A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of simulating that algorithm's logic can be constructed; the machine operates on an infinite memory tape divided into discrete "cells". The machine positions its "head" over a cell and "reads" or "scans" the symbol there; as per the symbol and its present place in a "finite table" of user-specified instructions, the machine writes a symbol in the cell either moves the tape one cell left or right either proceeds to a subsequent instruction or halts the computation. The Turing machine was invented in 1936 by Alan Turing, who called it an "a-machine". With this model, Turing was able to answer two questions in the negative: Does a machine exist that can determine whether any arbitrary machine on its tape is "circular", thus by providing a mathematical description of a simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem.
Thus, Turing machines prove fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalistic design makes them unsuitable for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory. Turing completeness is the ability for a system of instructions to simulate a Turing machine. A programming language, Turing complete is theoretically capable of expressing all tasks accomplishable by computers. A Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More it is a machine capable of enumerating some arbitrary subset of valid strings of an alphabet. A Turing machine has a tape of infinite length on which it can perform write operations. Assuming a black box, the Turing machine cannot know whether it will enumerate any one specific string of the subset with a given program.
This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways; this is famously demonstrated through lambda calculus. A Turing machine, able to simulate any other Turing machine is called a universal Turing machine. A more mathematically oriented definition with a similar "universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church–Turing thesis; the thesis states that Turing machines indeed capture the informal notion of effective methods in logic and mathematics, provide a precise definition of an algorithm or "mechanical procedure". Studying their abstract properties yields many insights into computer science and complexity theory. In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of:...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed.
At any moment there is one symbol in the machine. The machine can alter the scanned symbol, its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore have an innings; the Turing machine mathematically models a machine. On this tape are symbols, which the machine can read and write, one at a time, using a tape head. Operation is determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1. In the original article, Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly. More a Turing machine consists of: A tape divided into cells, one next to the other; each cell contains a symbol from some finite alphabet. The alphabet contains one or more other symbols.
The tape is assumed to be arbitrarily extendable to the left and to the right, i.e. the Turing machine is always supplied with as much tape as it needs for its computation. Cells that have not been written before are assumed to be filled with the blank symbol. In some models the tape has a left e
A quasiperiodic crystal, or quasicrystal, is a structure, ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, some twenty years they were found to apply to the study of natural quasicrystals; the discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, offered evidence for the existence of natural quasicrystals.
An ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions. Symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally, the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than two, four, or six. In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the scientific community's reaction, it took him two years to publish the results for which he was awarded the Nobel Prize in Chemistry in 2011.
In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically. Two years his student Robert Berger constructed a set of some 20,000 square tiles that can tile the plane but not in a periodic fashion; as further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane; these tilings displayed instances of fivefold symmetry. One year Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp'delta' peaks arranged in a fivefold symmetric pattern. Around the same time Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.
Mathematically, quasicrystals have been shown to be derivable from a general method that treats them as projections of a higher-dimensional lattice. Just as circles and hyperbolic curves in the plane can be obtained as sections from a three-dimensional double cone, so too various arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984; the tiling is formed by two tiles with rhombohedral shape. Shechtman first observed ten-fold electron diffraction patterns in 1982, as described in his notebook; the observation was made during a routine investigation, by electron microscopy, of a cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards. In the summer of the same year Shechtman related his observation to him. Blech responded. Around that time, Shechtman related his finding to John Cahn of NIST who did not offer any explanation and challenged him to solve the observation.
Shechtman quoted Cahn as saying: "Danny, this material is telling us something and I challenge you to find out what it is". The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, the existence of twins, was ruled out by his experiments. Since periodicity and twins were ruled out, unaware of the two-dimensional tiling work, was looking for another possibility: a new structure containing cells connected to each other by defined angles and distances but without translational periodicity. Blech decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material without long-range translational order but still not random, he termed this new structure multiple polyhedral. The idea of a new structure was the necessary paradigm shift to break the impasse.
