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.

# MazePuzzles

This blog was originally intended to publicize my maze books. In designing mazes for those books, I used many tessellations. Eventually I decided the tessellations were more interesting than the mazes and now the blog is mostly about tessellations.

## 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.

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.

## Tuesday, June 6, 2017

### A link to an note about tessellations

Here is a link to a short note of mine in the

https://goo.gl/LnJREO

The note asks how many different shapes formed on a square template using identical, asymmetric edges will tessellate. In simpler language, how many different four-edged puzzle pieces are there that will tessellate if all edges are shaped in the same way? If all edges are asymmetric but identical, there are only fifteen distinct shapes with two "out" edges and two "in" edges and all tessellate.

A more difficult question is how many of the different shapes formed based on the template of a regular hexagon using identical, asymmetric edges will tessellate. A total of 34 tile isohedrally as Heesch types and another 9 tile anisohedrally. The demonstration of this result is in

*Mathematical Gazette*that was discussed in an earlier post:https://goo.gl/LnJREO

The note asks how many different shapes formed on a square template using identical, asymmetric edges will tessellate. In simpler language, how many different four-edged puzzle pieces are there that will tessellate if all edges are shaped in the same way? If all edges are asymmetric but identical, there are only fifteen distinct shapes with two "out" edges and two "in" edges and all tessellate.

A more difficult question is how many of the different shapes formed based on the template of a regular hexagon using identical, asymmetric edges will tessellate. A total of 34 tile isohedrally as Heesch types and another 9 tile anisohedrally. The demonstration of this result is in

*Exploring Tessellations: A Journey through Heesch Types And Beyond.*## Sunday, January 8, 2017

### New for 2017

With the new year I am adding a new title,

All but one of my previous books have been either maze or coloring books and this new one contains a little of each plus much more. It is a combination of a traditional activity book and a book about tessellations.

I am not sure where the idea for this book came from. I was toying with the possibility of doing a maze book with only animal tessellations, but for reasons I no longer remember, I changed course and opted to expand the content to include more than mazes. The final draft of the book contains 21 mazes and ten coloring pages, all of them illustrating very short fables, mostly from Aesop.

I was not planning to develop new tessellations for the book but I found gaps between what I already had and what would fit well in the book. One of the additions was an attempt to do a scorpion tessellation that is used for a maze. It is not very realistic but I like the stinger part.

What else fits into a tessellation activity book? Matching and identification problems apply short explanations of topics such as symmetry, translation, rotation, and valence. These activities fill 19 pages and use almost half of tessellations designs that are in the book.

Mazes led me to tessellations because both use grids. I looked for other puzzle types that might fit a grid. Word searches were an obvious possibility, and there are ten pages of this type of puzzle. Below is the corner of one of them showing how tessellating turtles are used to frame the puzzle.

In addition to visual mazes, I have long been interested in what I call hidden-path mazes in which the challenge is to discover the maze. This type of puzzle occupies 13 pages. Below is a corner of one in which the path is on the ponies with letters that have mirror symmetry and the walls are ponies with letters that do not have mirror symmetry.

Sudoku puzzles are grid based but I thought the 9-by-9 variety might be too complex for the book, so I settled for mini-Sudoku puzzles that are based on grids of 16 and 36 cells. There are seven pages of them, with two per page. I used the extra space on some of these to point out a few features of tessellations. Below is part of a six-by-six puzzle framed with a design of tessellating elephants that I did for the book.

Decoder puzzles do not need a grid but can be put into one, as can word scrambles in which the order of letters is altered. There are eight pages of the former and three of the latter. Below is the corner of a decoder puzzle that contains two messages that are mixed together, one contained in letters that have rotational symmetry and the other in the rest of the letters. To decode, you must move back or forward from the given letter, with the dots telling you how far and in which direction. (This is the most complex of the decoder puzzles in the book.)

Searching through Amazon for something similar turned up one short book and I am not sure how similar it actually is. I suspect that the reason there is so little that is similar is that very few people find tessellations as nearly as interesting as I have found them. The suggested audience is anyone who enjoys tessellations and that may be a small group.

The main reason I designed this book is because it was fun. It would be nice if the book would also earn a bit of money, but at least it will not lose money thanks to on-demand printing.

