5609:"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines—made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's
5503:
were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by
Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."".
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three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of
Pythagorean triples, so as to make one square integer equal to the sum of two others."
5502:
that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar
Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions
4136:
is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of
1924:
verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated
5412:
consists of rules for finding
Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at
2988:
5619:)—was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that
2197:, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as
2298:
described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in
2333:
In
Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a
2912:
are concepts that describe when two shapes have similar characteristics. In
Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.
4321:
considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views. Since the 19th century, geometry has been used for solving problems in number theory, for example through the
2379:
In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies
2269:
With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.
3107:
held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of
1735:
used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to
2758:
Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit
2628:
defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two
2950:. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using
1498:, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in
2942:
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the
2246:
Points are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part", or in
1662:
in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in
4056:
and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies,
4191:
Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design. Applications of geometry to architecture include the use of
1896:, there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."
4476:
Until the 19th century, geometry was dominated by the assumption that all geometric constructions were
Euclidean. In the 19th century and later, this was challenged by the development of
5453:
4761:
4708:
4655:
4602:
2527:
1884:
contain "the earliest extant verbal expression of the
Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of
4549:
8007:
1564:
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—
1785:
and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the
5337:
3825:. It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of
1604:, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional
8743:
Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., ... & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American
Mathematical Soc.
3550:, and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as
4407:
8597:
Enumerative
Invariants in Algebraic Geometry and String Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6–11, 2005
4237:
and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.
3756:
to the subject, and illuminated the relations between complex geometry and algebraic geometry. The primary objects of study in complex geometry are
5022:
Ossendrijver, Mathieu (29 January 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph".
9222:
From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry
5245:
2046:
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with
10641:
10361:
1793:
was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.
5598:
2273:
However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is
1374:
10320:
6311:
6078:
A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity
4331:
2917:, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by
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Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner
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1767:, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of
1168:
3649:. From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by
3512:, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the
3116:, and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny.
10356:
5578:
5550:
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2972:
The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.
1449:
concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with
5117:
4492:
and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of
3249:
is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as
2937:
1771:, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose
1513:
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is
10187:. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.
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3569:. This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include
2093:
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of
268:
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in a purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in
11149:
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10392:
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that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann,
3081:
2274:
8799:
6289:
11221:
8708:
8644:
7238:
Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD Year, Berkeley, 1990–1991
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Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art,
1367:
234:
2662:, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a
2311:, a line may be an independent object, distinct from the set of points which lie on it. In differential geometry, a
1432:
10685:
6552:
5345:
3281:. The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as
1960:(b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.
11099:
10221:
4380:
3730:
3164:' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in
1700:(1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or
1638:
11197:
10093:
8969:
4272:
4106:
showed that there is more to geometry than just the metric properties of figures: perspective is the origin of
1161:
1115:
721:
180:
3062:
has received a number of apparently different definitions, which are all equivalent in the most common cases.
2928:, which studies the properties of geometric objects that are preserved by different kinds of transformations.
2368:
is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called
10627:
6699:
6636:
4114:
3184:
3024:
2740:
describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.
2621:
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
2212:
2109:(which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were
1949:
5310:
5153:
3561:
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of
3219:, and the result is an equally true theorem. A similar and closely related form of duality exists between a
1453:, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a
11209:
9570:
4375:
4306:
in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.
4182:
3634:
1360:
8835:(1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42).
8695:. Graduate Texts in Mathematics. Vol. 65. O. García-Prada (3rd ed.). New York: Springer-Verlag.
4713:
4173:. This is still used in art theory today, although the exact list of shapes varies from author to author.
3721:. Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of
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10680:
7274:
6631:
4660:
4412:
3863:
3761:
2079:
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over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and
10695:
9814:
6244:
5179:. Annals of Mathematics; Boston Studies in the Philosophy of Science. Vol. 240. pp. 211–231.
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4033:
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1315:
1305:
1154:
20:
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6436:
Complex analysis : an introduction to the theory of analytic functions of one complex variable
5583:
5555:
5527:
5517:
5341:
4485:
3973:
3938:
3934:
3830:
3153:
3149:
2227:(1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
2208:
1601:
1554:
1503:
123:
8752:
Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.
6549:
The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians
3645:. This led to a parallel development of algebraic geometry, and its algebraic counterpart, called
2097:
by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of
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4439:
4402:
4103:
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proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation
3016:
2981:
2925:
2585:
2573:
2497:
2223:(1777–1855) and others led to a revival of interest in this discipline, and in the 20th century,
2087:
1929:. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of
1643:
549:
229:
86:
8679:
Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.
5545:
4098:
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
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11034:
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11009:
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6315:
6194:
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3906:
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2083:
2005:
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1320:
1295:
1198:
625:
336:
214:
99:
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Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.
8233:
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7012:
6753:
6115:
5498:, "The Arabic Hegemony" pp. 241–242) "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an
3950:
2303:, a line in the plane is often defined as the set of points whose coordinates satisfy a given
11164:
11094:
10971:
10895:
10834:
10819:
10814:
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in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the
3000:
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2401:
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The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
9689:
9572:
7805:
6605:
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
6076:
5840:
4818:
3335:
Euclidean vectors are used for a myriad of applications in physics and engineering, such as
2137:" became something rich and varied, and the natural background for theories as different as
11144:
11024:
11019:
10943:
10844:
10417:
8558:
Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016
6626:
5623:
had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
5031:
4424:
4390:
4216:
4002:
3415:
2803:
2342:, where collinearity and ratios can be studied but not distances; it can be studied as the
2251:
2220:
2186:
1993:
1926:
1893:
1870:(3rd century BC) contains rules for ritual geometric constructions that are similar to the
1847:
1817:
1773:
1764:
1675:
1514:
1415:
1290:
1249:
1218:
1065:
988:
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741:
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158:
72:
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Cristiano Ceccato; Lars Hesselgren; Mark Pauly; Helmut Pottmann, Johannes Wallner (2016).
