Tuesday, November 15, 2016

Fun with puzzle pieces

If all edges placed on a square template are identical with mirror symmetry over the midpoint, there are two distinct shapes that will tessellate, one with two “outs” that are adjacent and one with two “outs” that are opposite. Note that the shape must have two “out” edges and two “in” edges; if the number of “outs” is not equal to the number of “ins”, the pieces will not tessellate. Below is an illustration of the two possibilities with an edge that forms puzzle pieces.

Suppose that the edge does not have the bump centered in the middle but rather offset to one side. How many distinctly different shapes with these identical asymmetric edges will tile the plane?
I answer this question in a note published in the November 2016 issue (Vol 100 Issue 549, pp 511-516) of the Mathematical Gazette: there are 15 distinctly different shapes and all will tile the plane. Thirteen will tile the plane as Heesch types and two will tile it in a non-isohedral pattern.

The note in the Mathematical Gazette is limited to the square template. Exploring Tessellations: A Journey Through Heesch Types and Beyond extends the analysis. If the template is a rhombus or diamond (a square squashed), there are 30 distinct shapes of which 20 will tile the plane. If the template is a regular hexagon, there are 108 distinct shapes that have three “ins” and three “outs” when identical, asymmetric edges are fitted to the template. Thirty-four tile isohedrally as Heesch types and another nine tile anisohedrally. Sixty-five will not tile.

Exploring Tessellations: A Journey Through Heesch Types and Beyond also considers equilateral templates fitted with edges that have central rotation and some of these results have been mentioned in past posts on this blog. There are two distinct shapes if the template is an equilateral triangle, four for a square, seven for a rhombus, and nine for a regular hexagon. One of the nine for the regular hexagon will not tessellate. Also, all Heesch types except three that will not fit as equilateral polygons (CC3C3, CC4C4, and C3C3C6C6) can be formed with identical edges of central rotation.

Wednesday, November 2, 2016

More Tesselmaniac fun

 A couple posts from October, 2015 (here and here) examined ways of using Tesselmaniac! to construct isohedral class tilings that that did not have a template in Tesselmaniac!. Three hexagonal tilings with opposite straight edges were not included and all three of these can be easily constructed without removing interior lines.

First is isohedral class IH14. It has four edges that are shaped the same. Each edge is glided and reflected, and the result is that all three opposite pairs are translated. It fits two Heesch types, TG1G2TG2G1 and TTTTTT. It also will tile as a non-Heesch (and non-isohedral) type with flips over the straight edges. It has cm symmetry. It can formed in Tesselmaniac! in the IH68 template by positioning a point to create a straight edge. Obviously, if that line is reduced to zero, the tiling will be an IH68 type, a mirrored G1G1G2G2 type.


Isohedral class IH15 is another mirrored tiling with two adjacent edges formed identically with center-point rotation and mirrored and the other two adjacent edges on the other side of the two straight edges also formed identically and mirrored. It satisfies the condition Heesch type TCCTCC with the straight edges serving as the translated pair. Because mirroring an edge formed with central rotation is the same as flipping it, IH15 also satisfies types TG1G1TG2G2, and TCCTGG. Finally, because the tiles can be flipped over their straight edges, it also tiles as a non-Heesch (and non isohedral) type. IH15 has pmg symmetry.

In Tesselmaniac! IH15 can be constructed with the mirrored C*CC*C or IH69 template by positioning a point to create a straight edge.


Isohedral class IH17 is a special case of both IH14 and IH15 (as well as IH8, IH9, IH12, and IH13). Two opposite edges are unshaped, straight lines. The other four edges are all shaped with identical center-point rotation and each is reflected (which is the same as gliding) both vertically and horizontally. In addition to tiling in a non-Heesch manner with flips over the straight edges, it satisfies the conditions of six of the seven hexagonal Heesch types, everything but the C3C3C3C3C3C3 type. It has cmm symmetry. Notice the the shape of the tile is restored when it is rotated 180 degrees, when it is flipped over its horizontal midpoint, and when it is flipped over its vertical midpoint.