*Tessellating Animals Activity Book*, to my books available on Amazon and CreateSpace.All but one of my previous books have been either maze or coloring books and this new one contains a little of each plus much more. It is a combination of a traditional activity book and a book about tessellations.

I am not sure where the idea for this book came from. I was toying with the possibility of doing a maze book with only animal tessellations, but for reasons I no longer remember, I changed course and opted to expand the content to include more than mazes. The final draft of the book contains 21 mazes and ten coloring pages, all of them illustrating very short fables, mostly from Aesop.

I was not planning to develop new tessellations for the book but I found gaps between what I already had and what would fit well in the book. One of the additions was an attempt to do a scorpion tessellation that is used for a maze. It is not very realistic but I like the stinger part.

Mazes led me to tessellations because both use grids. I looked for other puzzle types that might fit a grid. Word searches were an obvious possibility, and there are ten pages of this type of puzzle. Below is the corner of one of them showing how tessellating turtles are used to frame the puzzle.

Sudoku puzzles are grid based but I thought the 9-by-9 variety might be too complex for the book, so I settled for mini-Sudoku puzzles that are based on grids of 16 and 36 cells. There are seven pages of them, with two per page. I used the extra space on some of these to point out a few features of tessellations. Below is part of a six-by-six puzzle framed with a design of tessellating elephants that I did for the book.

Decoder puzzles do not need a grid but can be put into one, as can word scrambles in which the order of letters is altered. There are eight pages of the former and three of the latter. Below is the corner of a decoder puzzle that contains two messages that are mixed together, one contained in letters that have rotational symmetry and the other in the rest of the letters. To decode, you must move back or forward from the given letter, with the dots telling you how far and in which direction. (This is the most complex of the decoder puzzles in the book.)

Finally, there is one dot-to-dot puzzle making a total of 92 pages of puzzles and explanations using about 160 different tessellations patterns. More than half of the tessellations are of birds because I find them by far the easiest animal to tessellate. The final 14 pages of the book give solutions to the puzzles.

Searching through Amazon for something similar turned up one short book and I am not sure how similar it actually is. I suspect that the reason there is so little that is similar is that very few people find tessellations as nearly as interesting as I have found them. The suggested audience is anyone who enjoys tessellations and that may be a small group.

The main reason I designed this book is because it was fun. It would be nice if the book would also earn a bit of money, but at least it will not lose money thanks to on-demand printing.

## Thursday, January 5, 2017

### Questions no one is asking part four

(Part three is here. It begins the exercise of examining tilings with tiles that have all edges translated.)

Moving to the hexagonal types, a TTTTTT tiling by definition has translated edges.

Isohedral class IH8 fits both the TTTTTT type and the TCCTCC type. The tile has twofold rotational symmetry.

Isohedral class IH12 is simultaneously TG1G1TG2G2 and TTTTTT. IH12 reflects over the midpoint of its TT edges.

Isohedral class IH10 is both type C3C3C3C3C3C3 and TTTTTT. All edges are identical and each is rotated 120º to form the adjacent edge. (Also fitting both of these types are isohedral classes IH11 and IH18. IH11 is formed with identical edges that reflect over their midpoints and IH18 with identical edges of central rotation.)

Isohedral class IH14 fits both types TG1G2TG2G1 and TTTTTT but it has a pair of straight, unshaped edges that give the tile reflective symmetry over a diagonal. If the straight edges are replaced with asymmetric edges, the reflective symmetry of the tile is lost but the tile can still fit types TTTTTT and TG1G2TG2G1, though no longer simultaneously. As a TG1G2TG2G1 type it has a translation block of two.

Below is a tiling that fits type CG1CG2G1G2 and also type TCCTGG.

For both types a C edge must be paired with a G edge for the edges to translate so these pairs must be formed with center-point rotation. In the case of CG1CG2G1G2, the G1 edges are the edges that reflect over their midpoints. Because of symmetry, the translation block for both types is reduced to two.

Moving to the hexagonal types, a TTTTTT tiling by definition has translated edges.

Isohedral class IH8 fits both the TTTTTT type and the TCCTCC type. The tile has twofold rotational symmetry.