7807:
Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students
6570:
8:
11159:
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10991:
10890:
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10781:
10771:
10751:
10526:
10521:
10506:
10501:
10412:
10312:
10039:
9053:
5638:
5596:
Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed.,
5570:
5542:
5514:
4481:
4477:
4435:
4323:
4240:
4192:
4152:
4107:
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shapes in the Euclidean space and its more abstract analogues, often using techniques of
3871:
3753:
3678:
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3504:
with an alternative, or relaxing the metric requirement. In the former case, one obtains
3336:
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for nearly two centuries. One example of a mathematical use for higher dimensions is the
3012:
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2951:
2817:
2791:
2752:
2539:
2509:
2385:
2259:
2142:
2134:
2133:. As a consequence of these major changes in the conception of geometry, the concept of "
2071:
2040:
1885:
1866:
1777:, widely considered the most successful and influential textbook of all time, introduced
1749:
1737:
1633:
1589:
1546:
1070:
1014:
927:
781:
761:
686:
576:
447:
437:
300:
175:
170:
153:
128:
116:
68:
63:
44:
10247:
9806:
7684:
5175:
Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum".
5035:
3862:
for manipulating geometrical objects. Important problems historically have included the
2078:(1591–1661). Projective geometry studies properties of shapes which are unchanged under
1961:
1789:
were already known, Euclid arranged them into a single, coherent logical framework. The
11185:
11104:
11044:
10976:
10966:
10905:
10880:
10756:
10713:
10708:
10552:
10511:
10434:
10346:
10338:
9965:
9501:
9057:
8019:
5387:
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5198:
5090:
5055:
4972:
4960:
4861:
4497:
4457:
4385:
4351:
4248:
4196:
4057:
4032:, a recurring concept in convex geometry, was studied by the Greeks as well, including
3875:
3785:
3658:
3619:
3594:
3578:
3570:
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3547:
3501:
3489:
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3241:
3191:, among other fields. This meta-phenomenon can roughly be described as follows: in any
3133:
3055:
3028:
2943:
2869:
2861:
2744:
2690:
2651:
2596:
2543:
2335:
2308:
2247:
2203:
2173:
2166:
2126:
2102:
2009:
1953:
1930:
1861:
1778:
1717:
1713:
1577:
1569:
1558:
1550:
1526:
1460:
1350:
1254:
1029:
756:
596:
224:
148:
138:
109:
94:
9381:
The Everything Cartooning Book: Create Unique And Inspired Cartoons For Fun And Profit
3748:
in the early 1900s. Contemporary treatment of complex geometry began with the work of
1557:) can be developed without introducing any contradiction. The geometry that underlies
11243:
11180:
10900:
10885:
10829:
10776:
10608:
10496:
10454:
10444:
10378:
10200:
10161:
10120:
10097:
10074:
10055:
10044:
10003:
9976:
9931:
9892:
9853:
9828:
9818:
9773:
9734:
9695:
9656:
9617:
9578:
9539:
9512:
9467:
9424:
9385:
9346:
9307:
9268:
9226:
9187:
9148:
9106:
9067:
9021:
8979:
8933:
8894:
8855:
8832:
8820:
8791:
8781:
8714:
8704:
8650:
8640:
8601:
8562:
8515:
8507:
8476:
8434:
8395:
8356:
8317:
8278:
8239:
8200:
8161:
8122:
8083:
8044:
8023:
7974:
7935:
7896:
7857:
7811:
7744:
7705:
7645:
7635:
7603:
7564:
7521:
7479:
7440:
7401:
7362:
7323:
7284:
7242:
7203:
7135:
7096:
7057:
7018:
6979:
6949:
6922:
6882:
6843:
6801:
6759:
6728:
6707:
6670:
6660:
6608:
6584:
6556:
6511:
6501:
6450:
6440:
6398:
6370:
6360:
6281:
6271:
6165:
6121:
6082:
6040:
6032:
6001:
5968:
5961:
5924:
5885:
5846:
5807:
5768:
5729:
5687:
5648:
5477:
5467:
5391:
5315:
5188:
5158:
5059:
5047:
4976:
4945:
4928:
4895:
4868:
4824:
4370:
4359:
4345:
4299:
4244:
4140:
4065:
3818:
3801:
3749:
3662:
3509:
3443:
3318:
3109:
3076:
2996:
2884:
2865:
2849:
2780:
2748:
2639:
2569:
2300:
2255:
2047:
1977:
1934:
1825:
1753:
1732:
1728:
1704:. Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented
1581:
1520:
1455:
1346:
1100:
888:
866:
791:
650:
376:
305:
197:
143:
104:
9303:
Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry
7319:
Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science
5109:
4287:
3472:
Behavior of lines with a common perpendicular in each of the three types of geometry
2122:
2055:
1090:
1019:
816:
726:
11114:
11089:
10961:
10809:
10746:
10459:
10294:
10192:
10171:
10153:
8696:
7131:
The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle
6566:
6224:
5991:
5842:
Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing
5459:
5379:
5229:
5180:
5082:
5039:
4940:
4820:
Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age
4302:
in the form of functions and equations. This played a key role in the emergence of
4234:
4094:
4080:
Geometry has found applications in many fields, some of which are described below.
3942:
3886:
3773:
3757:
3737:
3714:
3709:
3686:
3582:
3330:
3282:
3121:
3051:
3035:
3031:. For instance, the configuration of a screw can be described by five coordinates.
3020:
2853:
2841:
2829:
2776:
2365:
2359:
2347:
2241:
2138:
2130:
2110:
2075:
2059:
1782:
1472:
1464:
1080:
821:
531:
409:
344:
202:
187:
52:
10286:
9011:
7928:
Russell M. Cummings; Scott A. Morton; William H. Mason; David R. McDaniel (2015).
