A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to a variety of geometries. A periodic tiling has a repeating pattern; some special kinds include regular tilings with regular polygonal tiles all of the same shape, semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation
Convex and Concave
Convex and Concave is a lithograph print by the Dutch artist M. C. Escher, first printed in March 1955, it depicts an ornate architectural structure with many stairs and other shapes. The relative aspects of the objects in the image are distorted in such a way that many of the structure's features can be seen as both convex shapes and concave impressions; this is a good example of Escher's mastery in creating illusions of "impossible architecture." The windows, roads and other shapes can be perceived as opening out in impossible ways and positions. The image on the flag is of reversible cubes. One can view these features as concave by viewing the image upside-down. All additional elements and decoration on the left are consistent with a viewpoint from above, while those on the right with a viewpoint from below: hiding half the image makes it easy to switch between convex and concave. Locher, J. L.. The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0
Dragon (M. C. Escher)
Dragon is a wood engraving print created by Dutch artist M. C. Escher in April 1952, depicting a folded paper dragon perched on a pile of crystals, it is part of a sequence of images by Escher depicting objects of ambiguous dimension, including Three Spheres I, Doric Columns, Drawing Hands and Print Gallery. Escher wrote "this dragon is an obstinate beast, in spite of his two-dimensions he persists in assuming that he has three". Two slits in the paper from which the dragon is folded open up like kirigami, forming holes that make the dragon's two-dimensional nature apparent, his head and neck pokes through one slit, the tail through the other, with the head biting the tail in the manner of the ouroboros. In Gödel, Bach, Douglas Hofstadter interprets the dragon's tail-bite as an image of self-reference, his inability to become three-dimensional as a visual metaphor for a lack of transcendence, the inability to "jump out of the system"; the same image has been called out in the scientific literature as a warning about what can happen when one attempts to describe four-dimensional space-time using higher dimensions.
A copy of this print is in the collections of U. S. National Gallery of Art and the National Gallery of Canada
Sky and Water II
Sky and Water II is a lithograph print by the Dutch artist M. C. Escher first printed in 1938, it is similar to the woodcut Sky and Water I, first printed only months earlier. Tessellation M. C. Escher—The Graphic Work. M. C. Escher—29 Master Prints. Publishers
Still Life with Spherical Mirror
Still Life with Spherical Mirror is a lithography print by the Dutch artist M. C. Escher first printed in November 1934, it depicts a setting with rounded bottle and a metal sculpture of a bird with a human face seated atop a newspaper and a book. The background is dark, but in the bottle can be seen the reflection of Escher's studio and Escher himself sketching the scene. Self-portraits in reflective spherical surfaces can be found in Escher's early ink drawings and in his prints as late as the 1950s; the metal bird/human sculpture was given to Escher by his father-in-law. This sculpture appears again in Escher's prints Another World Mezzotint and Another World. Printmaking Locher, J. L.. The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0
M. C. Escher
Maurits Cornelis Escher was a Dutch graphic artist who made mathematically-inspired woodcuts and mezzotints. Despite wide popular interest, Escher was for long somewhat neglected in the art world in his native Netherlands, he was 70. In the twenty-first century, he became more appreciated, with exhibitions across the world, his work features mathematical objects and operations including impossible objects, explorations of infinity, symmetry, perspective and stellated polyhedra, hyperbolic geometry, tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, conducted his own research into tessellation. Early in his career, he drew inspiration from nature, making studies of insects and plants such as lichens, all of which he used as details in his artworks, he traveled in Italy and Spain, sketching buildings, townscapes and the tilings of the Alhambra and the Mezquita of Cordoba, became more interested in their mathematical structure.
Escher's art became well known among scientists and mathematicians, in popular culture after it was featured by Martin Gardner in his April 1966 Mathematical Games column in Scientific American. Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums, he was one of the major inspirations of Douglas Hofstadter's Pulitzer Prize-winning 1979 book Gödel, Bach. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today, he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven. Although he excelled at drawing, his grades were poor, he took piano lessons until he was thirteen years old. In 1918, he went to the Technical College of Delft.
From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and the art of making woodcuts. He studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra and Ravello. In the same year, he traveled through Spain, visiting Madrid and Granada, he was impressed by the Italian countryside and, in Granada, by the Moorish architecture of the fourteenth-century Alhambra. The intricate decorative designs of the Alhambra, based on geometrical symmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of tessellation and became a powerful influence on his work. Escher returned to Italy and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924.
The couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta had two more sons – Arthur and Jan, he travelled visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the Amalfi coast in 1931 and 1934, Gargano and Sicily in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns, it was here that he became fascinated, to the point of obsession, with tessellation, explaining: It remains an absorbing activity, a real mania to which I have become addicted, from which I sometimes find it hard to tear myself away. The sketches he made in the Alhambra formed a major source for his work from that time on, he studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys.
His art correspondingly changed from being observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same his early work shows his interest in the nature of space, the unusual and multiple points of view. In 1935, the political climate in Italy became unacceptable to Escher, he had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d'Œx, where they remained for two years; the Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" in 1935, again in 1949 he designed Netherlands stamps. These were for the 75th anniversary of the Universal Postal Union. Escher, who had
Castrovalva (M. C. Escher)
Castrovalva is a lithograph print by the Dutch artist M. C. Escher, first printed in February 1930. Like many of Escher's early works, it depicts a place, it depicts the Abruzzo village of Castrovalva. The perspective is toward the northwest, from the narrow trail on the left which, at the point from which this view is seen, makes a hairpin turn to the right, descending to the valley. In the foreground at the side of the trail, there are several flowering plants, ferns, a beetle and a snail. In the expansive valley below there are cultivated fields and two more towns, the nearest of, Anversa degli Abruzzi, with Casale in the distance. In 1982 the name "Castrovalva" was used in a story in the BBC television series Doctor; the storyline relied on recursion, a favorite theme in Escher's and more famous works, used ideas taken from Belvedere and Descending, Relativity to trap the protagonists in the city of Castrovalva. The comic Kingdom of the Wicked is set in an imaginary world named Castrovalva.
Locher, J. L.. The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0