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. ![]() He blogs at thatsmaths.We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe. Peter Lynch is emeritus professor at the school of mathematics and statistics, University College Dublin. It is surprising how the simple concept of tessellating a region in terms of distance to a given set of points can be so powerful. These include network analysis, computer graphics, medical diagnostics, astrophysics, hydrology, robotics and computational fluid dynamics. There are numerous other applications of Voronoi diagrams. An engineering survey later showed that a poorly constructed drain was contaminating the pump water. When the pump handle was removed, death rates were greatly diminished and the epidemic quickly died out. This revealed that almost all fatalities were in houses supplied by a single pump in Broad Street, Soho. He plotted his data on a chart, effectively constructing a Voronoi diagram. He divided inner London into neighbourhoods, each having a separate water supply. Snow gathered statistics on the number of victims and locations of outbreaks. Proximity diagrams were used by many mathematicians, back to Descartes in the mid-17th century, but their theory was developed by Voronoi, who in 1908 defined and studied diagrams of this type in the general context of n-dimensional space, with n being the number of dimensions.Ī serious cholera outbreak in 1854 killed 500 people in five days. ![]() For any location, the nearest service can immediately be read off the diagram (see the accompanying figure). Voronoi diagrams are easily constructed and, with computer software, can be depicted as colourful charts, indicating the region associated with each service point or site. The cells are all convex polygons that is, they have boundaries made up of straight line segments and all corners have internal angles less than 180 degrees. In mathematical terms, we are given a finite set of points in the plane and, for each point, the corresponding Voronoi cell consists of all the locations closer to it than to any of the other points. Of course, numerous parameters other than distance must be considered, but access time is often the critical factor. A Voronoi diagram can be used to find the largest empty circle amid a collection of points, giving the ideal location for the new school. He is remembered today mostly for this diagram, also known as a Voronoi tessellation, decomposition or partition.Īnother practical problem is to choose a location for a new service, such as a school, which is as far as possible from existing schools while still serving the maximum number of families. Such a map is called a Voronoi diagram, named after Georgy Voronoi, a mathematician born in Ukraine in 1868. A map divided into cells, each cell covering the region closest to a particular centre, can assist us in our quest. We frequently need to find the nearest hospital, surgery or supermarket.
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