M.C. Escher has uncommon visions and intuitions. Many of Escher's drawings contain mathematical
ideas never seen elsewhere. One of the things that he did touched me deeply - the interlocking
shapes of people, birds and fish which repeat over a flat surface with no gaps. This group of
drawings are not his best contribution to mankind, but it is art for me because it stirred
my imagination immediately. This is something I can understand as an engineer and I think
I can develop these ideas further with computer graphics. The mathematics behind the
interlocking shapes on a flat surface was established by Heesch and Kienzle in 1963.
The following discussion is limited to regular pattern in a flat surface.
The best way to understand Escher's interlocking shapes is to dissect the figures. Take a piece of tracing paper and copy the outline of a shape (in the case of the horseman, the combined outline of the horse and the rider). This silhouette allows you to focus on the line properties between the shapes. Look in the original periodic drawing and locate the vertices: points where 3 (or more) lines joint together. The vertex is also a place where 3 or more neighboring shapes touch. There are at least 3 vertices and up to 6 vertices on the silhouette on the tracing paper. The line segments between the vertices holds the secrets of the mathematics. Look for rotation symmetry at these vertices. Let's discuss two famous Escher drawings to illustrate the structure: "Reptiles" where the lizards claw out of a book, and "Cycle" where a short person runs down the stairs and tumble into a hexagon form. In these figures, there are 6 vertices and 6 line segments. Upon closer examination, there are only 3 distinct line segments and 2 copies of each of these three lines. Such simplicity ! Now, the secret becomes lucent, half of the silhouette are made out of copies from the other half. Three of the 6 vertices have rotation symmetry C3, which means the line segments attached to these vertices can pivot about this point and radiate out 3 copies like the propeller of an electric fan. The two copies of identical line segments are connected at the rotation vertices C3. The other 3 vertices are the junctions of the dissimilar line segments. The line segments repeat across the page periodically and hook up with other line segments at the vertices to create the repetitive pattern. In all the drawings made by Escher there are 6 different kinds of repetition method for the line segments: translation T, half-turn C2, one-third turn C3, quarter turn C4, one-sixth turn C6 and gliding mirror reflection G. Only 6 different congruent lines and you can do everything ! The lines are equivalent to the sub-atomic particles in physics. The atoms are recognized forms which have an interesting meaning to our eyeballs: people, birds, fish, horse, animals. These atomic units may or may not nest on a surface. The molecules are the smallest unit that can interlock - what I call interlocking shapes. The molecules may be a silhouette of a horseman, which interlocks perfectly, and consists of two atoms: the horse and the rider. The atom has meaning to our eyeballs, but no mathematical interpretation. The silhouette, is formed by groups of congruent lines just like a molecule is build with shared orbits of electrons. The method of repetition of the interlocking shapes is rooted in the congruent movements of the lines. Thus, a silhouette that is build with C6 lines also repeat in 60 degree rotations all over the flat surface.
Crystallographers illustrate the molecular packing with regular drawings. These diagrams consists of geometric arrangement with no color or grayscale contrast. The Japanese quilts and Chinese lattice windows are also lacking in color (between adjacent shapes). The Islamic tiles have wonderful colors, which gives the picture depth and foreground/background. All the above still lack another characteristic: recognizable animal form and meaningful objects. The challenge Escher overcome is to build the shapes with familiar animals and things. One explanation for the purely abstract shapes in Islamic tile is that the religion forbid the use of familiar recognizable animals and human images.
Escher sees the beauty in structure and infinity. He forced the idea of meaningful lines into the mathematical framework of regular plane division. He likes to challenge the logic of seeing. You can see the white birds and regard the black as background. Or you can see the black fish and regard the white as background. When the mind jumps back and forth, that is when you are seeing two things existing together as interlocking shapes.
