In geometry , a dodecahedron (from Ancient Greek δωδεκάεδρον ( dōdekáedron ) ; from δώδεκα ( dṓdeka ) 'twelve' and ἕδρα ( hédra ) 'base, seat, face') or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid . There are also three regular star dodecahedra , which are constructed as stellations of the convex form. All of these have icosahedral symmetry , order 120.
116-433: In mathematics , two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a {\displaystyle a} and b {\displaystyle b} with a > b > 0 {\displaystyle a>b>0} ,
232-490: A = a b = φ . {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .} Thus, if we want to find φ {\displaystyle \varphi } , we may use that the definition above holds for arbitrary b {\displaystyle b} ; thus, we just set b = 1 {\displaystyle b=1} , in which case φ =
348-797: A {\displaystyle \varphi =a} and we get the equation ( φ + 1 ) / φ = φ {\displaystyle (\varphi +1)/\varphi =\varphi } , which becomes a quadratic equation after multiplying by φ {\displaystyle \varphi } : φ + 1 = φ 2 {\displaystyle \varphi +1=\varphi ^{2}} which can be rearranged to φ 2 − φ − 1 = 0. {\displaystyle {\varphi }^{2}-\varphi -1=0.} The quadratic formula yields two solutions: Because φ {\displaystyle \varphi }
464-539: A {\displaystyle a} is in a golden ratio to b {\displaystyle b} if a + b a = a b = φ , {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi ,} where the Greek letter phi ( φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) denotes
580-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
696-464: A simple continued fraction for the golden ratio: φ = [ 1 ; 1 , 1 , 1 , … ] = 1 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}} It
812-471: A symbol for the golden ratio. It has also been represented by tau ( τ {\displaystyle \tau } ), the first letter of the ancient Greek τομή ('cut' or 'section'). The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron / dodecahedron , and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling ,
928-455: A triangular gyrobianticupola. It has D 3d symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match. The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It
1044-437: A Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates φ {\displaystyle \varphi } . For example, These approximations are alternately lower and higher than φ {\displaystyle \varphi } , and converge to φ {\displaystyle \varphi } as
1160-432: A cube have the coordinates (±1, ±1, ±1). The coordinates of the 12 additional vertices are ( 0, ±(1 + h ), ±(1 − h ) ) , ( ±(1 + h ), ±(1 − h ), 0 ) and ( ±(1 − h ), 0, ±(1 + h ) ) . h is the height of the wedge -shaped "roof" above the faces of that cube with edge length 2. An important case is h = 1 / 2 (a quarter of the cube edge length) for perfect natural pyrite (also
1276-421: A cubic convex hull at one limit of collinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal. It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The endo-dodecahedron is concave and equilateral; it can tessellate space with
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#17327810991221392-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
1508-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
1624-453: A limiting case of the pyritohedron, and it has octahedral symmetry . The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling . There are numerous other dodecahedra . While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across
1740-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
1856-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
1972-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
2088-495: A pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. This gained in interest after Dan Shechtman 's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling. The golden ratio is an irrational number . Below are two short proofs of irrationality: This
2204-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
2320-1517: A result, one can easily decompose any power of φ {\displaystyle \varphi } into a multiple of φ {\displaystyle \varphi } and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ {\displaystyle \varphi } : If ⌊ 1 2 n − 1 ⌋ = m {\displaystyle {\bigl \lfloor }{\tfrac {1}{2}}n-1{\bigr \rfloor }=m} , then: φ n = φ n − 1 + φ n − 3 + ⋯ + φ n − 1 − 2 m + φ n − 2 − 2 m φ n − φ n − 1 = φ n − 2 . {\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}\\[5mu]\varphi ^{n}-\varphi ^{n-1}&=\varphi ^{n-2}.\end{aligned}}} The formula φ = 1 + 1 / φ {\displaystyle \varphi =1+1/\varphi } can be expanded recursively to obtain
2436-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
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#17327810991222552-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
2668-506: A square and a smaller rectangle with the same aspect ratio . The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets , in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature , including the spiral arrangement of leaves and other parts of vegetation. Some 20th-century artists and architects , including Le Corbusier and Salvador Dalí , have proportioned their works to approximate
2784-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
2900-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
3016-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
