Tessellations run the gamut from basic to boggling. The simplest ones consist of a single shape that covers a two-dimensional plane without leaving any gaps. From there, the sky's the limit, from complex patterns of multiple irregular shapes to three-dimensional solids that fit together to fill space or even higher dimensions.
Three regular geometric shapes tessellate with themselves: equilateral triangles, squares and hexagons. Other four-sided shapes do as well, including rectangles and rhomboids (diamonds). By extension, nonequilateral triangles tile seamlessly if placed back-to-back, creating parallelograms. Strangely enough, hexagons of any shape tessellate if their opposite sides are equal. Therefore, any four-sided shape can form a gapless mosaic if placed back-to-back, making a hexagon.
You can also tessellate a plane by combining regular polygons, or by mingling regular and semiregular polygons in particular arrangements. Polygons are two-dimensional shapes made up of line segments, such as triangles and rectangles. Regular polygons are special cases of polygons in which all sides and all angles are equal. Equilateral triangles and squares are good examples of regular polygons.
All tessellations, even shapely and complex ones like M.C. Escher's, begin with a shape that repeats without gaps. The trick is to alter the shape -- say, a rhomboid -- so that it still fits snugly together. One simple approach entails cutting a shape out of one side and pasting it onto another. This produces a shape that fits together with itself and stacks easily. The more sides you alter, the more interesting the pattern becomes.
If you're feeling more adventurous, try doodling a wavy line on one side, and then copying the same line to the opposite side. This approach may require some tweaking to get the pieces to interlock properly. For example, if your polygon has an odd number of sides, you might want to divide the leftover side in half and then draw mirror-image shapes on either side of the split. This creates a side that interlocks with itself.
Try your luck with two or more shapes that tessellate. You can do this geometrically, or simply fill the page with any shape that you like, and then imagine an image that fits the negative space. A related method entails filling a known tessellating shape with smaller shapes. There are even fractal tessellations -- patterns of shapes that fit together snugly and are self-similar at multiple scales.
Don't worry if your initial results seem a bit nonsensical. It took Escher years to master these mad mosaics, and even he had pairings that didn't always make sense.
Now that we've laid the groundwork, let's take a look at some of the special tessellations that researchers use to solve tricky theoretical and applied problems.