Escher said that he felt irresistible joy with putting multiple copies of an image on a drawing and making them fit together in a structure. Over a period of 40 years, he gradually learn about the mathematical rules and about techniques used by other cultures, and the science of packing molecules. Within each interlocking shape, half of the boundary is a clone of the other half. These identical sections of the boundary make it possible for neighboring shapes to mate together without gaps. This pre-condition often prevents an arbitrary shape from mating together, or it makes a good fitting shape look meaningless (absurd silhouette). Here is another rule, the boundary of a silhouette cannot cross itself. The vertices of Escher's lizard drawing form a regular hexagon. Three of the vertices are rotation points for the 3 pairs of line segments. When you replace the straight edges of the hexagon with these curved lines, the lines wind in and out of the hexagon. It is difficult to let the lines come to close proximity (to form the legs of the lizard) and yet never intersect another boundary line. Benoit Mandelbrot stated this as "self avoiding curves" in his discussion of Koch Island. Through 40 year of practice, Escher found a way to mediate the need for the eyeballs to find meaning and beauty, and the need for the mathematical rules to be followed. This is a tension which haunts every student who tries to use these computer programs to draw interlocking shapes. You lack the freedom to draw the clone half of the silhouette. Escher has a wonderful description of this thinking process in his book "Exploring the Infinite". Here is a direct quote: "A contour line between two interlocking figures has a double function, and the act of tracing such a line therefore presents a special difficulty. On either side of it, a figure takes shape simultaneously. But, as the human mind can't be busy with two things at the same moment, there must be a quick and continuous jumping from one side to the other. The desire to overcome this fascinating difficulty is perhaps the very reason for my continuing activity in this field".
Escher's repeating lizards have 4 properties: (1)The drawing is periodic and can extend to infinity. (2) The lizards can rotate 1/3 turn and superimpose on another lizard of a different color. (3) There are no gaps between the adjacent lizards, so the tiles interlock. (4)There is only one tile with 3 different colors. The first 2 aspects are symmetry properties of the aggregate. The 3rd and 4th are properties of the tile boundary (line segments of the tile). The study of symmetry in crystallography give rise to the 17 wallpaper groups. There is an international notation (IUC) which describes pattern properties such as m(mirror), g(gliding mirror), p1(translation). Many people said the mathematics of symmetry describes the properties of Escher drawings. But the 3rd and 4th properties are missing from the "Wallpaper groups" which is based on aggregates and not tiles. A tile is an enclosure of finite area. There is a close relationship between the tile symmetry and the aggregate symmetry. Robert Fathauer remarked to me that all tilings are patterns, but a patterns is not necessary a tile. A pattern(aggregate) may not have boundary or enclosure or finite size. Most people interested in Escher-like drawings want to design tiles by a step-by-step cookbook method. The mathematics guides and restricts design, but the mathematics will not replace vision. This is why so few interlocking shapes have ever been designed successfully. Heesch studied tile properties and defined 28 basic tile formation based on line segments. Here we have moved one level down and talk about movements of the line segments. The tile itself is an assembly of 3 to 6 line segments which always come in pairs. Computer programs are available that speed up the design of such tiles and patterns using the mathematics of symmetry.
Now let us look at how the pattern repeats to infinity. Each regular pattern has tile properties and cluster properties. The tile, which is similar to a molecule, may repeat in rotation, translation and gliding mirror. We can build a cluster of molecules, similar to a small crystal that repeats only in translation. If you take the Escher example of the lizard and the tumbling man, a cluster consist of one of each color lizard (3 in the cluster). This cluster can shed all rotation and reproduce by translation. Linear translation is one of the easiest things for the computer to do: increment the x and y co-ordinates by the period. These cluster contains 1 to 6 molecules. There are at least 2 translation axes and sometimes 3 translation axes. Doing tessellation with JAVA program means finding a cluster of lines or polygons and the shift vectors for this group. Usually, a x-vector is used to repeat the shape from left to right. The second vector may be at 60 degrees to x-vector when the next row has an offset in the x-direction. This offset is typically half the regular x displacement. To make a webpage wallpaper, the two vectors must be orthogonal (perpendicular). The x and y vectors must have units equal to the number of pixels in the width and height of the GIF or JPEG file. This is done by regrouping the cluster, cut some off a shape and patch that onto the other side. As Hop David said "If you want the tiles to appear seamless, take care that the left side matches the right side and that the top matches the bottom." The wallpaper tiles are straightly rectangular. Once you have the tile, to activate this tile on the webpage is easy, because HTML has a build-in tiling function in BODY statement with the parameter "BACKGROUND=catcat3.gif". It is interesting to look for the cluster in the JAVA graphics examples.