3132-433: Is flat " and "a field is always a ring ". Dodecahedron Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron , a common crystal form in pyrite , has pyritohedral symmetry , while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron can be seen as
3248-461: Is a proof by infinite descent . Recall that: If we call the whole n {\displaystyle n} and the longer part m {\displaystyle m} , then the second statement above becomes To say that the golden ratio φ {\displaystyle \varphi } is rational means that φ {\displaystyle \varphi }
3364-576: Is a dodecahedron with chiral tetrahedral symmetry (T). Like the regular dodecahedron , it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form. Abstractions sharing
3480-585: Is a fraction n / m {\displaystyle n/m} where n {\displaystyle n} and m {\displaystyle m} are integers. We may take n / m {\displaystyle n/m} to be in lowest terms and n {\displaystyle n} and m {\displaystyle m} to be positive. But if n / m {\displaystyle n/m}
3596-400: Is a ratio between positive quantities, φ {\displaystyle \varphi } is necessarily the positive root. The negative root is in fact the negative inverse − 1 / φ {\displaystyle -1/\varphi } , which shares many properties with the golden ratio. According to Mario Livio , Some of
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3712-461: Is also a parallelohedral spacefiller . Another important rhombic dodecahedron, the Bilinski dodecahedron , has twelve faces congruent to those of the rhombic triacontahedron , i.e. the diagonals are in the ratio of the golden ratio . It is also a zonohedron and was described by Bilinski in 1960. This figure is another spacefiller, and can also occur in non-periodic spacefillings along with
3828-406: Is also closely related to the polynomial x 2 + x − 1 {\displaystyle \textstyle x^{2}+x-1} , which has roots − φ {\displaystyle -\varphi } and φ − 1 {\displaystyle \textstyle \varphi ^{-1}} . As
3944-407: Is based on one that is itself created by enlarging 24 of the 48 faces of the disdyakis dodecahedron .) The crystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges. The following points are vertices of a tetartoid pentagon under tetrahedral symmetry : under
4060-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
4176-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
4292-414: Is dual to the quasiregular cuboctahedron (an Archimedean solid ) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces. The rhombic dodecahedron has several stellations , the first of which
4408-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
4524-428: Is equal to the sum of the two immediately preceding powers: φ n = φ n − 1 + φ n − 2 = φ ⋅ F n + F n − 1 . {\displaystyle \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.} As
4640-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
4756-1832: Is in fact the simplest form of a continued fraction, alongside its reciprocal form: φ − 1 = [ 0 ; 1 , 1 , 1 , … ] = 0 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle \varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}} The convergents of these continued fractions, 1 1 {\displaystyle {\tfrac {1}{1}}} , 2 1 {\displaystyle {\tfrac {2}{1}}} , 3 2 {\displaystyle {\tfrac {3}{2}}} , 5 3 {\displaystyle {\tfrac {5}{3}}} , 8 5 {\displaystyle {\tfrac {8}{5}}} , 13 8 {\displaystyle {\tfrac {13}{8}}} , ... or 1 1 {\displaystyle {\tfrac {1}{1}}} , 1 2 {\displaystyle {\tfrac {1}{2}}} , 2 3 {\displaystyle {\tfrac {2}{3}}} , 3 5 {\displaystyle {\tfrac {3}{5}}} , 5 8 {\displaystyle {\tfrac {5}{8}}} , 8 13 {\displaystyle {\tfrac {8}{13}}} , ..., are ratios of successive Fibonacci numbers . The consistently small terms in its continued fraction explain why
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4872-408: Is in lowest terms, then the equally valued m / ( n − m ) {\displaystyle m/(n-m)} is in still lower terms. That is a contradiction that follows from the assumption that φ {\displaystyle \varphi } is rational. Another short proof – perhaps more commonly known – of the irrationality of
4988-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
5104-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
5220-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
5336-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
5452-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
5568-515: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
5684-515: Is the greater to the lesser. The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers . Luca Pacioli named his book Divina proportione ( 1509 ) after
5800-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
5916-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
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#17327810991226032-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
6148-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
6264-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
6380-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
6496-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
6612-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
6728-451: The divine proportion by Luca Pacioli ; and also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon 's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron . A golden rectangle —that is, a rectangle with an aspect ratio of φ {\displaystyle \varphi } —may be cut into
