![]() ![]() 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. If they do, the straight sides must remain straight and there is no longer flexibility to make a recognizable figure.We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe. However, these mirror symmetries should not lie on the straight sides of the polygon tiles. ![]() To create a tessellation by bilaterally symmetric tiles, we need to start with a geometric pattern that has mirror symmetries. The less common triangle systems are easily identified because three or six motifs will meet at a point, and the entire tessellation will have order 3 or order 6 rotation symmetry.įigures with bilateral symmetry are naturally easier to make into recognizable figures, because many natural forms have bilateral symmetry. The bulk of Escher’s tessellations are based on quadrilaterals, which the novice will find much easier to work with. All of Escher’s tessellations by recognizable figures are derived from just a handful of geometric patterns.Įscher created his tessellations by using fairly simple polygonal tessellations, which he then modified using isometries. Escher organizes his tessellations into two classes: systems based on quadrilaterals, and triangle systems built on the regular tessellation by equilateral triangles. ![]() He used these figures to tell stories, such as the birds evolving from a rigid mesh of triangles to fly free into the sky in Liberation. Though Escher’s goal was recognizability, his tessellations began with geometry, and as he grew more accomplished at creating these tessellations he returned to geometry to classify them. He wanted to create tessellations by recognizable figures, images of animals, people, and other everyday objects that his viewers would relate to. The most common tessellations today are floor tilings, using square, rectangular, hexagonal, or other shapes of ceramic tile. Escher’s primary interest in tessellations was as an artist. It also explains how they can be transformed using translation, rotation and glide reflection to create shapes like fish.Ī tessellation, or tiling, is a division of the plane into figures called tiles. It shows a simple visual demonstration of tessellating triangles, squares and hexagons. Escher inspired Tessellation Art, which explains the basic principles behind tessellating shapes and patterns. What is Tessellation? An educational video animation by M.
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