FabTwo?

The recent post on the FabFours typefaces examined tiles based on a square template with identical edges formed with central rotation. These tiles could take only four distinct shapes and these four shapes would, when arranged vertex to vertex, produce voids that must have the same four shape as the tiles.

What happens if we replace the square template with the template of an equilateral triangle? For these triangular-based tiles only two distinct shapes are possible. One of the two shapes fits the criteria for isohedral class IH90. The other is formed by taking one edge of this shape and flipping it. Many of the visually-interesting edges from the tiles with square templates will fail when put on the triangle template because adjacent edges will overlap.

These two shapes form two sets of tiles, each with eight orientations. In each set of shapes, the IH90 shape has two orientations and the other has six. One set has its base on the bottom and the other is a 180º rotation of the first set. When one set is used, the other set forms the voids.  Below are examples of one set, with the IH90 shape in the first two positions.

 Here is the second set.

 In the first example below the red tiles leave voids that have shape of the IH90 tile. Notice that the red tiles are from the the first set and the voids are from the second.

The next example uses white tiles, so the tiles are not readily distinguishable from the voids.

If a rhombus (diamond) is fitted with identical sides of center-point rotation, there are exactly seven distinct shapes that result. The pattern above from a triangle template has all seven shapes based on the 60º-120º rhombus if we can erase some of the lines that separate a tile and an adjoining void. The seven edges that need to be erased to reveal the seven possible rhombus shapes are numbered in the picture above. (If we squeeze the rhombus template until it becomes a square, which is a special case of a rhombus, shapes 2 and 3 converge, as do shapes 4 and 6, and 5 and 7.)

Below are a couple more examples with different edges. They are taken from pages of Delightful Designs: A Coloring Book of Magical Patterns.

The visual interest from patterns formed with tiles of these sets may be more in seeing these rhombus-based shapes than in the patterns of the triangular shapes. Other interesting shapes result from combining blocks of six that share a common vertex and blocks of four centered on a middle tile.

Will what works for square and triangular templates work for hexagonal templates? The answer is,"No". Patterns based on the square and triangular templates have an even number of lines meeting at each vertex, four and six. The allows the tiles and the voids to alternate, a TVTV or TVTVTV sequence. Patterns based on a hexagonal template have three edges meeting at each vertex, so the alternating sequence of tiles and voids is not possible. In addition, all the shapes that can be formed with an identical side of center-point rotation using either a square or triangular template will tessellate. The same is not true when shapes are formed in the same way with a hexagonal template.

Anisohedral tiles and the 3G method

When I started working on Exploring Tessellations: A Journey Through Heesch Types and Beyond, I realized that tessellations could be classified by Heesch type and I understood that there were some that did not fit that classification, but knew almost nothing about those that did not fit the Heesch classification. As I sorted patterns and searched the Internet for more information, I discovered the isohedral classification of Grunbaum and Shepard.  It was not until I began to sort letter tessellations done for several maze books that I found I needed to learn what lay beyond isohedral tilings.

Below are two two tilings of the letter F, one on the right and one on the left. (They differ in how the backs of the letter F are positioned.) They do not fit into the Heesch types and they do not fit into the isohedral classification. The problem is that the tiles do not fit their neighbors in the same way throughout the pattern. Consider the top of the F. In some cases it is fit between the top of a neighbor and the middle bar, and in other cases it fits above the top of a neighboring F. The requirement for an isohedral tiling is that a tile fit its neighbors in the same way throughout the pattern. (There is a mathematical way to express this that is undoubtedly more precise but those without a mathematical background struggle to understand it.) Mathematicians call this type of tiling anisohedral.

The website that I found very helpful in trying to understand anisohedral tilings is www.angelfire.com/mn3/anisohedral/index.html. From that site I learned that mathematicians make a distinction between anisohedral tilings and anisohedral tiles. The two patterns above are anisohedral tilings, which means they are not isohedral. However, the shape used for the letter F is not an anisohedral tile because it can be fit into an isohedral tiling, such as that below, which is a Heesch type TCCTCC. The very bottom of the letter F connects to the top of the letter and these two edges are the TT edges, edges that are translated. The other four edges are all edges with center-point rotation. Notice how in each the edge fits into the same edge of a neighbor.

As I sorted through the various letter tessellations, I found a number of examples of anisohedral tilings but no anisohedral tiles. However, I was very close. I needed only to put a slant on the ends of the bars of the letter F and the rotation shown above would not be possible. Below are two modified letter Fs that are anisohedral tiles. (The way the ends of the letters are sloped differs from  right to left.)

When I was doing letter tilings, I had no interest in this shape because it is an inferior way of representing the letter F compared to the shape in the first figure.