The “Eureka moment” came when the computer simulation showed sharp ten-fold diffraction patterns, similar to the observed ones, emanating from the three-dimensional structure devoid of periodicity. The multiple polyhedral structure was termed by many researchers as icosahedral glass but in effect it embraces any arran
Scientific American is an American popular science magazine. Many famous scientists, including Albert Einstein, have contributed articles to it, it is the oldest continuously published monthly magazine in the United States. Scientific American was founded by inventor and publisher Rufus M. Porter in 1845 as a four-page weekly newspaper. Throughout its early years, much emphasis was placed on reports of what was going on at the U. S. Patent Office, it reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, the universal joint which now can be found in nearly every automobile manufactured. Current issues include a "this date in history" section, featuring excerpts from articles published 50, 100, 150 years earlier. Topics include humorous incidents, wrong-headed theories, noteworthy advances in the history of science and technology. Porter sold the publication to Alfred Ely Beach and Orson Desaix Munn a mere ten months after founding it.
Until 1948, it remained owned by Company. Under Munn's grandson, Orson Desaix Munn III, it had evolved into something of a "workbench" publication, similar to the twentieth-century incarnation of Popular Science. In the years after World War II, the magazine fell into decline. In 1948, three partners who were planning on starting a new popular science magazine, to be called The Sciences, purchased the assets of the old Scientific American instead and put its name on the designs they had created for their new magazine, thus the partners—publisher Gerard Piel, editor Dennis Flanagan, general manager Donald H. Miller, Jr.—essentially created a new magazine. Miller retired in 1979, Flanagan and Piel in 1984, when Gerard Piel's son Jonathan became president and editor. In 1986, it was sold to the Holtzbrinck group of Germany. In the fall of 2008, Scientific American was put under the control of Nature Publishing Group, a division of Holtzbrinck. Donald Miller died in December 1998, Gerard Piel in September 2004 and Dennis Flanagan in January 2005.
Mariette DiChristina is the current editor-in-chief, after John Rennie stepped down in June 2009. Scientific American published its first foreign edition in 1890, the Spanish-language La America Cientifica. Publication was suspended in 1905, another 63 years would pass before another foreign-language edition appeared: In 1968, an Italian edition, Le Scienze, was launched, a Japanese edition, Nikkei Science, followed three years later. A new Spanish edition, Investigación y Ciencia was launched in Spain in 1976, followed by a French edition, Pour la Science, in France in 1977, a German edition, Spektrum der Wissenschaft, in Germany in 1978. A Russian edition V Mire Nauki was launched in the Soviet Union in 1983, continues in the present-day Russian Federation. Kexue, a simplified Chinese edition launched in 1979, was the first Western magazine published in the People's Republic of China. Founded in Chongqing, the simplified Chinese magazine was transferred to Beijing in 2001. In 2005, a newer edition, Global Science, was published instead of Kexue, which shut down due to financial problems.
A traditional Chinese edition, known as Scientist, was introduced to Taiwan in 2002. The Hungarian edition Tudomány existed between 1984 and 1992. In 1986, an Arabic edition, Oloom Magazine, was published. In 2002, a Portuguese edition was launched in Brazil. Today, Scientific American publishes 18 foreign-language editions around the globe: Arabic, Brazilian Portuguese, Simplified Chinese, Traditional Chinese, Dutch, German, Hebrew, Japanese, Lithuanian, Romanian and Spanish. From 1902 to 1911, Scientific American supervised the publication of the Encyclopedia Americana, which during some of that period was known as The Americana, it styled itself "The Advocate of Industry and Enterprise" and "Journal of Mechanical and other Improvements". On the front page of the first issue was the engraving of "Improved Rail-Road Cars"; the masthead had a commentary as follows: Scientific American published every Thursday morning at No. 11 Spruce Street, New York, No. 16 State Street, No. 2l Arcade Philadelphia, by Rufus Porter.