6036:
The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective
4102:
Mathematics and art are related in a variety of ways. For instance, the theory of
2875:
In a different direction, the concepts of length, area and volume are extended by
11054:
10981:
10910:
10703:
10578:
10516:
10449:
10332:
10116:
10090:
Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences
9997:
9925:
9886:
9847:
9767:
9728:
9611:
9340:
9301:
9262:
9220:
9181:
9142:
9100:
9061:
9015:
8973:
8965:
8927:
8888:
8849:
8771:
8690:
8634:
8595:
8556:
8509:
8470:
8428:
8389:
8350:
8311:
8272:
8231:
8194:
8116:
8077:
8038:
7968:
7890:
7851:
7597:
7473:
7356:
7278:
7129:
7090:
6973:
6876:
6837:
6791:
6654:
6495:
6434:
6392:
5956:
5801:
5762:
5719:
5642:
5184:
4889:
4069:
4053:
4006:
3990:
3985:
3964:, which is a geometric representation of a group. Other important topics include
3894:
3882:
3833:, the Kneser-Poulsen conjecture, etc. It shares many methods and principles with
3822:
3741:
3642:
3497:
3493:
3439:
3419:
3286:
3258:
3100:
3008:
2887:, where the measures follow rules similar to those of classical area and volume.
2845:
2825:
2644:
2531:
2381:
2328:
2304:
2290:
2118:
1805:
1597:
1593:
1534:
1468:
1272:
503:
366:
209:
192:
133:
39:
10278:
9648:
7599:
Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra
4125:
for the human figure. These concepts have been used and adapted by artists from
2216:
1075:
1044:
978:
826:
771:
706:
11132:
11059:
10766:
10476:
10235:
8552:
8548:
7678:
7554:
6833:
6650:
6600:
6388:
6248:
6229:
6212:
5611:
5132:
4314:
3965:
3826:
3811:
3717:
studies the nature of geometric structures modelled on, or arising out of, the
3450:(where the object under study is a part of some ambient flat Euclidean space).
3408:
3396:
3161:
3104:
3039:
2992:
2876:
2857:
2724:
2630:
2589:
2521:
2501:
1989:
1985:
1300:
1131:
1039:
983:
948:
856:
766:
736:
696:
601:
10088:
Hayashi, Takao (2003). "Indian Mathematics". In Grattan-Guinness, Ivor (ed.).
9832:
8700:
8546:
5803:
Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century
5463:
5383:
2617:
2070:. The second geometric development of this period was the systematic study of
1105:
716:
11237:
10920:
10852:
10804:
10335:, 3 October 2007 (available for MP3 and MP4 download as well as a text file)
8795:
8718:
8654:
6544:
6350:
6285:
4327:
4310:
4252:
4204:
4200:
4018:
3998:
3834:
3726:
3718:
3697:
3674:
3666:
3654:
3615:
3566:
3551:
3043:
2836:
The concept of length or distance can be generalized, leading to the idea of
2600:
2577:
2505:
2343:
2263:
2224:
1997:
1872:
1659:
1225:
1110:
1095:
1024:
841:
801:
751:
526:
489:
456:
294:
290:
9995:
6674:
6374:
5481:
5043:
4267:
2760:
2720:
2189:, one of the most influential books ever written. Euclid introduced certain
10862:
10857:
10761:
10542:
10365:
10143:
10051:
6515:
6454:
6430:
6072:
5455:
The development of Arabic mathematics : between arithmetic and algebra
5051:
4318:
4148:
4133:
4126:
4010:
3961:
3736:
Complex geometry first appeared as a distinct area of study in the work of
3693:
3540:
3404:
3352:
3310:
3274:
3266:
3220:
3145:
3113:
2837:
2667:
2339:
1981:
1745:
1697:
1667:
1541:, that is, as stand-alone spaces, and has been expanded into the theory of
1495:
1049:
998:
811:
666:
581:
371:
10619:
10302:
10262:
8775:
7552:
6479:
Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.
6359:. Vol. 2 (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc.
6265:
5458:. Boston Studies in the Philosophy of Science. Vol. 156. p. 35.
5286:
5073:
Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early Trigonometry".
2542:
is a three-dimensional object bounded by a closed surface; for example, a
1913:
1839:
11064:
10728:
10651:
10197:
Euclid's Window – The Story of Geometry from Parallel Lines to Hyperspace
7511:
6969:
6939:
6354:
6312:"geodesic – definition of geodesic in English from the Oxford dictionary"
5367:
5305:
4295:
4283:
4144:
3477:
3262:
3117:
3047:
2686:
2682:
2663:
2373:
2369:
2106:
2032:
1945:
1907:
1902:
1724:
established a system of geometry including early versions of sun clocks.
1655:
1592:
that consider only alignment of points but not distance and parallelism,
1446:
1263:
1189:
1085:
958:
776:
711:
639:
611:
586:
8823:(1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.
6583:
Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals."
2764:
2161:
11049:
10928:
10723:
9420:
The Golden Ratio: The Story of PHI, the World's Most Astonishing Number
8593:
5241:
5202:
5094:
4282:
was strongly influenced by geometry. For instance, the introduction of
4061:
4025:
3994:
3916:
3781:
3278:
3224:
3173:
3157:
2028:
1969:
1864:
mathematicians also made many important contributions in geometry. The
1794:
1741:
1712:
within time-velocity space. These geometric procedures anticipated the
1450:
1268:
1258:
943:
922:
912:
902:
861:
806:
701:
691:
591:
442:
4158:
3367:
1654:
The earliest recorded beginnings of geometry can be traced to ancient
1588:), etc.—or on the properties of Euclidean spaces that are disregarded—
10547:
8964:
5603:
5323:
4493:
4222:
4118:
3911:
3881:
Although being a young area of geometry, it has many applications in
3855:
3789:
3431:
3254:
3250:
2812:
2787:
2194:
1957:
1897:
1705:
1671:
1663:
1549:. Later in the 19th century, it appeared that geometries without the
1530:
1459:. Until the 19th century, geometry was almost exclusively devoted to
1438:
1421:
1404:
953:
671:
634:
498:
470:
10111:
Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.).
