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.

A link to an note about tessellations

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

New for 2017

With the new year I am adding a new title, 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.

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.)

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.

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.

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.

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.

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

Part four will continue with hexagonal types.