There are even fractal tessellations - patterns of shapes that fit together snugly and are self-similar at multiple scales.ĭon't worry if your initial results seem a bit nonsensical. A related method entails filling a known tessellating shape with smaller shapes. 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. Try your luck with two or more shapes that tessellate. This creates a side that interlocks with itself.

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 approach may require some tweaking to get the pieces to interlock properly. If you're feeling more adventurous, try doodling a wavy line on one side, and then copying the same line to the opposite side. The more sides you alter, the more interesting the pattern becomes. This produces a shape that fits together with itself and stacks easily. One simple approach entails cutting a shape out of one side and pasting it onto another. The trick is to alter the shape - say, a rhomboid - so that it still fits snugly together. Escher's, begin with a shape that repeats without gaps. Equilateral triangles and squares are good examples of regular polygons.Īll tessellations, even shapely and complex ones like M.C. Regular polygons are special cases of polygons in which all sides and all angles are equal. Polygons are two-dimensional shapes made up of line segments, such as triangles and rectangles. You can also tessellate a plane by combining regular polygons, or by mingling regular and semiregular polygons in particular arrangements. In this article, we'll show you what these mathematical mosaics are, what kinds of symmetry they can possess and which special tessellations mathematicians and scientists keep in their toolbox of problem-solving tricks. Beyond the transcendent beauty of a mosaic or engraving, tessellations find applications throughout mathematics, astronomy, biology, botany, ecology, computer graphics, materials science and a variety of simulations, including road systems.

Mathematics, science and nature depend upon useful patterns like these, whatever their meaning. Tessera in turn may arise from the Greek word tessares, meaning four. In fact, the word 'tessellation' derives from tessella, the diminutive form of the Latin word tessera, an individual, typically square, tile in a mosaic. Escher, or the breathtaking tile work of the 14th century Moorish fortification, the Alhambra, in Granada, Spain. Like π, e and φ, examples of these repeating patterns surround us every day, from mundane sidewalks, wallpapers, jigsaw puzzles and tiled floors to the grand art of Dutch graphic artist M.C. Science, nature and art also bubble over with tessellations. It even bears a relationship to another perennial pattern favorite, the Fibonacci sequence, which produces its own unique tiling progression. The golden ratio (φ) formed the basis of art, design, architecture and music long before people discovered it also defined natural arrangements of leaves and stems, bones, arteries and sunflowers, or matched the clock cycle of brain waves. Euler's number (e) rears its head repeatedly in calculus, radioactive decay calculations, compound interest formulas and certain odd cases of probability. Pick apart any number of equations in geometry, physics, probability and statistics, even geomorphology and chaos theory, and you'll find pi (π) situated like a cornerstone. Tessellations - gapless mosaics of defined shapes - belong to a breed of ratios, constants and patterns that recur throughout architecture, reveal themselves under microscopes and radiate from every honeycomb and sunflower.

Mathematics achieves the sublime sometimes, as with tessellations, it rises to art. Within its figures and formulas, the secular perceive order and the religious catch distant echoes of the language of creation. We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe.

Animal translation tessellation