9052:
8508:
James Carlson; James A. Carlson; Arthur Jaffe; Andrew Wiles (2006).
6033:
Pat Herbst; Taro Fujita; Stefan Halverscheid; Michael Weiss (2017).
5233:
5086:
4271:
The Pythagoreans discovered that the sides of a triangle could have
3637:
that establishes a strong correspondence between algebraic sets and
10953:
10799:
10469:
9811:
Geometry of Quantum States: An Introduction to Quantum Entanglement
4430:
4279:
4170:
3535:
3530:
3427:
3392:
3375:
3348:
3298:
3165:
3092:
3071:
2821:
2772:
2768:
2678:
2659:
2565:
2561:
2555:
2493:
2406:
2316:
2312:
2153:
The following are some of the most important concepts in geometry.
2098:
2063:
2051:
1921:
1813:
1674:, and various crafts. The earliest known texts on geometry are the
1542:
1476:
1240:
1034:
993:
963:
851:
846:
796:
521:
480:
428:
322:
285:
9927:
Number Theory and Geometry: An Introduction to Arithmetic Geometry
9726:
9455:
4772:
The ancient Greeks had some constructions using other instruments.
3446:, which determines how distances are measured near each point) or
3027:
of a physical system, which has a dimension equal to the system's
2338:
without reference to distances or angles; it can be studied as an
2062:(1601–1665). This was a necessary precursor to the development of
10738:
10464:
10157:
9687:
5335:
4501:
4294:
marked a new stage for geometry, since geometric figures such as
4291:
4049:
3611:
3400:
3270:
3192:
2987:
2954:, parabolas and other curves, or mechanical devices, were found.
2914:
2828:
500–200 BC. The Pythagorean theorem is a consequence of the
2655:
2508:, respectively. In algebraic geometry, surfaces are described by
2067:
2013:
1976:
of geometrical quantities, and contributed to the development of
1701:
1235:
968:
681:
475:
419:
219:
8232:
Nihat Ay; Jürgen Jost; Hông Vân Lê; Lorenz Schwachhöfer (2017).
6250:
Analytic Geometry of the Point, Line, Circle, and Conic Sections
5920:
Using History to Teach Mathematics: An International Perspective
2488:
is a two-dimensional object, such as a sphere or paraboloid. In
10246:
9098:
8157:
Mathematics of Bioinformatics: Theory, Methods and Applications
7199:
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
6267:
Handbook of incidence geometry : buildings and foundations
4792:
4230:
4162:
4045:
4041:
3806:
3626:. Algebraic geometry became an autonomous subfield of geometry
3314:
3096:
2737:
2729:
2714:
2706:
2625:
2295:
2182:
1829:
917:
907:
786:
731:
606:
569:
557:
512:
465:
383:
48:
9845:
7740:
Constructivism and Science: Essays in Recent German Philosophy
7053:
Mathematics for Elementary School Teachers: A Process Approach
4143:, or tessellations, have been used in art throughout history.
3780:, and these spaces find uses in string theory. In particular,
2751:, the length of a line segment can often be calculated by the
10573:
10351:
9649:
Marian Moffett; Michael W. Fazio; Lawrence Wodehouse (2003).
8079:
Differential Forms with Applications to the Physical Sciences
4496:
in the 17th century, all the way back to the implicit use of
4489:
4161:
advanced the theory that all images can be built up from the
4037:
3692:
Algebraic geometry has applications in many areas, including
3677:
whose solution uses scheme theory and its extensions such as
3599:
3485:
3356:
3294:
2947:
2918:
2612:
2564:
is a generalization of the concepts of curve and surface. In
2410:
A sphere is a surface that can be defined parametrically (by
2207:
geometry. At the start of the 19th century, the discovery of
2190:
2177:
1973:
1965:
1846:. Illustration at the beginning of a medieval translation of
1721:
1609:
1605:
1488:
1480:
973:
897:
831:
676:
280:
275:
9535:
Integrating the Arts Across the Elementary School Curriculum
7014:
Euclidean and Transformational Geometry: A Deductive Inquiry
3467:
2526:
10370:
5881:
Mathematical Thought From Ancient to Modern Times: Volume 3
4226:
4166:
2733:
2710:
1809:
1647:
564:
414:
10357:
Dynamic Geometry Sketches (with some Student Explorations)
10266:
8309:
8153:
5718:
4793:"Geometry - Formulas, Examples | Plane and Solid Geometry"
4028:
gave the first known precise definition of convexity. The
3610:
Algebraic geometry is fundamentally the study by means of
2496:, surfaces are described by two-dimensional 'patches' (or
1752:, though the statement of the theorem has a long history.
10271:
9996:
Gary Cornell; Joseph H. Silverman; Glenn Stevens (2013).
9765:
9498:
9449:
9342:
Geometry of Design: Studies in Proportion and Composition
8958:
7736:
7439:. Springer Science & Business Media. pp. 158ff.
6191:
Handbook of incidence geometry: buildings and foundations
5218:"The Pentagram and the Discovery of an Irrational Number"
5002:
3414:
In particular, differential geometry is of importance to
2931:
8114:
7888:
7743:. Springer Science & Business Media. pp. 127–.