Isohedral class IH12 is simultaneously TG1G1TG2G2 and TTTTTT. IH12 reflects over the midpoint of its TT edges.

Isohedral class IH10 is both type C3C3C3C3C3C3 and TTTTTT. All edges are identical and each is rotated 120º to form the adjacent edge. (Also fitting both of these types are isohedral classes IH11 and IH18. IH11 is formed with identical edges that reflect over their midpoints and IH18 with identical edges of central rotation.)

Isohedral class IH14 fits both types TG1G2TG2G1 and TTTTTT but it has a pair of straight, unshaped edges that give the tile reflective symmetry over a diagonal. If the straight edges are replaced with asymmetric edges, the reflective symmetry of the tile is lost but the tile can still fit types TTTTTT and TG1G2TG2G1, though no longer simultaneously. As a TG1G2TG2G1 type it has a translation block of two.

Below is a tiling that fits type CG1CG2G1G2 and also type TCCTGG.

For both types a C edge must be paired with a G edge for the edges to translate so these pairs must be formed with center-point rotation. In the case of CG1CG2G1G2, the G1 edges are the edges that reflect over their midpoints. Because of symmetry, the translation block for both types is reduced to two.

In the beginning of this exercise I stated that I would use an asymmetric edge when possible. If an asymmetric edge is used for the TCCTGG type, the tile has no symmetry and the result is a translation block of four rather than the two in the above figure.

## Monday, January 2, 2017

### Questions no one is asking part three

Two previous posts (here and here) explored tessellations formed using only glided edges. This post looks at tessellations formed with only translated edges. Like glided edges, translated edges must be paired and the pairs must be equally long, which limits them to quadrilateral and hexagonal types. Glided edges, however, can be aligned in several ways while translated edges must be opposite and parallel to each other. Hence, the possibilities when using tiles with only translated edges are even more limited than those when using only glided edges.

In what follows tilings that have a translation block of one are presented when possible. If asymmetric edges are possible, they are used, then edges with reflective symmetry over their midpoints, with edges formed using center-point rotation only as a last resort.

First, a TTTT type.

Isohedral class IH68 is both a TTTT type and a G1G1G2G2 type. All edges are identical and the tile mirrors over a diagonal.

In isohedral class IH57 edges are formed with center-point rotation and opposite edges translate. It fits three types: TTTT, CCCC, and TCTC.

IH62 is similar to IH57 but it requires all four edges to be identical. Each edge is rotated 90º to form the adjacent edge. It fits type C4C4C4C4 in addition to the three types that IH57 fits.

Isohedral class IH74 is another tile that uses identical edges formed with central rotation. The tile mirrors over both diagonals. In addition to types TTTT and CCGG, it fits types CCCC, TCTC, and G1G1G2G2.

Although the topic of this exercise is translated edges, the last three examples highlight the flexibility of edges with center-point rotation. If we want an all-translated version of C4C4C4C4 formed with asymmetric edges, it will have a translation block of four. (In this and in some other cases below, it should be obvious how the tile can arranged as a TTTT pattern with a translation block of one.)

Similarly, if we want to use an asymmetric edge for the TT pair of the TCTC type, the tiling will have a translation block of two.

A TGTG tile with translated edges also has a translation block of two. Edges can be both glided and translated only when they mirror over their midpoints.

A CGCG type with all edges translated results in another tiling with a translation block of two.

An all-translated G1G2G1G2 type has a translation block of four.

Below is a C3C3C6C6 type in which the tiles have translated edges. For the edges to both translate to opposite edges and rotate to form adjacent edges, they must be formed with reflection over their midpoints. The tile is symmetrical over its long diagonal and the tiling fits isohedral class IH68.

The figure below shows the tile from the above figure arranged as a TTTT type. In this arrangement it fits isohedral class IH68.

In what follows tilings that have a translation block of one are presented when possible. If asymmetric edges are possible, they are used, then edges with reflective symmetry over their midpoints, with edges formed using center-point rotation only as a last resort.

First, a TTTT type.

Isohedral class IH68 is both a TTTT type and a G1G1G2G2 type. All edges are identical and the tile mirrors over a diagonal.

