illusion mugs, t-shirts and other gifts Free online jigsaw puzzles

Patterns - tilings
& Comic Book Picture 18, The Makeover
View Comic-Book Pictures

It's not obvious, but the wall-paper pattern in comic book picture 18 could be made up of interlocking tiles, all identical, except that the tiles in each row are reversals left to right of the ones immediately above and below. That is shown to the left, where individual tiles are slightly different in tone. A shape with irregular edges like this will only tile, or tessellate, so that the tiles fit together like jigsaw pieces, with no overlaps or gaps, if the boundary of the tile has special characteristics.

To get an idea of the characteristics, notice that in the middle to the left only the boundary of a tile is shown. Now imagine it snipped into six segments at the points indicated by arrows. At bottom left, the segments have been just eased apart, and now you can see that they form three pairs. The pair marked A are identical, with the right hand one just shifted along (translated) from the left hand one. Now note that the pair marked B and B (reflected) are also identical, except that the lower one has not only been shifted down from the upper one, but mirror reflected horizontally as well. The same goes for the line pair marked C and C (reflected). Lines that are repeated like the B and C pairs, reflected horizontally but shifted vertically (or vice versa), are glide reflections.

Now it is a fact that any design whose edge is made up of three pairs of segments, one pair a translation and two pairs glide reflections parallel to one another, connected in the sequence of these pairs, will tile the plain, or tessellate, without overlaps or gaps. So this specification, of a number of segments along with how they repeat, can be thought of as a recipe for one kind of tile.

There are actually 28 recipes like this for tessellating tiles, each specifying:

the number of pairs of segments that make up the boundary of the tile;

how each segment repeats to form a pair, whether by translation or glide reflection (as here), or by rotation through a specified angle;

the sequence in which they connect;

constraints on orientation of the axes of the glide reflections, or of axes through points of rotation. Axes must for some recipes be parallel to one another, for others at right angles;

There is a diagram of the 28 recipes, (not very easily deciphered), reproduced from the work of mathematician Heinrich Heesch, in Doris Schattschneider: M.C.Escher, Visions of Symmetry, (Thames and Hudson, London, 2004), page 326. (Hint: Heesch's diagrams of recipes 27 and 28 will not tile as they appear, because the axes of the glide reflections as shown are not at right angles). But this book is generally a wonderful introduction to tiling, and to Escher's researches in particular.

There's also loads on the so-called tessellating patterns on my other site at