6844-703: The square root of 5 {\displaystyle 5} , must also be rational. This is a contradiction as the square roots of all non- square natural numbers are irrational. The golden ratio is also an algebraic number and even an algebraic integer . It has minimal polynomial x 2 − x − 1. {\displaystyle x^{2}-x-1.} This quadratic polynomial has two roots , φ {\displaystyle \varphi } and − φ − 1 {\displaystyle \textstyle -\varphi ^{-1}} . The golden ratio
6960-460: The tetartoid with tetrahedral symmetry : A pyritohedron is a dodecahedron with pyritohedral (T h ) symmetry. Like the regular dodecahedron , it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure). However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of
7076-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
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#17327810991227192-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
7308-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
7424-608: The Fibonacci and Lucas numbers increase. Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
7540-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
7656-689: The approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations , which states that for every irrational ξ {\displaystyle \xi } , there are infinitely many distinct fractions p / q {\displaystyle p/q} such that, | ξ − p q | < 1 5 q 2 . {\displaystyle \left|\xi -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.} This means that
7772-586: The basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry ; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons . According to one story, 5th-century BC mathematician Hippasus discovered that
7888-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
8004-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
8120-872: The constant 5 {\displaystyle {\sqrt {5}}} cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers . A continued square root form for φ {\displaystyle \varphi } can be obtained from φ 2 = 1 + φ {\displaystyle \textstyle \varphi ^{2}=1+\varphi } , yielding: φ = 1 + 1 + 1 + ⋯ ) . {\displaystyle \varphi ={\sqrt {1+{\sqrt {\textstyle 1+{\sqrt {1+\cdots {\vphantom {)}}}}}}}}.} Fibonacci numbers and Lucas numbers have an intricate relationship with
8236-598: The convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular great stellated dodecahedron where all edges and angles are equal again, and the faces have been distorted into regular pentagrams . On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces. A tetartoid (also tetragonal pentagonal dodecahedron , pentagon-tritetrahedron , and tetrahedric pentagon dodecahedron )
8352-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
8468-1186: The defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with φ {\displaystyle \varphi } : φ 2 = φ + 1 = 2.618033 … , 1 φ = φ − 1 = 0.618033 … . {\displaystyle {\begin{aligned}\varphi ^{2}&=\varphi +1=2.618033\dots ,\\[5mu]{\frac {1}{\varphi }}&=\varphi -1=0.618033\dots .\end{aligned}}} The sequence of powers of φ {\displaystyle \varphi } contains these values 0.618033 … {\displaystyle 0.618033\ldots } , 1.0 {\displaystyle 1.0} , 1.618033 … {\displaystyle 1.618033\ldots } , 2.618033 … {\displaystyle 2.618033\ldots } ; more generally, any power of φ {\displaystyle \varphi }
8584-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
8700-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
8816-611: The face of a perfect crystal (which is rarely found in nature). Height = 5 2 ⋅ Long side {\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}} Width = 4 3 ⋅ Long side {\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}} Short sides = 7 12 ⋅ Long side {\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}} The eight vertices of
8932-461: The figure that return to the original point without crossing over any other corner. The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}. The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex. The convex regular dodecahedron also has three stellations , all of which are regular star dodecahedra. They form three of
9048-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
9164-481: The following conditions: The regular dodecahedron is a tetartoid with more than the required symmetry. The triakis tetrahedron is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.) A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular anticupola connected base-to-base, called
9280-514: The four Kepler–Poinsot polyhedra . They are the small stellated dodecahedron {5/2, 5}, the great dodecahedron {5, 5/2}, and the great stellated dodecahedron {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the great icosahedron {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of
9396-536: The golden ratio ; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about 0.6180340 {\displaystyle 0.6180340} " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle , which combines
9512-518: The golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: lim n → ∞ F n + 1 F n = lim n → ∞ L n + 1 L n = φ . {\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\varphi .} In other words, if
9628-482: The golden ratio makes use of the closure of rational numbers under addition and multiplication. If φ = 1 2 ( 1 + 5 ) {\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}} is assumed to be rational, then 2 φ − 1 = 5 {\displaystyle 2\varphi -1={\sqrt {5}}} ,
