Thursday, May 12, 2016

Z Z Z Z and IH9

I have struggled to understand isohedral class IH9. I could see that it fit Heesch type TG1G1TG2G2 but was not sure it always fit TG1G2TG2G1. Playing with it in the process of yet another update to Exploring Tessellations: A Journey through Heesch Types and Beyond, I realized that IH9 is the class for tiles that simultaneously fit TG1G1TG2G2 and TG1G2TG2G1. It can fit both because flipping a IH9 tile over a horizontal line give the same result as flipping it over a vertical line. Both TG1G1TG2G2 and TG1G2TG2G1 have pg symmetry, but IH9 has pgg symmetry, so that the tiling can be rotated 180 degrees and it will reproduce itself.

An earlier post reported that a way to construct IH9 tiles in Tesselmaniac! is to use the CGCG template and ignore one of the C lines. An alternative is to use the CCGG template, also ignoring one of the C lines. Below are two ways of tessellating the letter “Z” with screen shots from Tesselmaniac! showing how each can be created in CGCG and CCGG. 

In this first picture showing CGCG tiles, the green line is eliminated to combine two tiles into one IH9 tile.
In the CCGG case, the yellow line is eliminated to combine two tiles into one IH9 tile. In all the cases shown here, the midpoints of the C lines are the centers of two-fold rotation.
The second Z fits in a different way. Below the yellow line is the line to eliminate to get the IH9 tile.
In this fourth picture, the yellow line that is eliminated connects across the stem of the letter. 
To see why IH9 can be both TG1G1TG2G2 and TG1G2TG2G1, consider how the glide lines in each are oriented. Below is a simple tile that tessellates as TG1G1G2G2. The purple lines are lines of glide reflection. They are perpendicular to the edges that are the TT edges. 
In contrast, the lines of glide reflection are parallel to the two TT edges in this TG1G2TG2G1 tiling.
The shape of a class IH9 tile allows both sets of glide-reflection lines. Flipping the tile horizontally gives the same result as flipping it vertically.

The May revision of Exploring Tessellations: A Journey through Heesch Types and Beyond was substantial. I corrected dozens of mistakes, most small but a few larger (such as misclassifications of tessellation patterns) and added about a dozen pages. I also reorganized the book, moving sections and dividing the Explorations chapter into two chapters. I doubt if this will be the last revision; I am sure that many mistakes remain.

Thursday, April 7, 2016

More chevron tilings

A previous post on vertex-to-vertex tiling with chevrons and tiles based on the chevron template looked at patterns in which the tiles met at five vertices. It is easy to arrange chevrons so that four vertices meet with no common edges and this pattern can be adapted to the chevron templates fitted with identical centro-symmetric sides.
The chevron pattern at the top fits  Heesch types: TTTT and TGTG.

I think a more interesting pattern with four common vertices is one that uses a G1G1G2G2 arrangement. Of the 64 possible tiles with the identical centro-symmetric sides, only eight shapes will form patterns that are true tessellations. Two are shown below.
In the figure below all eight are shown. Six tiles of each of the eight surround two voids of the same shape. Can you identify them?
There are only eight because three of the sides of the chevron are determined by the other three. (The chevron template has six sides but only four edges in this configuration. An edge is the border with a neighbor and a side is a straight line in the template.) One G1 edge determines the other and one G2 edge determines the other. Each of the three sides that can be independently set can have two states, and 2x2x2=8.

Fitting together other shapes results in voids that are different than the surrounding tiles.
I have not found a use for any of these chevron tilings, though as mentioned in the previous post I could include a few examples in Delightful Designs: A Coloring Book of Magical Patterns if I ever revise it.

