("writing a book" is not an accurate description of the process) organized my exploration of the topic and the coloring book format was quick and easy. As I worked on sorting patterns into groups, I developed additional patterns and some of these are included in the book. The title is Exploring Symmetry Coloring Book and the book is available from Amazon and CreateSpace with a list price of $5.99.

The book starts with six pages that give a short explanation of each of the 17 symmetry groups with feet used to illustrate the patterns. This is how feet illustrates the p31m group:

 p31m is one of the more difficult groups to identify because there is another group with very similar symmetry called p3m1.  Both groups repeat themselves if rotated 120 degrees around their centers of rotation and both have mirror reflection.

Below feet illustrate a group called p4g. It needs to be distinguished from a group called p4m.

 After the short introduction explaining how one can identify groups of symmetry, the book has almost 100 pages of patterns that can be colored or decorated. The pages may also inspire or help readers to create new designs and patterns.

I designed a maze book with a railroad theme and used the pattern of rails shown below for a maze. It has reflection symmetry around both vertical and horizontal lines. It can reproduce itself it rotated 180 degrees. Finally, it has a staggered pattern that is caused by what is called glide reflection.The group that has these attributes is called cmm.

In designing a book of pirate mazes, I used a pattern with swords. In working on this book I found several other ways to use swords in patterns, such as this one which is p3. A benefit of working out thoughts in a book form is that it led to new patterns.

I tried to limit the number of designs that are fairly well known though I have used quite a number of them in mazes. The one below I found years ago in Tilings and Patterns by Branko Grunbaum and Geoffrey Shepard, It is another example of p4g.

There are other coloring books of geometrical and symmetrical designs that feature more complex and intricate designs and are probably better choices for those who want only a coloring book. Exploring Symmetry Coloring Book is not only a coloring book, it is also an introduction to the topic of symmetry. It is an educational coloring book.

Although young children can color the pages, only older people will appreciate the explanation of the symmetry groups. Supplementing the introduction, there is a page of notes at the end that identifies the groups used on each page.

Available from Amazon and Create Space.