7676:
7517:
Strings, Conformal Fields, and Topology: An Introduction
7475:
Beyond Geometry: Classic Papers from Riemann to Einstein
6120:. Springer Science & Business Media. pp. 254–.
5838:
3500:, non-Euclidean geometry arises by either replacing the
3019:). Furthermore, mathematicians and physicists have used
2595:
Manifolds are used extensively in physics, including in
10352:
Interactive geometry reference with hundreds of applets
10257:. Vol. 11 (11th ed.). 1911. pp. 675–736.
9884:
7358:
Symmetry as a Developmental Principle in Nature and Art
7234:
6000:. Springer Science & Business Media. pp. 29–.
5434:
5432:
4225:, especially as it relates to mapping the positions of
4147:
makes frequent use of tessellations, as did the art of
3788:
predicts that the extra 6 dimensions of 10 dimensional
3744:. Work in the spirit of Riemann was carried out by the
3546:
Topology is the field concerned with the properties of
2643:
of the angle. The size of an angle is formalized as an
2250:. In modern mathematics, they are generally defined as
1952:
contributed to the development of geometry, especially
1821:
10182:
9564:
7810:. Springer Science & Business Media. pp. 6–.
7701:
Bundles of Topological Vector Spaces and Their Duality
7520:. Springer Science & Business Media. p. 151.
7432:
7202:. Springer Science & Business Media. p. 367.
6500:. Vol. 2. Englewood Cliffs, N.J.: Prentice-Hall.
6081:. Springer Science & Business Media. pp. 6–.
4203:
in constructing domes and similar objects, the use of
3399:
to study problems in geometry. It has applications in
3015:
or surface), and 3 (our ambient world conceived of as
9989:
9923:
9299:
8118:
Applications of Differential Geometry to Econometrics
6755:
History and Measurement of the Base and Derived Units
5728:. Springer Science & Business Media. p. 43.
5686:. Springer Science & Business Media. p. 95.
4716:
4663:
4610:
4557:
4519:
2700:
1906:(499) includes the computation of areas and volumes.
1525:("remarkable theorem") that asserts roughly that the
9179:
9046:
8594:
Marcos Marino; Michael Thaddeus; Ravi Vakil (2008).
8352:
Transformation Geometry: An Introduction to Symmetry
7882:
7092:
Transformation Geometry: An Introduction to Symmetry
6758:. Springer International Publishing. pp. 101–.
6162:
Euclid's Elements – All thirteen books in one volume
5832:
5429:
4341:
3492:. As Euclidean geometry lies at the intersection of
2637:
of the angle, sharing a common endpoint, called the
9917:
9804:
9492:
9293:
8890:
The Cube – A Window to Convex and Discrete Geometry
8587:
8501:
8108:
7853:
Architecture and Geometry in the Age of the Baroque
7845:
7843:
7841:
7839:
7166:
6659:. Mark E. Saul. Boston: Birkhäuser. pp. 1–20.
6193:. North-Holland. pp. 1015–1031. Archived from
5963:
The King of Infinite Space: Euclid and His Elements
4255:makes use of several variants of geometry, as does
2039:, that by the 19th century led to the discovery of
1941:triangles with rational sides and rational areas).
1561:is a famous application of non-Euclidean geometry.
10043:
9964:
9878:
9839:
9500:
8270:
8006:Schmidt, W.; Houang, R.; Cogan, Leland S. (2002).
8005:
7921:
7683:. McGraw-Hill book Company, Incorporated. p.
7546:
7426:
6263:
5960:
4929:"Methods and traditions of Babylonian mathematics"
4860:
4755:
4702:
4649:
4596:
4543:
3673:is a famous example of a long-standing problem of
3520:that have also been called non-Euclidean geometry.
3038:, the concept of dimension has been extended from
2856:. Other important examples of metrics include the
9730:Recent Developments in Pseudo-Riemannian Geometry
9720:
9642:
8464:
8462:
7634:
7393:
6164:, Based on Heath's translation, Green Lion Press
5986:
5984:
5679:
5568:
5540:
5512:
5336:O'Connor, J.J.; Robertson, E.F. (February 1996).
3960:Geometric group theory often revolves around the
3784:of strings are modelled by Riemann surfaces, and
3183:A different type of symmetry is the principle of
2012:continued by later European geometers, including
1820:, and gave remarkably accurate approximations of
11235:
9531:
8426:
8420:
8348:
8147:
8040:Multivariable Calculus and Differential Geometry
8008:"A Coherent Curriculum: The Case of Mathematics"
7999:
7849:
7836:
7088:
7010:
6751:
6026:
5990:
4854:
4852:
4816:
3810:Discrete geometry includes the study of various
3160:that Klein's idea to 'define a geometry via its
1748:, which is credited with the first proof of the
1708:procedures for computing Jupiter's position and
9962:
9681:
9609:
9463:M. C. Escher's Legacy: A Centennial Celebration
9256:
9254:
9173:
8925:
8540:
8075:
8036:
7049:
6790:
5955:
5884:. US: Oxford University Press. pp. 1010–.
5416:
5408:, p. 198): "The arithmetic content of the
5215:
4887:
3933:uses large-scale geometric techniques to study
2879:, which studies methods of assigning a size or
2315:is a generalization of the notion of a line to
2113:(1826–1866), working primarily with tools from
1828:bearing his name and obtained formulas for the
1808:used the method of exhaustion to calculate the
1510:, and remained unsolved for several centuries.
9377:
9260:
9218:
8886:
8468:
8459:
7730:
7315:
7127:
7017:. Jones & Bartlett Learning. p. 255.