Each number will be furnished with from two to five original Engravings, many of them elegant, illustrative of New Inventions, Scientific Principles, Curious Works. Improvements and Inventions; this paper is entitled to the patronage of Mechanics and Manufactures, being the only paper in America, devoted to the interest of those classes. As a family newspaper, it will convey more useful intelligence to children and young people, than five times its cost in school instruction. Another important argument in favor of this paper, is that it will be worth two dollars at the end of the year when the volume is complete, (Old volumes of the New York Mechanic, being now worth double th
Microsoft Entertainment Pack
Windows Entertainment Pack is a collection of 16-bit casual computer games for Windows. There were four Entertainment Packs in the original series. Many of the games were released in the Best of Microsoft Entertainment Pack; these games were somewhat unusual for the time, in that they would not run under MS-DOS. All games being 16-bit run on modern 32-bit versions of Windows but not on 64-bit Windows. Support for all versions of Microsoft Entertainment Pack ended on January 31, 2003. Minesweeper from pack 1 was bundled with Windows 3.1, FreeCell was included in Windows 95. WinChess and Taipei, both written by David Norris, received remakes in Windows Vista, called Chess Titans and Mahjong Titans, respectively. Mahjong Titans was replaced with Microsoft Mahjong in Windows 8. Microsoft Solitaire Collection includes versions of Tut's Tomb and TriPeaks. Microsoft Entertainment Pack was designed by the company's “Entry Business” team, whose job was to make Windows more appealing to homes and small businesses.
Ex-Microsoft product manager Bruce Ryan said the company did this because it "was concerned that the operating system’s high hardware requirements meant that people would only see it as a tool for large enterprises". The project had "almost no budget", no major video game publishers got involved because they doubted Windows' legitimacy as a gaming platform. According to Microsoft FreeCell developer Jim Horne, the packs were not copy protected so customers could distribute copies to friends, to encourage using Windows for games; as payment, each author received ten shares of Microsoft stock. Microsoft advertised Entertainment Packs for casual gaming on office computers; the boxes had slogans like "No more boring coffee breaks" and "Only a few minutes between meetings? Get in a quick game of Klotski"; the marketing succeeded. The original Microsoft Windows Entertainment Pack titles include: Cruel Golf Minesweeper, written by Rob Donner Pegged, written by Mike Blaylock Taipei Tetris TicTactics IdleWild, written by Brad Christian FreeCell Jigsawed Pipe Dream, written by Eric Geyser Rattler Race Rodent's Revenge Stones, developed by Michael C. Miller Tut's Tomb IdleWild - 8 new screen savers for this pack Fuji Golf Klotski Life Genesis SkiFree, written by Chris Pirih TetraVex TriPeaks WordZap IdleWild - 8 new screen savers for this pack Chess Chip's Challenge, written by Chuck Sommerville Dr. Black Jack, a card game created by Mike Blaylock, based on the game of the same name Go Figure!
JezzBall Maxwell's Maniac Tic Tac Drop, a form of Connect Four with quadrilateral and plus-shaped boards and customizable win pattern and number of rows and columns For much of the early 1990s, the Gamesampler, a subset of the Entertainment Pack small enough to fit on a single high-density disk, was shipped as a free eleventh disk added to a ten-pack of Verbatim blank 3.5" microfloppy diskettes. Games on the sampler included Jezzball, Rodent's Revenge and Skifree. A "Best of" disk of several of the games was available at times as a mail-in premium from Kellogg's cereals. In the copies of Windows NT 4.0 and Windows 2000 source code which leaked back in 2004, there are 32-bit versions of Cruel, Pegged, Snake and TicTactics. However, FreeCell and Minesweeper have had official 32-bit versions bundled with early versions of Windows NT; the original game developers of some of the games such as SkiFree, TriPeaks, WordZap now offer 32-bit versions. Third party developers have created 32-bit freeware clones of Klotski, TetraVex Rodent's Revenge, Tetris.