9744-412: The golden ratio was neither a whole number nor a fraction (it is irrational ), surprising Pythagoreans . Euclid 's Elements ( c. 300 BC ) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so
9860-436: The golden ratio with the Pythagorean theorem . Kepler said of these: Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel. Eighteenth-century mathematicians Abraham de Moivre , Nicolaus I Bernoulli , and Leonhard Euler used a golden ratio-based formula which finds
9976-400: The golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle. Two quantities a {\displaystyle a} and b {\displaystyle b} are in the golden ratio φ {\displaystyle \varphi } if a + b
10092-523: The golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence 0 , 1 {\displaystyle 0,1} : The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences , of which this is part) is like the Fibonacci sequence, in which each term is the sum of the previous two, however instead starts with 2 , 1 {\displaystyle 2,1} : Exceptionally,
10208-412: The golden ratio. The constant φ {\displaystyle \varphi } satisfies the quadratic equation φ 2 = φ + 1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid , and
10324-470: The greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece , through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler , to present-day scientific figures such as Oxford physicist Roger Penrose , have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated
10440-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
10556-491: The interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to
10672-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
10788-609: The length ratio taken in reverse order (shorter segment length over longer segment length, b / a {\displaystyle b/a} ). This illustrates the unique property of the golden ratio among positive numbers, that 1 φ = φ − 1 , {\displaystyle {\frac {1}{\varphi }}=\varphi -1,} or its inverse, 1 1 / φ = 1 φ + 1. {\displaystyle {\frac {1}{1/\varphi }}={\frac {1}{\varphi }}+1.} The conjugate and
10904-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
11020-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
11136-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
11252-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
11368-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
11484-478: The pyritohedron in the Weaire–Phelan structure ). Another one is h = 1 / φ = 0.618... for the regular dodecahedron . See section Geometric freedom for other cases. Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the compound of two dodecahedra . The pyritohedron has a geometric degree of freedom with limiting cases of
11600-460: The ratio; the book, largely plagiarized from Piero della Francesca , explored its properties including its appearance in some of the Platonic solids . Leonardo da Vinci , who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that
11716-405: The rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra. There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing
11832-707: The root of a quadratic polynomial, the golden ratio is a constructible number . The conjugate root to the minimal polynomial x 2 − x − 1 {\displaystyle \textstyle x^{2}-x-1} is − 1 φ = 1 − φ = 1 − 5 2 = − 0.618033 … . {\displaystyle -{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .} The absolute value of this quantity ( 0.618 … {\displaystyle 0.618\ldots } ) corresponds to
11948-431: The same abstract regular polyhedron ; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron. In crystallography , two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry , and
12064-576: The same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes. Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite , and it may be an inspiration for the discovery of the regular Platonic solid form. The true regular dodecahedron can occur as a shape for quasicrystals (such as holmium–magnesium–zinc quasicrystal ) with icosahedral symmetry , which includes true fivefold rotation axes. The name crystal pyrite comes from one of
12180-415: The solid's topology and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In Conway polyhedron notation this is a gyro tetrahedron.) A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron . (The tetartoid shown here
12296-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
12412-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
12528-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
12644-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
12760-419: The two common crystal habits shown by pyrite (the other one being the cube ). In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for
12876-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
12992-543: The value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet , for whom it was named "Binet's formula". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875. By 1910, inventor Mark Barr began using the Greek letter phi ( φ {\displaystyle \varphi } ) as
13108-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
13224-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
13340-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
13456-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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