6832:
5981:
5916:
5174:
4442:, to understand the concept of four dimensions
4262:
1596:that omits the concept of angle and distance,
1529:of a surface is independent from any specific
10635:
10386:
9956:
9759:
9603:
9525:
9338:
9010:
8844:
8838:
8264:
7121:
7004:
6747:
6745:
5949:
5712:
5637:
5599:Encyclopedia of the History of Arabic Science
5311:An Introduction to the History of Mathematics
5154:An Introduction to the History of Mathematics
4965:"Chap. IV Egyptian Mathematics and Astronomy"
4849:
3821:is a subject that has close connections with
3007:Traditional geometry allowed dimensions 1 (a
2924:Congruence and similarity are generalized in
2771:, area and volume can be defined in terms of
2185:took an abstract approach to geometry in his
1368:
1162:
9688:Robin M. Green; Robin Michael Green (1985).
9507:. Sterling Publishing Company, Inc. p.
9251:
8475:. Translated by Judith D. Sally. CRC Press.
8303:
8192:
8069:
8030:
7966:
7960:
7387:
7228:
6497:Differential geometry of curves and surfaces
6157:
6155:
6153:
6151:
6149:
6071:
5923:. Cambridge University Press. pp. 45–.
5877:
5633:
5631:
5177:Classics in the History of Greek Mathematics
5021:
4513:Pythagorean triples are triples of integers
2890:
2844:measures the distance between points in the
2372:and those in 3-dimensional space are called
1612:provided by Euclidean geometry; presently a
1537:. This implies that surfaces can be studied
10649:
10149:A Participatory Approach to Modern Geometry
9967:Pythagoras' Revenge: A Mathematical Mystery
9727:Dmitriĭ Vladimirovich Alekseevskiĭ (2008).
9416:
9371:
9136:
9134:
9004:
8919:
8342:
8225:
7934:. Cambridge University Press. p. 449.
7670:
7644:. Cambridge University Press. p. 272.
7510:
7471:
7354:
7273:
7267:
7195:
7043:
6968:
6113:
5799:
5760:
5673:
4504:and to navigate the oceans since antiquity.
3614:methods of some geometrical shapes, called
3442:whose geometric structure is governed by a
2957:
2534:, a ball is the volume bounded by a sphere.
1518:
10642:
10628:
10393:
10379:
10341:at the Stanford Encyclopedia of Philosophy
9410:
9099:Xianfeng David Gu; Shing-Tung Yau (2008).
9092:
8692:Differential analysis on complex manifolds
8632:
8186:
7628:
7595:
7563:. American Mathematical Soc. p. 132.
7400:. Cambridge University Press. p. 34.
7309:
7280:Why Beauty Is Truth: A History of Symmetry
7241:. American Mathematical Soc. p. 111.
7082:
6874:
6826:
6784:
6742:
6725:Riemannian Geometry and Geometric Analysis
6539:
6537:
6345:
6343:
6341:
6339:
6337:
6243:
6065:
5683:The Historical Development of the Calculus
5534:
5360:
4959:
4408:List of important publications in geometry
3438:(meaning that the spaces it considers are
2307:, but in a more abstract setting, such as
1375:
1361:
1169:
1155:
38:
10002:. Springer Science & Business Media.
9846:Harley Flanders; Justin J. Price (2014).
9694:. Cambridge University Press. p. 1.
9655:. Laurence King Publishing. p. 371.
9332:
9306:. Springer Science & Business Media.
9212:
9140:
9066:. Springer Science & Business Media.
9020:. Springer Science & Business Media.
8932:. Springer Science & Business Media.
8880:
8854:. Springer Science & Business Media.
8770:
8433:. Springer Science & Business Media.
8387:
8355:. Springer Science & Business Media.
8277:. Springer Science & Business Media.
7797:
7691:
7589:
7465:
7189:
7095:. Springer Science & Business Media.
6800:. American Mathematical Soc. p. 11.
6706:, 7th ed., Brooks Cole Cengage Learning.
6228:
6181:
6146:
6107:
5871:
5806:. Springer Science & Business Media.
5754:
5628:
5270:, "The Age of Plato and Aristotle" p. 92)
4944:
4309:Another important area of application is
4024:Convex geometry dates back to antiquity.
3900:
3840:
3792:may be modelled by Calabi–Yau manifolds.
3517:
3453:
2008:, were part of a line of research on the
1984:(1048–1131) found geometric solutions to
1925:his famous theorem on the diagonals of a
9131:
8381:
7803:
7697:
7504:
7348:
7160:
6868:
6649:
6489:
6487:
6485:
6387:
6189:. In Buekenhout, F.; Kantor, W. (eds.).
5910:
5793:
5725:The Geometrical Work of Girard Desargues
5398:
5222:The Two-Year College Mathematics Journal
4881:
4863:Geometry: the language of space and form
4266:
4093:
3910:
3805:
3598:
3534:
3466:
3366:
3362:
3230:
3075:
2986:
2980:For broader coverage of this topic, see
2811:
2616:
2525:
2405:
2160:
1933:), as well as a complete description of
1838:
1637:
1506:, a problem that was stated in terms of
10110:
10087:
9999:Modular Forms and Fermat's Last Theorem
9574:Advances in Architectural Geometry 2010
9144:Geometric Group Theory: An Introduction
9063:Computational Geometry: An Introduction
8636:Complex geometry : an introduction
7602:. World Scientific Publishing Company.
6962:
6921:, American Mathematical Society, 2001,
6797:Geometry of Lengths, Areas, and Volumes
6693:
6624:
6543:
6534:
6439:(3rd ed.). New York: McGraw-Hill.
6429:
6349:
6334:
5579:MacTutor History of Mathematics Archive
5551:MacTutor History of Mathematics Archive
5523:MacTutor History of Mathematics Archive
5439:
5423:
5107:
5072:
4926:
4810:
4398:List of formulas in elementary geometry
3689:, is a question in algebraic geometry.
2797:
2786:Other geometrical measures include the
2230:
1877:
1650:practicing geometry in the 15th century
11236:
9971:. Princeton University Press. p.