In isohedral class IH57 edges are formed with center-point rotation and opposite edges translate. It fits three types: TTTT, CCCC, and TCTC.

IH62 is similar to IH57 but it requires all four edges to be identical. Each edge is rotated 90º to form the adjacent edge. It fits type C4C4C4C4 in addition to the three types that IH57 fits.

Isohedral class IH74 is another tile that uses identical edges formed with central rotation. The tile mirrors over both diagonals. In addition to types TTTT and CCGG, it fits types CCCC, TCTC, and G1G1G2G2.

Although the topic of this exercise is translated edges, the last three examples highlight the flexibility of edges with center-point rotation. If we want an all-translated version of C4C4C4C4 formed with asymmetric edges, it will have a translation block of four. (In this and in some other cases below, it should be obvious how the tile can arranged as a TTTT pattern with a translation block of one.)

Similarly, if we want to use an asymmetric edge for the TT pair of the TCTC type, the tiling will have a translation block of two.

A TGTG tile with translated edges also has a translation block of two. Edges can be both glided and translated only when they mirror over their midpoints.

A CGCG type with all edges translated results in another tiling with a translation block of two.

An all-translated G1G2G1G2 type has a translation block of four.

Below is a C3C3C6C6 type in which the tiles have translated edges. For the edges to both translate to opposite edges and rotate to form adjacent edges, they must be formed with reflection over their midpoints. The tile is symmetrical over its long diagonal and the tiling fits isohedral class IH68.

The figure below shows the tile from the above figure arranged as a TTTT type. In this arrangement it fits isohedral class IH68.

Type C3C3C6C6 cannot be formed with translated edges because the template cannot be a parallelogram.

Part four will continue with hexagonal types.

## Monday, December 19, 2016

### Qustions no one is asking part two

(This post continues a previous post.)

Moving on to hexagonal types, the way to form an all-glided type TTTTTT using as many asymmetric edges as possible is with isohedral class IH12, which is also type TG1G1TG2G2 with mirroring over the midpoint of the TT edges. The mirroring requires that the G1 and G2 edges be identical.

The all-glided type TG1G1TG2G2 below is similar to the tiling above but uses a different shape for the G1 and G2 edges. It no longer fits TTTTTT.

Below is an example of TG1G2TG2G1 with the TT edges glided with reflective symmetry.

The type TCCTCC example below formed with glided edges also fits type TG1G1TG2G2. If the translated edges were straight, it would be an example of isohedral class IH15.

An alternative way of forming a glided TCCTCC will fit type TG1G2TG2G1 and is an example of IH13. As such, it also fits CG1CG2G1G2.

TCCTGG has a translation unit of 4x1. This example uses all three types of edges that can be glided: asymmetric edges, edges with reflection over the midpoint, and edges with center-point rotation.

Because the tile used to form this all-glide version of TCCTGG has the CC part glided, it will also tile as TG1G1TG2G2.

To form an all-glided version of type C3C3C3C3C3C3, the edges are formed with edges of mirror reflection arranged as G1G1G2G2G3G3.

Notice how the two CC edges are mirrored in this all-glided version of CG1CG2G1G2.

Moving on to hexagonal types, the way to form an all-glided type TTTTTT using as many asymmetric edges as possible is with isohedral class IH12, which is also type TG1G1TG2G2 with mirroring over the midpoint of the TT edges. The mirroring requires that the G1 and G2 edges be identical.

The all-glided type TG1G1TG2G2 below is similar to the tiling above but uses a different shape for the G1 and G2 edges. It no longer fits TTTTTT.

Below is an example of TG1G2TG2G1 with the TT edges glided with reflective symmetry.

The type TCCTCC example below formed with glided edges also fits type TG1G1TG2G2. If the translated edges were straight, it would be an example of isohedral class IH15.

An alternative way of forming a glided TCCTCC will fit type TG1G2TG2G1 and is an example of IH13. As such, it also fits CG1CG2G1G2.

TCCTGG has a translation unit of 4x1. This example uses all three types of edges that can be glided: asymmetric edges, edges with reflection over the midpoint, and edges with center-point rotation.

Because the tile used to form this all-glide version of TCCTGG has the CC part glided, it will also tile as TG1G1TG2G2.

