Thursday, January 24, 2019

Tessellating Typefaces (part 4)

The addition of two more fonts of Escher-like tessellations at brings the total of the new tessie series to eleven. Four are of birds, one of other animals, one of bugs,  one of puzzle pieces, and four of everything else.

One of the recent additions is TessieBugs. It contains tessellating butterflies, moths, ants, and other creepy, crawly insects.

The other is TessieOddsNends, a hodgepodge of things that did not make it into any of the other ten faces.
Each contains two styles, a solid style that must be colored in order to see the shapes (after all, a tessellation fills the plane with no gaps or overlaps), and an outline style that can be used alone or layered over the solid style.

Thursday, December 6, 2018

Tessellating Typefaces (part 3)

Three more tessellating typefaces have been added to Included is a fourth typeface of birds, TessieXtraBirds. When playing with TesselManiac and before it TesselMania, I have had the tendency to see bird possibilities much more readily than other shapes.
The other two typefaces have a variety of shapes and no clear theme as their names indicate. Below is a poster for TessieMoreStuff. The plane shape has four edges that are all shaped with identical center-point rotation. The poster has them in three different tiling arrangements.
The musical notes from TessieMiscellaneous are a based on a triangle. It is a pattern unlike any that I have seen elsewhere.
Each family contains two styles, a solid style that must be colored and an outline style that can be used alone or layered over the solid style (as shown in the two top illustrations). Each has a sample file that shows what key combination are needed to produce the various patterns that are possible.

(Cross posted at

Tuesday, November 20, 2018

Tessellating Typefaces (part 2)

Three more tessellating typefaces have been added to Included is another typeface of birds, TessieMoreBirds. Many of the bird shapes in this typeface are of swimming birds. I noticed that combining two of the swimming birds in the tiling below would result in both swimming and flying birds (shown on the right).  (The flying birds will not tessellate alone.)

 TessieSpinners has a variety of shapes that have suggest spinning or rotating.

Finally, TessieAnimals has tessellating shapes that resemble fish and mammals.

Part 1, with three other tessellating typefaces, is here.

(Cross posted to

Friday, November 9, 2018

Tessellation typefaces (part 1)

Over the past twenty years I have developed/created/found a lot of tessellations patterns as I have designed maze books, coloring books, an activity book, and a book about tessellations. A few years ago I decided to make a couple of typefaces from some of this material, but those typefaces were only outlines. This past year I again decided to try to use this material, which has since grown, in typefaces. After a dead end or two, I have mostly finished the effort and am now putting the resulting typefaces on In this new effort I have both outline and solid versions of each pattern so that the user has much greater flexibility than with the first efforts. To aid the user in using the fonts, for each there is a gallery file in a pdf format that shows what characters are needed to get each tessellation pattern.

TessiePuzzlePieces contain puzzle pieces. The reason I created these shapes can be found at the link.
 TessieStandingBirds was a result from a quest to see how many different Heesch types I could illustrate with birds standing on the backs of other birds.
 TessieFlyingBirds has a large variety of flying birds. I tried to limit the use of characters that are not readily reached from the standard keyboard, so some flying bird shapes are on other Tessie fonts that are in the process of being put on
To get a tessellation pattern can require as few as one character but usually two, three, four, or six are required. The pattern above requires two characters because alternate rows are indented.

This illustration below shows three ways these fonts can be used. On the left is the solid style by itself, in the middle the solid style is overlaid with the outline style, and on the right the outline style is alone This shape is unusual because there are two ways to tile it. In the top six row of the picture, birds are flying both to the left and to the right. In the bottom four rows birds are flying only to the left.

For those who like the technical details of tessellations, the top birds are tiled as Heesch type TGGTCC while those on the bottom are tiled as Heesch type TG1G1TG2G2.

(Cross posted to

Friday, August 3, 2018

Butterflies in Mazes Escher Would Like

Here are some further notes on the mazes in Mazes Escher Would Like.

One surprise I found was a hexagonal pattern that looked like it fit into a rectangular grid but when I made a maze with it, I found that it drifted. Below is a moth pattern that does not drift. The edges are labeled clockwise, with 1 being the horizontal edge on the right. Notice that to go down the column, one follows the pattern of 35263526 etc.

Below is a different butterfly pattern with its six edges labeled in the same way. Notice following the 35263526 route takes down and to the right.

One of my maze-making programs allows a correction to be made so that the maze will line up correctly.

Perhaps the most interesting pattern that I have found in the past few years is of butterflies such as the ones below.

The shape of the tile has mirror symmetry but it does not fit into any of the symmetrical types of Gr├╝nbaum and Shepard's classification. The tiling above is type CG1CG2G1G2. Because of the symmetry of the the shape, the translation block appears to be two rather than four. 

There is another tiling possible, a TCCTCC type, shown below. To see the difference between the two, follow a diagonal from southwest to northeast. Below all the shapes have the same orientation along this diagonal. Above there are two oriented the same, then two different, etc. 

Some of the other shapes that are included in the book have appeared in posts on this blog during the past few years and many have appeared in coloring books or in Exploring Tessellations, but not in other maze books.

Seventy-nine of the 83 mazes use tessellations. Ten are geometric or abstract shapes, one is a symbol, 13 are mammals including people, 7 are other vertebrates, 11 are invertebrates, 6 are non-living objects, and 32 are bird shapes.

Wednesday, July 18, 2018

Further maze book revisions

After updating tessellating maze books in May, I updated some other books. While doing that I discovered patterns that made simple mazes quite difficult to solve. I could not resist the temptation to include some and so did a second round of revisions and updating. Below is a sample, though not an actual maze from either book. All the paths go through the corner swirls.
The updating resulted in the elimination of a number of common geometric tessellations that, while they made decent mazes, were not of much interest to those looking for visually-interesting tessellation. Tantalizing Tessellating Mazes had four additional pages changed and More Tessellating Mazes had 12 additional pages changed. As a result of the two 2018 revisions, both books have 39 new mazes and 45 holdover mazes.

In the maze above there are two shapes that tessellate. The top one is an example of isohedral class IH71. All edges are shaped the same and there is mirroring over a diagonal line. The bottom shape is an example of class IH61. The tile has two-fold rotational symmetry and uses the same edge as the top shape.

(If the maze above is presented with big openings through the sides rather than the little openings through the curved corners, it is trivially easy to solve. See below.)

Sunday, July 8, 2018


The introduction to A Final Coloring Book of Tessellations noted that "as I stumble on additional patterns ... I can use the better discoveries to update and revise this and the previous books." In the past few weeks I have revised A Tessellating Coloring Book, More Tessellations Coloring Book, and A Final Coloring Book. New patterns (and thus deleted patterns) were six for the first book, eleven for the second, and seven for the last.

The three books contain over 300 tessellation designs and each is unique to one book. The first two of the books listed above were designed before the adult-coloring-book fad hit and were meant for children. The last book was intended for adults but it lacks the fussiness that many of the adult colorers seem to desire.

They all still have some weak designs and some geometric tilings that are in the public domain so if in the future I find additional interesting tessellating shapes, there may be more revisions.