8391:A Concise Course in Algebraic Topology
8160:. John Wiley & Sons. p. 106.
6039:. Taylor & Francis. pp. 20–.
5451:
4332:Wiles's proof of Fermat's Last Theorem
3488:closely related to those that specify
3434:. Differential geometry can either be
3378:to study problems involving curvature.
2938:Compass and straightedge constructions
2932:Compass and straightedge constructions
2588:is a space where each neighborhood is
2066:and a precise quantitative science of
1720:, by 14 centuries. South of Egypt the
269:Straightedge and compass constructions
10623:
10374:
10315:Pegs and Ropes Geometry at Stonehenge
10068:
10038:
10016:from the original on 30 December 2019
9944:from the original on 27 December 2019
9866:from the original on 24 December 2019
9786:from the original on 24 December 2019
9747:from the original on 28 December 2019
9708:from the original on 21 December 2019
9669:from the original on 27 December 2019
9630:from the original on 24 December 2019
9591:from the original on 25 December 2019
9480:from the original on 22 December 2019
9437:from the original on 30 December 2019
9398:from the original on 27 December 2019
9359:from the original on 31 December 2019
9320:from the original on 29 December 2019
9281:from the original on 1 September 2021
9239:from the original on 28 December 2019
9200:from the original on 24 December 2019
9161:from the original on 29 December 2019
9119:from the original on 24 December 2019
9080:from the original on 28 December 2019
9034:from the original on 28 December 2019
9017:Classical Topics in Discrete Geometry
8992:from the original on 27 December 2019
8946:from the original on 24 December 2019
8907:from the original on 23 December 2019
8868:from the original on 27 December 2019
8688:
8614:from the original on 27 December 2019
8575:from the original on 27 December 2019
8489:from the original on 25 December 2019
8447:from the original on 27 December 2019
8408:from the original on 23 December 2019
8369:from the original on 22 December 2019
8330:from the original on 26 December 2019
8310:Charles Nash; Siddhartha Sen (1988).
8291:from the original on 28 December 2019
8252:from the original on 24 December 2019
8213:from the original on 26 December 2019
8174:from the original on 27 December 2019
8154:Matthew He; Sergey Petoukhov (2011).
8135:from the original on 1 September 2021
8096:from the original on 1 September 2021
8057:from the original on 27 December 2019
7948:from the original on 1 September 2021
7909:from the original on 31 December 2019
7870:from the original on 25 December 2019
7824:from the original on 25 December 2019
7785:from the original on 27 December 2019
7757:from the original on 1 September 2021
7718:from the original on 27 December 2019
7658:from the original on 26 December 2019
7616:from the original on 26 December 2019
7577:from the original on 29 December 2019
7534:from the original on 24 December 2019
7492:from the original on 1 September 2021
7453:from the original on 24 December 2019
7414:from the original on 27 December 2019
7336:from the original on 29 December 2019
7297:from the original on 25 December 2019
7255:from the original on 28 December 2019
7216:from the original on 24 December 2019
7177:from the original on 27 December 2019
7148:from the original on 27 December 2019
7070:from the original on 28 December 2019
7031:from the original on 25 December 2019
6992:from the original on 27 December 2019
6895:from the original on 25 December 2019
6856:from the original on 24 December 2019
6814:from the original on 31 December 2019
6772:from the original on 30 December 2019
6493:
6482:
6210:
6134:from the original on 1 September 2021
6095:from the original on 1 September 2021
6053:from the original on 1 September 2021
6014:from the original on 1 September 2021
5937:from the original on 1 September 2021
5898:from the original on 1 September 2021
5859:from the original on 28 December 2019
5781:from the original on 28 December 2019
5742:from the original on 27 December 2019
5700:from the original on 29 December 2019
5661:from the original on 26 December 2019
5546:"Al-Sabi Thabit ibn Qurra al-Harrani"
5495:
5405:
5366:
5293:
5280:
5267:
5248:from the original on 9 September 2022
5142:, "Ionia and the Pythagoreans" p. 43)
5139:
5009:
4908:from the original on 1 September 2021
4858:
4837:from the original on 20 February 2021
4446:List of interactive geometry software
3671:Wiles' proof of Fermat's Last Theorem
3588:
3261:, and many technical fields, such as
3235:
11204:
10591:List of differential geometry topics
9766:Shing-Tung Yau; Steve Nadis (2010).
9552:from the original on 7 December 2019
9499:Robert Capitolo; Ken Schwab (2004).
8313:Topology and Geometry for Physicists
7987:from the original on 7 December 2019
7737:Robert E. Butts; J.R. Brown (2012).
7109:from the original on 7 December 2019
6938:
6722:
6494:Carmo, Manfredo Perdigão do (1976).
5820:from the original on 7 December 2019
5370:(1999). "Greek and Vedic Geometry".
4867:. Infobase Publishing. p. xiv.
4756:{\displaystyle 12^{2}+35^{2}=37^{2}}
3795:
3746:Italian school of algebraic geometry
3513:
3484:consists of two geometries based on
3324:
2101:as the central consideration in the
11216:
10331:, lecture by Robin Wilson given at
10113:The Blackwell Companion to Hinduism
9905:from the original on 1 January 2020
8975:Discrete and Computational Geometry
8780:. Oxford: Oxford University Press.
8115:Paul Marriott; Mark Salmon (2000).
7889:P. Vanícek; E.J. Krakiwsky (2015).
7781:. Moses King. 1886. pp. 181–.
7375:from the original on 1 January 2020
5764:Introduction to Projective Geometry
5075:The Journal of Egyptian Archaeology
4989:from the original on 14 August 2020
4703:{\displaystyle 8^{2}+15^{2}=17^{2}}
4290:and the concurrent developments of
4113:Artists have long used concepts of
3703:
2546:is the volume bounded by a sphere.
