1.
Typeface
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In metal typesetting, a font is a particular size, weight and style of a typeface. Each font was a set of type, one piece for each glyph. In modern usage, with the advent of digital typography, font is frequently synonymous with typeface, in particular, the use of vector or outline fonts means that different sizes of a typeface can be dynamically generated from one design. The word font derives from Middle French fonte melted, a casting, the term refers to the process of casting metal type at a type foundry. In a manual printing house the word font would refer to a set of metal type that would be used to typeset an entire page. Unlike a digital typeface it would not include a definition of each character. A font when bought new would often be sold as 12pt 14A 34a, meaning that it would be a size 12-point font containing 14 uppercase As, given the name upper and lowercase because of which case the metal type was located in, otherwise known as majuscule and minuscule. The rest of the characters would be provided in quantities appropriate for the distribution of letters in that language. Some metal type characters required in typesetting, such as dashes, spaces and line-height spacers, were not part of a specific font, line spacing is still often called leading, because the strips used for line spacing were made of lead. In the 1880s–90s, hot lead typesetting was invented, in which type was cast as it was set, either piece by piece or in entire lines of type at one time. In European alphabetic scripts, i. e. Latin, Cyrillic and Greek, the main properties are the stroke width, called weight, the style or angle. The regular or standard font is sometimes labeled roman, both to distinguish it from bold or thin and from italic or oblique. The keyword for the default, regular case is often omitted for variants and never repeated, otherwise it would be Bulmer regular italic, Bulmer bold regular, Roman can also refer to the language coverage of a font, acting as a shorthand for Western European. Different fonts of the same typeface may be used in the work for various degrees of readability and emphasis. The weight of a font is the thickness of the character outlines relative to their height. A typeface may come in fonts of many weights, from ultra-light to extra-bold or black, four to six weights are not unusual, many typefaces for office, web and non-professional use come with just a normal and a bold weight which are linked together. If no bold weight is provided, many renderers support faking a bolder font by rendering the outline a second time at an offset, the base weight differs among typefaces, that means one normal font may appear bolder than some other normal font. For example, fonts intended to be used in posters are often quite bold by default while fonts for long runs of text are rather light, therefore, weight designations in font names may differ in regard to the actual absolute stroke weight or density of glyphs in the font
2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Typewriter
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A typewriter is a mechanical or electromechanical machine for writing characters similar to those produced by printers movable type. A typewriter operates by means of keys that strike a ribbon to transmit ink or carbon impressions onto paper, typically, a single character is printed on each key press. The machine prints characters by making ink impressions of type elements similar to the used in movable type letterpress printing. At the end of the century, the term typewriter was also applied to a person who used a typing machine. After its invention in the 1860s, the quickly became an indispensable tool for practically all writing other than personal handwritten correspondence. It was widely used by writers, in offices. As with the automobile, telephone, and telegraph, a number of people contributed insights, historians have estimated that some form of typewriter was invented 52 times as thinkers tried to come up with a workable design. Some of the early typing instruments, In 1575 an Italian printmaker, Francesco Rampazzetto, invented the scrittura tattile, in 1714, Henry Mill obtained a patent in Britain for a machine that, from the patent, appears to have been similar to a typewriter. In 1802 Italian Agostino Fantoni developed a particular typewriter to enable his blind sister to write, in 1808 Italian Pellegrino Turri invented a typewriter. He also invented carbon paper to provide the ink for his machine, in 1823 Italian Pietro Conti di Cilavegna invented a new model of typewriter, the tachigrafo, also known as tachitipo. In 1829, William Austin Burt patented a machine called the Typographer which, the Science Museum describes it merely as the first writing mechanism whose invention was documented, but even that claim may be excessive, since Turris invention pre-dates it. Even in the hands of its inventor, this machine was slower than handwriting, Burt and his promoter John D. Sheldon never found a buyer for the patent, so the invention was never commercially produced, because the typographer used a dial, rather than keys, to select each character, it was called an index typewriter rather than a keyboard typewriter. Index typewriters of that era resemble the squeeze-style embosser from the 1960s more than they resemble the modern keyboard typewriter, by the mid-19th century, the increasing pace of business communication had created a need for mechanization of the writing process. Stenographers and telegraphers could take down information at rates up to 130 words per minute, from 1829 to 1870, many printing or typing machines were patented by inventors in Europe and America, but none went into commercial production. Charles Thurber developed multiple patents, of which his first in 1843 was developed as an aid to the blind, in 1855, the Italian Giuseppe Ravizza created a prototype typewriter called Cembalo scrivano o macchina da scrivere a tasti. It was a machine that let the user see the writing as it was typed. In 1861, Father Francisco João de Azevedo, a Brazilian priest, made his own typewriter with basic materials and tools, such as wood, in that same year the Brazilian emperor D
4.
