Tessellations

Fall Leaves, by Jill Ethridge

Many, many years ago, when I first became a high school art instructor, I was somewhat startled to find that our high school math department also taught art. They were having students create tessellations. I'm ashamed to say it now, but at the time, I had never even heard of such things, much less how to create one. Yet students were coming to me for help in making theirs. I wasn't much help. I kept my mouth shut and observed (a tactic I'd advise for anyone knowing little or nothing about any subject). What I saw was really quite amazing. Students were coming up with incredibly complex designs (similar to the one above), some just short of mind-boggling. They were well-instructed. I don't recall for sure, but I think these were probably plane geometry students, which is not exactly higher mathematics but neither is it a playground for imbeciles. Mathematics being a rigidly left-brain subject, it would be fair to say this was an example of left-brain art...an art form ruled by rules.

 

Semi-regular floor tiling--hexagons, squares, and equilateral
triangles. Notice that all outside dimensions are equal.
(Ignore the cracks in the tiles.)

For those who are as uninformed as I was on the subject, a tessellation is a design created on a flat surface using tiling of one or more geometric shapes (called tiles), with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A common checkerboard would probably be considered the simplest example of a tessellation. The shape (perhaps literally a tile) would be the square (probably, but not necessarily) of two contrasting color values. This would be considered a periodic tiling since it has a repeating pattern. Such patterns can be broken down into two types: regular tilings utilizing a regular polygonal shape, each identical (the checkerboard); and Semi-regular tilings, regular tiles of more than one shape and with every corner identically arranged (above).

 

Bird Tessellation, ca. 1938, M.C. Escher

Undoubtedly the most famous artist to utilize tessellations was M.C. Escher, famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects (above). If the right contrasting colors are chosen for the tiles of the various shapes, amazing patterns are formed, and these can be used to decorate physical surfaces. Tessellations tiles need not have straight edges. Each tile may contain non-tessellating decorative elements as well. Notice the faint vertical and diagonal guidelines used to align the tiles.

Cone mosaic pattern columns, ca. 3300-3000 BC.

Tessellations are not the artistic stepchild of modern-day mathematics. They go back some five-thousand years to the Sumerian culture (above) of around 3300-3000 BC (located in modern-day Iraq). Since that time, they've been an element in virtually every civilization having developed an advanced decorative culture. In 1619 the German mathematician, Johannes Kepler, made one of the earliest documented study of tessellations. He wrote about regular and semi-regular tessellations in his Harmonices Mundi. He was the first to explore and explain the hexagonal structures of honeycombs and snowflakes. About two-hundred years later, in 1891, the Russian scientist, Yevgraf Fyodorov, in studying the arrangement of atoms in the crystalline solids, proved that every periodic tiling features one of seventeen different groups of isometries (reflections, rotations, and translations). Fyodorov was the first to engage in a mathematical study of tessellations.

An Alhambra tessellation as drawn by M.C. Escher, 1936.

Being the farthest thing from a mathematician or scientist, from this point on tessellation go way over my head. Suffice to say, many other types of tessellation are possible under different sets of rules. For example, there are eight types of semi-regular tessellations, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can also be made from shapes such as pentagons and polyominoes (a plane geometric figure formed by joining one or more equal squares edge to edge, as in the electronic game of Tetris). Besides floor tiling, tessellations are also sometimes used in wallpaper design. In fact, tilings with translational symmetry in two independent directions are categorized in what's they call wallpaper groups. Unless you're a mathematician, on a par with Yevgraf Fyodorov, you don't want to know the details, other than to recall his magic number of seventeen such groups. It has been claimed that all seventeen are represented in the Alhambra palace in Granada, Spain (above), though there is some dispute in this regard.

A Republican tessellation

If you're a RINO
(Republican in name only)
you might like this tessellation more.