Let's summarize the mathematics. The lines are congruent and composed of an array of pixels. The forms make sense to the eye but has no mathematical significance. The tile is the interlocking shape and may be composed of several sub-divisions which are recognizable forms. Clusters of 1-6 tiles can form a translational repeating unit. For this translational unit to be suitable for wallpaper on a webpage, make sure that the tile is rectangular and the two axes of repetition are same as the width and height of the tile.
Engineers have used interlocking shapes to conserve materials in stamping . The links and references at the bottom of this page will show many uses of tessellation in different art and science professions.
I developed a set of computer programs at the University of Illinois using the above mathematics. In 1978, the programs were used for students to drawing Escher-like designs. People interested in this kind of work can check the technical papers. "Automatic Generation of Interlocking Shapes" Computer Graphics and Image Processing April 1979, Volume 9, Number 4, pp. 333-353. "Interlocking Shapes in Art and Engineering" Computer Aided Design, January 1980, Volume 12 Number 1, pp.29-34. These journals can be found in engineering libraries at major universities.
"Flachenschluss" by H. Heesch and O. Kienzle, Springer-Verlag, Berlin, 1963.
"Visions of Symmetry - Notebooks, Periodic Drawings, and Related Work of M. C. Escher" by Doris Schattschneider, W. H. Freeman and Company, New York, 1990, 354 pages.
"Tilings and Patterns" by Branko Grunbaum & G.C. Shepard, W.H. Freeman and Company, New York, 1986, 700 pages. This is a classic book and very best I have seen.
"The Symmetry of Things" by Conway, Burgiel and Goodman-Strauss, 2008.
"Regulare Parkettierungen - Mit Anwendungen in Kristallograhie, Industrie, Baugewerbe, Design und Kunst",by Hans-GŁnther Bigalke and Heinrich Wippermann, BI Wissenschaftsverlag Mannheim-Leipzig-Wien-Zurich, 1994, 460 pages, ISBN 3-411-16711-4.
"Handbook of Regular Patterns" by Peter S. Stevens, MIT Press, 1981, 1996, 454 illustrations, 400 pages.
"Escher on Escher, Exploring the Infinite", by M.C. Escher, published by Harry N. Abrams, 1989, 158 pages.
"Pattern Design" by Archibald H. Christie, Dover Publications Inc. New York, 1910, 1929, U.S. Edition 1969, 359 illustrations, 313 pages.
"Chinese Lattice Designs" by Daniel Sheets Dye, Dover Publications, Inc. New York, 1937, 1949, 1974, 1239 illustrations, 469 pages.
"3000 Decorative Patterns of the Ancient World" by Flinders Petrie, Dover Book 1986, 106 pages.
"Groups and Symmetry" by David W. Farmer, Mathematical World Volume 5, American Mathematical Society, 1996, 102 pages.
"Shapes, space and symmetry," by Alan Holden, Dover Publications, New York, 1991, 200 pages, 319 illustrations.
"Andrew Glassner's Notebook - Recreational Computer Graphics," by Andrew Glassner, Morgan Kaufmann Publishers, 1999, 304 pages.
"Symmetries of Culture," by Dorothy K. Washburn and Donald W. Crowe, University of Washington Press, 1988, 300 pages. This is considered an archaeology book, it has good discussion of symmetry in human history from different cultures.
"The Grammer of Ornament">by Owen Jones, A Dorling Kindersley Book, 2001, 504 pages