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.

A bit more on tessellating birds (Revised)

A previous post mentioned that by using TesselManiac! it is possible to get reasonable standing bird tessellations with 17 of the 28 Heesch types. In addition, some of the types allow more than one final product. For example, below is a simple tessellation of a quadrilateral using a TGTG type, with the top of the bird on a G side.

 If we do the top of the bird on the T side, we get a different design and arrangement. It is still a TGTG type because the Heesch identification always begins with a T if there is a T side. (If there is no T side, it will begin with a C if there is a C side. The other sides follow in clockwise order.)

 The CG1CG2G1G2 pattern in the previous post made the G2 sides small. Below is a design from the type with the C sides made very small. You can see them in the beaks and in the small part of the tails that overlap

Another variant that looked promising was taking the hexagonal TG1G2TG2G1 type and drop the legs on one of the G edges rather than on the T edge. My limited playing with it was disappointing and not worth developing.

Finally, the pentagon type CG1G2G1G2 in the last post had the C side placed on the tail. It could also be placed on the bill, as below.

Sometimes the end result hides the TesselManiac! starting point. The pattern below looks like TGTG. However, it was created using a hexagonal TG1G2TG2G1 type. The lines were manipulated so that one of the G pairs was eliminated.

The stork pattern below appears to be a TG1G2TG2G1 type, but it was constructed using a TGTG type in TesselMania. (It was done in 2012.)

In the original posts about these tessellations (which this and another post replaced) I did not pay enough attention to that possibility that the starting point in Tesselmaniac might not identify the final result. I originally used the design below as an illustration of TGTG, but it touches six adjacent birds, so it is more correctly labeled a type TG1G2TG2G1.

My original example of a CGCG type also touched six others and should be classified as CG1CG2G1G2

Also a CG1CG2G1G2 type is the example originally presented as G1G2G1 and built from that starting point in TesselManiac!.

The example I gave originally for a pentagonal CG1G2G1G2 type was constructed in TesselManiac! from that type, but I overlapped lines on the bills to get what is actually a CG1CG2G1G2 type.

I have learned a lot about Heesch types from this exercise and am beginning to get comfortable with the terminology. I corrected a lot of mistakes in these revisions of the original posts and hopefully have not left or introduced any major ones.

(Revised 02/20/2015)

Exploring tessellating birds (Revised)

In my a previous post I mentioned that I had updated  A Tessellating Coloring Book with about 25 new designs and that I would do a follow-up post on bird tessellations.

The design below was used in the original edition of A Tessellating Coloring Book. It is a simple type TTTT tessellation of a quadrilateral. The top is repeated on the bottom and the right side is repeated on the left side. M. C. Escher had a tessellating bird that is used the same TTTT tessellation and is similar.

(People doing tessellations use a terminology named after German mathematician Herman Heersch. A T or translation moves one side to the opposite side. A G or glide moves a side to the opposite side with a flip or to an adjacent side with a flip and a rotation. A C or center point rotation has one half of a side rotated to form the other half. There are also corner rotations of 60, 90, and 120 degrees. One place that this terminology is explained with examples is here.)

Using TesselManiac! I found some other ways to tessellate a standing bird. Below is one based on a hexagon using a type TTTTTT tessellation. Each of the six sides of the hexagon is repeated on the opposite side: top to bottom, etc.

Below is a type TGTG tessellation. The left is repeated on the right, but the top is flipped and shifted to the bottom.

The tessellation below is formed from a hexagon and TesselManiac! tells me that this is a TG1G2TG1G2 transformation. The top of the image is repeated on the bottom, the T part. The upper right side is flipped vertically and horizontally to form the upper left side, and the lower right side is also flipped vertically and horizontally to form the lower right side.

The above patterns appear in A Tessellating Coloring Book. However, once started, I wondered how many other Heersch types could be made to work with a similar standing chicken/duck tessellation (bird standing on the back of a bird).

 Below is a type CGCG transformation of a quadrilateral. The top is flipped horizontally to form the bottom. Both tops of sides are rotated 180 degrees to form the bottoms of the sides.

Another pattern I found that worked used a G1G2G1G2 transformation. The top is flipped horizontally to form the bottom and the right side is flipped vertically to form the left side. (The sides look like they could be C sides, but if they were, the birds would be head-to-head and tail-to-tail.)

Here is a TCTC type on a quadrilateral.

I was not happy with what I got from a pentagon type TCCTC in TesselManiac!, but I later got a much better result of the type in TesselManiac! with a TCTC starting point. The better result is below.

Type TCTGG, another pentagon type, is one of the patterns that lines up four different positions of the shape in a row rather than in a box.

A third pentagon type, a CG1G2G1G2, will work better for mazes than the two above. The C part is in the tail.

Below are four more based on hexagons. The first of type CG1CG2G1G2 with very short G2 pairs.

It took a some time and effort to get something decent from the TG1G1TG2G2 type.

A type TCCTGG has four different positions of the birds in a row.

A hexagonal TCCTCC standing bird turned out better than I expected. Two of the C sides are very small, the one that forms the top of the bill and the one that forms the back of the tail. (More interesting shapes are possible if one does not constrain the results to line up in a square grid as I did below.)

Finally, it is even possible to get standing birds with a triangle CGG tessellation.

A similar shape will tessellate for a quadrilateral CCGG

 and also for a quadrilateral G1G1G2G2.

There are 28 Heersh types and at least 17 will produce recognizable birds standing on the backs of other birds: seven of the quadrilaterals, three of the pentagons, and six of the hexagons, and one of the triangles. When I began this exploration, I did not expect to get that many. All eleven Heesch types not included above are composed solely of various types of rotations, either center point rotations or corner rotations of 60 degrees, 90 degrees, or 120 degrees.

Some of the Heesch types above can be yield additional ways of tessellating a standing bird and often when using TesselManiac! the end result is not the Heesch type that the template says it is. Those are topics for another post.

(Revised 04-20-1015)

Updating A Tessellating Coloring Book

While working on Holiday Mazes, I discovered that functionality of TesselMania had gotten new life in the form of a new program, TesselManiac!. TesselMania no longer worked well with new operating systems--even with Windows XP one had to adjust the screen to 256 colors to make it work. Playing with TesselManiac! I developed several designs that I used to update The Big Book of Princess Mazes, but I had lots of material left over. Because I have not thought of another theme that would work well for a maze book, the obvious place to use some of these designs was in A Tessellating Coloring Book.

The new and improved A Tessellating Coloring Book replaced about a quarter of the original designs with new ones. I added a some bugs.

 A second one resembles a pattern of spiders, though the two middle legs are joined.

 Birds seem to keep coming up when I play with Tesselmanic.

I will do a post later that explores more bird tessellations.

A Tessellating Coloring Book can serve as a coloring book for a child or a source of ideas for tessellation projects. It is available on Amazon and CreateSpace. (Also available is a coloring book that is devoted to tessellations of the letters of the alphabet, A Tessellating Alphabet Coloring Book.)