If we analyze this shape and a large variety of similar shapes that are anisohedral tiles, we find that the key side is made from three glide moves. In the pattern above, we start with the bent line that makes up most of the very top of the letter F. We copy this line, flip it, and move it downward (1). After connecting it to the original line, we copy it,  flip it, and move it downward again (2). Again we connect this line that has been flipped twice (so it has the original orientation), then copy it, flip it and move it for one last time (3). When we connect this last line to the rest of the figure, we have formed the front part of the letter F (4).  We have a shape that has been formed with three glides and this method is the key element in a great number of the almost 150 anisohedral examples shown on the site mentioned above, displayed here. (Note that in a true glide the segments would be moved downward until they connected. In the picture below they are move out of position to make the segments more visible.)

I used puzzle pieces to illustrate various things in Exploring Tessellations. Below is a puzzle-piece example of the 3G method. Ignore the top and bottom edges. The sides have two edges. One of those edges has one side of a puzzle piece and the other edge has three identical sides flipped. The tile is anisohedral.

The angelfire site has instructions on making anisohedral tiles that involve combining two tiles and cutting them. There are several ways to to this. I noticed that in an example I used to illustrate isohedral class IH12 there were four sides lined up with three glides. Combining two of the tiles and splitting them results in an anisohedral tile and tiling.

 Unlike isohedral tiles that can be sorted into a limited number of types or classes, there is no catalog of classes for isohedral tiles. There may be an infinite number possible. Instead of classifying the patterns, mathematicians classify them by the number of roles the tile plays and the number of neighbors it has in each role. (A tiling in which the tile plays one role is isohedral.) In addition, the number of alike and unalike neighbors is counted. In the F example above that has anisohedral tiles, each tile has six neighbors. Two of the neighbors are alike and four unalike. The way this tile would be classified is 6₄6₄. In the puzzle example each tile has five neighbors. For each kind, three are alike and two are unalike, so the classification is 5₂5₂ as is the last one of the doubled birds.

Right now anisohedral tiles and tilings are of interest only to a very few mathematicians. As far as I know, no one has created an Escher-like tiling with them and until someone finds a way to use some of them to make something other than abstract, geometric shapes, there will be little interest in them.

I have some other ways to create anisohedral tiles that may be the subject of a future post.

Delightful Designs: A Coloring Book of Magical Patterns

Coloring books for adults are a new fad. As I write this, nine of the top twenty best selling books on Amazon are coloring books. That is not nine of the top twenty craft books or design books. That is nine of the top twenty books of any kind.

I have released a few coloring books as offshoots of the maze books I have designed, and this fall these offshoots are selling better than the maze books by a considerable margin. (For me a book that sells five copies in a month is a spectacular hit. Only rarely does one of my books crack the top 100,000 list on Amazon.) So when I finished up the FabFours typefaces, I decided to add another book to my coloring book offerings. It is titled Delightful Designs: A Coloring Book of Magical Patterns. Below is the cover.

Here are examples of what is inside. I have kept the lines thin to aid coloring.

 Each of the designs in made from a family of four tiles that share the same identically shaped edge.

 Some of the adult coloring books have incredibly intricate designs. The designs in my coloring books are fairly simple, but they have interesting geometric properties that the intricate designs often lack.

The book features a result I discovered while studying tessellations. There are sets of four shapes that can be tiled in a checkerboard-like pattern of tiles and voids, and the voids must take the same four shapes that the set of tiles has. The result is two-, three- and four-tile tessellations, that is, patterns that tessellate with two shapes, or three shapes, or four shapes. The tiling above, for example, has two distinct shapes in it, while the first two have three distinct shapes in them. When I discovered the patterns that these sets of shapes could make, it seemed magical, and hence the name of the book.

Delightful Designs is available on CreateSpace and Amazon.

The FabFours family of typefaces

While working on Exploring Tessellations: A Journey through Heesch Types And Beyond, I discovered that if a square template is shaped with identical center-point rotation edges, there are exactly four distinct shapes possible. Two of those shapes can be in two orientations, one in four orientations, and the last in eight orientations, for a total of 16 orientations.

If the shapes are connected not edge to edge but vertex to vertex, they form a checkboard-like pattern of tiles and voids. The voids have four edges that are formed by the four adjacent tiles and thus they too must have the same four shapes in 16 orientations that the tiles have. By arranging the tiles in different ways, a huge number of different patterns become possible. I tried to find some place on the web that explored these patterns, but could not. However, it is unclear what the best search terms would be.

I have taken this insight and made a series of eleven typefaces that are now available on MyFonts.com. A person using these typefaces can, by simply typing in letters from A to P or a to p, explore possible patterns that these four shapes can generate. Each typeface has two different designs on it. Below are a couple examples.

I was able to use these patterns to illustrate 11 of the 17 wallpaper groups of symmetry.

They are available at www.myfonts.com/fonts/ingrimayne/fab-fours/.

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.