Sunday, March 13, 2016

A chevron tiling

I became interested in tessellating shapes based on the template of a chevron because of an oddity in Grunbaum and Shepard’s classification of tessellating isohedral types. In their 93 classes there are twenty four that have at least one edge that is shapeable and at least one edge that is an unshapeable straight line. Nineteen of them can be viewed as a bisection or trisection of other classes with symmetry that both tiling and tile share and four have symmetry that allows them to be bisected. The one that does not fit into the pattern is IH58. It lacks a parent, but it if had one, it would be a tile that has the template of a six-edged chevron.
(An example of IH58 is shown below on the top. It has four edges. Two opposite edges are straight lines. The other two opposite edges are formed with center-point rotation and are translated. The tiling has pmg symmetry. The bottom part of the figure shows the parent tile and tiling. It also has pmg symmetry and is a special case of IH15. In IH15 each pair of adjacent edges formed with center-point rotation is shaped identically, so two different shaped edges can be used in their construction. IH15 can have a template of a convex hexagon; the parent of IH58 must have one interior angle equal or greater than 180 degrees. Bisecting IH15 tilings on the axis of symmetry results in IH49 tilings.)
An earlier post mentioned how equilateral chevrons with six edges could tessellate and how the template of that equilateral template could be fitted with identical centro-symmetric edges. Tiles based on equilateral triangles, equilateral rhombuses, and regular hexagons with identical centro-symmetric edges can be placed so that only vertices meet and no edges, resulting in patterns of tiles and voids. (See the posts on Fab FoursFab Twos?, and An Unfinished Journey. ) There seems to be no way that equilateral chevrons can be arranged so all six vertices of the chevron will meet with no common edges, but in the pattern below five vertices meet with no common edges. The tiles are equilateral chevrons and the voids are equilateral triangles and hexagons.
If the chevrons are replaced with tiles with identical centro-symmetric sides, patterns such as the following result. The voids are shapes with identical centro-symmetric sides formed on templates of equilateral triangles and regular hexagons.
Below is another arrangement of chevrons with five corners touching. There are several others similar.
I have not found a use for the above, though perhaps I could include a few examples in Delightful Designs: A Coloring Book of Magical Patterns if I ever revise it. Here is what the previous figure would look like as a coloring page. How would you color it?
(Retrieved after being accidentally deleted.)

Monday, February 15, 2016

Presidents' Day

Today is Presidents' Day, one of the lesser Federal Holidays in the United States. Government offices and financial markets will be closed.

A year ago I was working on a maze book, the last one I have done, called Holiday Mazes. Presidents' Day is included. Part of the maze for the day and an explanation of the design is shown below.

Have a nice Presidents Day.

Friday, January 8, 2016

An unfinished journey?

Print on demand makes updating an existing book very easy. There are both good and bad aspects of this feature. Easy updating allows one to correct mistakes in a published work without calling back an inventory of unsold books, but it also encourages publication before the work is completely finished.

I recently updated Exploring Tessellations: A Journey through Heesch Types And Beyond again and this time the changes were substantial. The book is now organized into five chapters: Symmetry, The 28 Heesch Types, The Isohedral Classes, Explorations, and Themed Examples. The Explorations chapter has most of the material that I have developed since the book was published about three months ago. (I plead guilty to rushing to publish.) In the original version I attempted to see how many Heesch types would allow a puzzle piece with identical sides and 90 degree angles. I found 13 distinct shapes that fit the quadrilateral Heesch types that lack an edge of center-point rotation. For some reason I decided to follow up with the question of how many distinct shapes with identically shaped asymmetric edges could be formed with a square template. I was surprised to find 15 and discovered two shapes that did not fit Heesch types. They were anisohedral.

It was an easy extension to ask how many distinct shapes could be formed if the edge was not asymmetric but had center-point rotation. Only four distinct shapes are possible and all tessellate. Playing with these results, I found that by placing the shapes vertex to vertex rather than edge to edge, a pattern of tiles and voids results in which the voids necessarily have the same set of shapes as the tiles. This result was presented on this blog in two posts, here and here. There are only two distinct shapes when the exercise is done with triangles and seven when done with rhombuses, results that were the topic of the previous post on this blog.

When the template of a regular hexagon is shaped with identical sides with center-point rotation, there are nine distinct shapes possible, but one of them does not tessellate. There is only one way to arrange them vertex to vertex so that all vertexes meet another vertex, and the voids then form shapes based on the triangular template. Although there are a huge number of possible arrangements, the results are not as visually interesting as those with the square template. (In the figure you below you can see examples of the nine distinct shapes. The one on the bottom left, which has the most symmetry, is the one that will not tessellate.)

Out of curiosity I decided to see what was possible with a template of an equilateral chevron when it was tiled with six adjacent tiles. (Some odd tilings are possible with the chevron, one of which spirals out to infinity.)  I found that there were sixteen distinct shapes that tessellated (and sixteen distinct shapes that do not tessellate). All 16 tessellated as TCCTCC types, four as TG1G1TG2G2, and 12 as TCCTGG with four of them tessellating in two distinct ways. This particular exploration seems to be a dead end; I cannot see any extension that looks interesting. (In the tiling below the first four rows are tiled TCCTCC and the last four TG1G1TG2G2. Notice that the edge used is identical to the edge used in the previous figure.)
The latest update of Exploring Tessellations is about twenty pages longer than the previous version. I will not be surprised if there are more revisions in the future, though I am not currently working on anything that would lead to an update. I know that errors remain in the text but I do not know if I can find them.

Friday, December 4, 2015


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.

Friday, November 27, 2015

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 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.