To form an all-glided version of type C3C3C3C3C3C3, the edges are formed with edges of mirror reflection arranged as G1G1G2G2G3G3.

Notice how the two CC edges are mirrored in this all-glided version of CG1CG2G1G2.

We have finished looking at tiles with all edges formed with glides. We will continue in part three to look at another question no one is asking, how many of the Heesch types can be formed with tiles in which all edges are translated.

## Wednesday, December 7, 2016

### Questions no one is asking, part 1

A previous post mentioned that 25 of the 28 Heesch types could be formed using only identical edges of central rotation. The other three can also be formed using edges of central rotation, but they require at least two sizes of edges. Edges formed with center-point rotation can serve not just as C edges in the Heesch classification of types but also as T, G, C3, C4, and C6 edges, that is, any type of edge.

Mirroring an edge formed with central rotation is equivalent to gliding it. Translating an edge that mirrors over its midpoint is also equivalent to gliding it. Hence, some edges that are translated or mirrored can be seen as glided. How many of the Heesch types can be formed with all edges that are glided whether or not the edge fits as a glided edge?

Glided edges have less flexibility than edges formed with center-point rotation partly because they must be paired and not all edges pair in types formed on triangular and pentagonal templates. The closest we can get to an all-glided version of a triangular type, for example, is shown below. Notice that it can satisfy the CGG type in two ways, but not three. This tiling satisfies CCC and CC6C6 as well as CGG.

However, all edges of quadrilateral and hexagonal types can have edges paired and all of these can be formed with edges that are glided. In constructing them below, I have used asymmetric edges when the edges serve as G edges in the Heesch type and whenever else they can be used. When the type calls for C edges, I have used an edge with center-point rotation and mirrored it. All other edges have mirror symmetry over their midpoints. When there two or more pairs of the same type, different shapes are used to differentiate them if it is possible.Let us start with the two quadrilateral types that must be formed with glided edges, G1G1G2G2 and G1G2G1G2. Below is an example of G1G1G2G2 with matching pairs differentiated.

Next is an example of G1G2G1G2.

Some of the other quadrilateral types formed with glided edges also fit either G1G1G2G2 or G1G2G1G2. To form TGTG with only glided edges, the TT pair must be formed with reflective symmetry over the midpoint. It also fits type G1G2G1G2.

Type TCTC can be formed with glided edges if the TT pair of edges reflects over their midpoints and the CC pair of edges is identical and mirrors. The tile has symmetry over the translated edges and fits isohedral class IH66. It also fits G1G2G1G2.

The CGCG type has the CC pair of edges formed identically and mirrored rather than translated. Like the previous two, it also fits G1G2G1G2.

The CC edges can also be mirrored when they are adjacent, as this example of a CCGG type shows. The tiling also fits G1G1G2G2.

A CCCC type that uses glided edges can reflect over the diagonal and fit isohedral class IH69. It is type G1G1G2G2 formed with edges of central rotation.

Alternatively, the edges can mirror as opposite edges in which case it is also type G1G2G1G2.

Two of the eleven quadrilateral Heesch types can be formed with asymmetric edges that will simultaneously fit either G1G2G1G2 or G1G1G2G2. Isohedral class IH68 is simultaneously TTTT and G1G1G2G2. All edges are shaped identically and there is mirroring over one diagonal

Isohedral class IH71 can be seen as either G1G2G1G2 or as C4C4C4C4. All edges must be identically formed and there is mirroring over one diagonal.

An alternative way of getting a glided C4C4C4C4 tiling is with an arrangement that is simultaneously G1G1G2G2 and C4C4C4C4.

(By accident I discovered that this shape will tile anisohedrally.)

Finally, two of the eleven quadrilateral Heesch types can be formed with glided edges but the tilings are not G1G1G2G2 or G1G2G1G2. To form type C3C3C3C3 with glided edges, the edges must reflect over their midpoints.

The tile in the tiling above can be arranged in a G1G1G2G2 tiling.

As with type C3C3C3C3, to form type C3C3C6C6 with glided edges, the edges must reflect over their midpoints.

The tile can also be arranged in a G1G1G2G2 pattern as illustrated below.

Part two will consider the hexagonal types.

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