A multicart containing seven games was released for the Game Boy Color as The Best Of Entertainment Pack. Four of the games on the multi-cart were ported from Best of Microsoft Entertainment Pack, while the remaining three were ported from other Microsoft Entertainment Pack compilations. Digital Trends noted, "For many, the simple but enjoyable games found in the Entertainment Pack provided a first taste of early PC gaming and served as a gateway to more complex classics." PC World described the pack as having "standout time-wasters". Microsoft Entertainment Pack: The Puzzle Collection – a 32-bit collection for Windows 95 Microsoft Pinball Arcade Microsoft Arcade Microsoft Entertainment Pack at MobyGames
Dominoes is a family of tile-based games played with rectangular "domino" tiles. Each domino is a rectangular tile with a line dividing its face into two square ends; each end is blank. The backs of the dominoes in a set are either blank or having some common design; the domino gaming pieces make up a domino set, sometimes called a pack. The traditional Sino-European domino set consists of 28 dominoes, featuring all combinations of spot counts between zero and six. A domino set is a generic gaming device, similar to playing cards or dice, in that a variety of games can be played with a set; the earliest mention of dominoes is from Song dynasty China found in the text Former Events in Wulin by Zhou Mi. Modern dominoes first appeared in Italy during the 18th century, but how Chinese dominoes developed into the modern game is unknown. Italian missionaries in China may have brought the game to Europe; the name "domino" is most from the resemblance to a kind of carnival costume worn during the Venetian Carnival consisting of a black-hooded robe and a white mask.
Despite the coinage of the word polyomino as a generalization, there is no connection between the word "domino" and the number 2 in any language. European-style dominoes are traditionally made of bone or ivory, or a dark hardwood such as ebony, with contrasting black or white pips. Alternatively, domino sets have been made from many different natural materials: stone; these sets have a more novel look, the heavier weight makes them feel more substantial. Modern commercial domino sets are made of synthetic materials, such as ABS or polystyrene plastics, or Bakelite and other phenolic resins. Modern sets commonly use a different color for the dots of each different end value to facilitate finding matching ends. One may find a domino set made of card stock like that for playing cards; such sets are lightweight and inexpensive, like cards are more susceptible to minor disturbances such as a sudden breeze. Sometimes, dominoes have a metal pin in the middle; the traditional set of dominoes contains one unique piece for each possible combination of two ends with zero to six spots, is known as a double-six set because the highest-value piece has six pips on each end.
The spots from one to six are arranged as they are on six-sided dice, but because blank ends having no spots are used, seven faces are possible, allowing 28 unique pieces in a double-six set. However, this is a small number when playing with more than four people, so many domino sets are "extended" by introducing ends with greater numbers of spots, which increases the number of unique combinations of ends and thus of pieces; each progressively larger set increases the maximum number of pips on an end by three, so the common extended sets are double-nine, double-12, double-15, double-18. Larger sets such as double-21 can theoretically exist, but are seen in retail stores, as identifying the number of pips on each domino becomes difficult, a double-21 set would have 253 pieces, far more than is necessary for most domino games with eight players; the oldest confirmed written mention of dominoes in China comes from the Former Events in Wulin written by the Yuan Dynasty author Zhou Mi, who listed pupai, as well as dice as items sold by peddlers during the reign of Emperor Xiaozong of Song.
Andrew Lo asserts that Zhou Mi meant dominoes when referring to pupai, since the Ming author Lu Rong explicitly defined pupai as dominoes. The earliest known manual written about dominoes is the 《宣和牌譜》 written by Qu You, but some Chinese scholars believe this manual is a forgery from a time. In the Encyclopedia of a Myriad of Treasures, Zhang Pu described the game of laying out dominoes as pupai, although the character for pu had changed, yet retained the same pronunciation. Traditional Chinese domino games include Tien Gow, Pai Gow, Che Deng, others; the 32-piece Chinese domino set, made to represent each possible face of two thrown dice and thus have no blank faces, differs from the 28-piece domino set found in the West during the mid 18th century. Chinese dominoes with blank faces were known during the 17th century. Many different domino sets have been used for centuries in various parts of the world to play a variety of domino games; each domino represented one of the 21 results of throwing two six-sided dice.
One half of each domino is set with the pips from one die and the other half contains the pips from the second die. Chinese sets introduce duplicates of some throws and divide the dominoes into two suits: military and civil. Chinese dominoes are longer than typical European dominoes; the early 18th century had dominoes making their way to Europe, making their first appearance in Italy. The game changed somewhat in the translation from Chinese to the European culture. European domino sets contain neither sui
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
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