In Tesselmaniac! IH17 can be constructed with the mirrored C*C*C*C *or IH74 template by positioning a point to create a straight edge.

The tiles in IH14 and IH15 (and also IH16) have symmetry over one diagonal. The tiles in IH17 have symmetry over one diagonal and one edge, which also gives them twofold rotational symmetry.

Saturday, October 29, 2016

A final coloring book of tessellations?

In the process of working on Exploring Tessellations: A Journey through Heesch Types and Beyond, I keep stumbling on new tessellations patterns. Mostly because it is fun to design books, I decided in October to collect material that I had not put into one of my past coloring books and create a new coloring book from it. I may in the future regret the choice of title, but it seemed catchy. Thus was born A Final Coloring Book of Tessellations.
The cover has a pattern that is similar to some in Delightful Designs: A Coloring Book of Magical Properties. There are two related tilings formed with a single edge, one in the upper left and in the lower right and the flipped version of this in the upper right and lower left. The seams where they meet have shapes formed with the same edge but with a different arrangement on the rhombus frame.

A Final Coloring Book of Tessellations  has similar content to A Tessellating Coloring Book and More Tessellations: A Coloring Book. There are a variety of Escher-like tilings and a few abstract, geometric tessellations.  What is different is that the size of the tilings is smaller. The two books mentioned above were done thinking that the most likely audience would be children. This book is aimed at an older audience.

The graphic on the back cover features of what I call arrowplanes. There is also a page in the book with the design.
It is rare when I have to add details to the interior to suggest what the tiling represents, but I am not sure that this shape would suggest a person to everyone.
The post Take Outs showed a crude cup that I had removed from Exploring Tessellations. A revision made it into A Final Coloring Book. Both the ladies above and the cups or chalices below fit the IH12 isohedral class with its reflective symmetry over the translated edges and with the other four edges formed with glide reflection.
Two versions of these gyrating women are included on one page. They are an example of Heesch type CGCG.
Maybe this book will be my final coloring book of tessellations and maybe not. Who knows what the future will bring?

The book has over 100 pages to color. It is printed by CreateSpace and pages are printed on both sides of the page. Some people who color may object the this format and if you are one of them, do not purchase it.

Monday, August 29, 2016

Take outs

A slightly revised edition of Exploring Tessellations is now available. The revision corrects a number of mistakes and adds some new tessellations. In order to make room for the new illustrations, I removed some that were in earlier versions of the book.

Some of the removals were geometric patterns that I had used for mazes. Although they have made good mazes, they showed nothing special as far as tessellations are concerned. These first two illustrated isohedral class IH71, a special case of C4C4C4C4 with all edges the same and symmetry over one of the diagonals. I have no shortage of other examples.

This next figure was used to illustrate CC4C4C4C4. It actually is more than that. It is a bisection of isohedral class IH73 and that is where I should have included it, as an example of IH29. However, I had other examples that were at least as interesting.
 This geometric design of TG1G2TG2G1 made very good mazes but it is geometric rather than representational.
 I used this TCCTCC tiling to illustrate the type because I had few examples in the first draft. Since then I have found more interesting examples to replace it.
The head and wings of this stylized bird are OK, but the tail end is all wrong. With the addition of some better illustrations, it was expendable. 

These crude cups of type TTTTTT were used in a maze book. They are not very interesting or attractive.
 These birds were a second way to illustrated Heesch type CG1CG2G1G2, this one with one of the C edges on the beak of the bird. I found a better way to draw the bird so dropped this one.
 Another illustration of Heesch type CG1CG2G1G2 is the next tiling of fish or dolphins. This was my version of a much better dolphin tessellation by Andrew Crompton. If I had discovered it independently I would have kept it, but I did not.
Will there be more revisions? Probably. I am confident that many errors remain. If I find enough of them to justify a revision, I will update.

(Looking for a link to IH29, I found that a site that I have relied on as a reference, http://freespace.virgin.net/tom.mclean/index.html, is no longer responding. There is a mirror, however, at http://www.jaapsch.net/tilings/mclean/index.html.)

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.)