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.
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.

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,, is no longer responding. There is a mirror, however, at

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.