Blackboard
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A blackboard is a reusable writing surface on which text or drawings are made with sticks of calcium sulfate or calcium carbonate, known, when used for this purpose, as chalk. Blackboards were originally made of smooth, thin sheets of black or dark grey slate stone, a blackboard can simply be a piece of board painted with matte dark paint. Black plastic sign material, using the trade name sintra is also used to create custom chalkboard art, a more modern variation consists of a coiled sheet of plastic drawn across two parallel rollers, which can be scrolled to create additional writing space while saving what has been written. The highest grade blackboards are made of a rougher version porcelain enamelled steel, porcelain is very hard wearing and blackboards made of porcelain usually last 10–20 years in intensive use. Manufacturing of slate blackboards began by the 1840s, green chalkboards, generally made of porcelain enamel on a steel base, first appeared in the 1960s. Lecture theatres may contain a number of blackboards in a grid arrangement, the lecturer then moves boards into reach for writing and then moves them out of reach, allowing a large amount of material to be shown simultaneously. The chalk marks can be wiped off with a damp cloth. However, chalk marks made on some types of wet blackboard can be difficult to remove, sticks of processed chalk are produced especially for use with blackboards in white and also in various colours. These are often not from chalk rock but from calcium sulfate in its dihydrate form. Chalk sticks containing calcium carbonate typically contain 40-60% of CaCO3, as compared to whiteboards, blackboards have a variety of advantages, Chalk requires no special care, whiteboard markers must be capped or else they dry out. Chalk is an order of magnitude cheaper than whiteboard markers for a amount of writing. It is easier to draw lines of different weights and thicknesses with chalk than with whiteboard markers, dashed lines can be drawn very quickly using a technique involving the friction of the chalk and blackboard. Chalk has a smell, whereas whiteboard markers often have a pungent odor. Chalk writing often provides better contrast than whiteboard markers, Chalk can be easily erased, writing which has been left on a whiteboard for a prolonged period may require a solvent to remove. Chalk can be removed from most clothing, whiteboard markers often permanently stain fabric. On the other hand, chalk produces dust, the amount depending on the quality of chalk used, the dust also precludes the use of chalk in areas shared with dust-sensitive equipment such as computers. The writing on chalkboards is difficult to read in the dark, Chalk sticks are notorious for shrinking through use, and breaking in half unless inserted in a writing utensil designed to store chalk. According to a run by Michael Oehler, a professor at the University of Media and Communication in Cologne, Germany
5.
Donald Knuth
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Donald Ervin Knuth is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the author of the multi-volume work The Art of Computer Programming and he contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process he also popularized the asymptotic notation, Knuth strongly opposes granting software patents, having expressed his opinion to the United States Patent and Trademark Office and European Patent Organization. Knuth was born in Milwaukee, Wisconsin, to German-Americans Ervin Henry Knuth and his father had two jobs, running a small printing company and teaching bookkeeping at Milwaukee Lutheran High School. Donald, a student at Milwaukee Lutheran High School, received academic accolades there, for example, in eighth grade, he entered a contest to find the number of words that the letters in Zieglers Giant Bar could be rearranged to create. Although the judges only had 2,500 words on their list, Donald found 4,500 words, as prizes, the school received a new television and enough candy bars for all of his schoolmates to eat. Knuth had a time choosing physics over music as his major at Case Institute of Technology. He also joined Beta Nu Chapter of the Theta Chi fraternity, while studying physics at the Case Institute of Technology, Knuth was introduced to the IBM650, one of the early mainframes. After reading the manual, Knuth decided to rewrite the assembly and compiler code for the machine used in his school. In 1958, Knuth created a program to help his schools basketball team win their games and he assigned values to players in order to gauge their probability of getting points, a novel approach that Newsweek and CBS Evening News later reported on. Knuth was one of the editors of the Engineering and Science Review. In 1963, with mathematician Marshall Hall as his adviser, he earned a PhD in mathematics from the California Institute of Technology, after receiving his PhD, Knuth joined Caltechs faculty as an associate professor. He accepted a commission to write a book on computer programming language compilers and he originally planned to publish this as a single book. As Knuth developed his outline for the book, he concluded that he required six volumes and he published the first volume in 1968. Knuth then left this position to join the Stanford University faculty, Knuth is a writer as well as a computer scientist. Knuth has been called the father of the analysis of algorithms, in the 1970s, Knuth described computer science as a totally new field with no real identity. And the standard of available publications was not that high, a lot of the papers coming out were quite simply wrong. So one of my motivations was to put straight a story that had been very badly told, by 2013, the first three volumes and part one of volume four of his series had been published
6.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
7.
William Rowan Hamilton
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Sir William Rowan Hamilton PRIA FRSE was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and his best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the study of classical field theories such as electromagnetism. In pure mathematics, he is best known as the inventor of quaternions, Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, This young man, I do not say will be, but is, Hamilton also invented icosian calculus, which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once. Hamilton was the fourth of nine born to Sarah Hutton and Archibald Hamilton. Hamiltons father, who was from Dunboyne, worked as a solicitor, by the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. His uncle soon discovered that Hamilton had an ability to learn languages. At the age of seven he had made very considerable progress in Hebrew. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, in September 1813 the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, an older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor, in reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, which he entered at age 18. He studied both classics and mathematics, and was appointed Professor of Astronomy in 1827, prior to his taking up residence at Dunsink Observatory where he spent the rest of his life. Hamilton made important contributions to optics and to classical mechanics and his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of Caustics in 1824 to the Royal Irish Academy. It was referred as usual to a committee, while their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, but it also became more intelligible, and the features of the new method were now easily to be seen. Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics and he proposed for it when he first predicted its existence in the third supplement to his Systems of Rays, read in 1832
8.
Jean-Pierre Serre
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Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000, born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951, from 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France and his wife, Professor Josiane Heulot-Serre, was a chemist, she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil, the French mathematician Denis Serre is his nephew. Serres thesis concerned the Leray–Serre spectral sequence associated to a fibration, together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, Serre subsequently changed his research focus. However, Weyls perception that the place of classical analysis had been challenged has subsequently been justified. In the 1950s and 1960s, a collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldnt capture as much topology as singular cohomology with integer coefficients, amongst Serres early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important and this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA5, from 1959 onward Serres interests turned towards group theory, number theory, in particular Galois representations and modular forms. In his paper FAC, Serre asked whether a finitely generated module over a polynomial ring is free. This question led to a deal of activity in commutative algebra. This result is now known as the Quillen-Suslin theorem, Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000 and he has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre. He is a member of several scientific Academies and has received many honorary degrees
9.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
10.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0