1624:on which some geometry is defined.
1411: 'land measurement'; from
13:
10135:
10092:. Vol. 1. Baltimore, MD: The
9885:Jon Rogawski; Colin Adams (2015).
7931:Applied Computational Aerodynamics
7235:Bill Jacob; Tsit-Yuen Lam (1994).
6217:Notre Dame Journal of Formal Logic
5621:Pseudo-Tusi's Exposition of Euclid
4121:developed a complicated theory of
3979:
3949:, which included the proof of the
2701:Measures: length, area, and volume
1920:in 628. Chapter 12, containing 66
1445: 'a measure') is a branch of
14:
11255:
10215:
9733:. European Mathematical Society.
9345:. Princeton Architectural Press.
8802:from the original on 1 March 2023
8725:from the original on 1 March 2023
8661:from the original on 1 March 2023
7677:Linnaeus Wayland Dowling (1917).
7433:B. Rosenfeld; Bill Wiebe (2013).
7397:Introduction to Symmetry Analysis
7322:. World Scientific. p. 144.
7056:. Cengage Learning. p. 614.
6975:An Introduction to Measure Theory
6681:from the original on 1 March 2023
6522:from the original on 1 March 2023
6461:from the original on 1 March 2023
6411:from the original on 1 March 2023
6394:From Affine to Euclidean Geometry
6292:from the original on 1 March 2023
6211:Clark, Bowman L. (January 1985).
5839:Eduardo Bayro-Corrochano (2018).
4650:{\displaystyle 3^{2}+4^{2}=5^{2}}
4597:{\displaystyle a^{2}+b^{2}=c^{2}}
4151:. Escher's work also made use of
3060:dimension of an algebraic variety
2767:of various geometric objects. In
235:Noncommutative algebraic geometry
11215:
11203:
11192:
11191:
11179:
10604:
10603:
10324:– Geometric Areas of Mathematics
10073:. New York: Wiley-Interscience.
9798:
9538:. Cengage Learning. p. 55.
9423:. Crown/Archetype. p. 166.
9102:Computational Conformal Geometry
8826:
8814:
8764:
8755:
8746:
8737:
8682:
8673:
8626:
8528:from the original on 30 May 2016
7553:Mladen Bestvina; Michah Sageev;
6878:A Primer of Lebesgue Integration
6264:Francis Buekenhout, ed. (1995).
5326:, has been more widely used...."
5296:, "Euclid of Alexandria" p. 104)
5283:, "Euclid of Alexandria" p. 119)
5120:from the original on 5 June 2011
4344:
3752:, who introduced the concept of
3725:, and has found applications to
3459:This section is an excerpt from
2148:
1888:, which are particular cases of
1463:, which includes the notions of
1428: 'earth, land' and
1345:
11100:Computational complexity theory
10183:Nikolai I. Lobachevsky (2010).
10152:. World Scientific Publishing.
9849:Calculus with Analytic Geometry
9652:A World History of Architecture
9180:John Morgan; Gang Tian (2014).
8394:. University of Chicago Press.
7856:. University of Chicago Press.
7769:
6948:. University of Chicago Press.
6932:
6907:
6716:
6704:Calculus: Early Transcendentals
6643:
6618:
6594:
6577:
6473:
6423:
6381:
6304:
6257:
6237:
6204:
6175:
6117:Geometry: Our Cultural Heritage
5590:
5562:
5506:
5488:
5445:
5329:
5299:
5273:
5260:
5209:
5168:
5145:
5108:Slayman, Andrew (27 May 1998).
5101:
5066:
4969:The Exact Sciences in Antiquity
4766:
4507:
4381:Category:Differential geometers
4176:
4075:
3565:, in which transformations are
3144:an analogous role is played by
2275:Whitehead's point-free geometry
2086:, especially as they relate to
1818:summation of an infinite series
10094:Johns Hopkins University Press
9924:Álvaro Lozano-Robledo (2019).
9300:Jürgen Richter-Gebert (2011).
8978:. Princeton University Press.
8893:. Cambridge University Press.
8199:. Princeton University Press.
8121:. Cambridge University Press.
5015:
4953:
4920:
4785:
4538:
4520:
4470:
4129:to modern comic book artists.
4017:and important applications in
4005:. It has close connections to
3140:and rigid motions, whereas in
1608:of the physical world and its
628:- / other-dimensional
1:
10054: ed.). New York: Wiley.
9930:. American Mathematical Soc.
9384:. Adams Media. pp. 82–.
9225:. American Mathematical Soc.
9186:. American Mathematical Soc.
9183:The Geometrization Conjecture
8851:Lectures on Discrete Geometry
8514:. American Mathematical Soc.
8511:The Millennium Prize Problems
8472:History of Algebraic Geometry
7173:. The Academy. pp. 62–.
6978:. American Mathematical Soc.
4891:Geometry and Its Applications
4779:
3937:. It is closely connected to
3627:
2852:measures the distance in the
2725:Volume § Volume formulas
2213:Nikolai Ivanovich Lobachevsky
2017:
1950:mathematics in medieval Islam
1851:
1798:
1757:
1686:
10400:
10050:(Second edition, revised by
8929:Convex and Discrete Geometry
8196:General Theory of Relativity
7167:Charles Jasper Joly (1895).
5644:History of Analytic Geometry
5372:Journal of Indian Philosophy
5185:10.1007/978-1-4020-2640-9_11
5114:Archaeology Magazine Archive
4946:10.1016/0315-0860(81)90069-0
4376:Category:Algebraic geometers
4183:Mathematics and architecture
3046:, for example) and positive
2975:
2721:Area § List of formulas
2693:can be calculated using the
2549:
2165:An illustration of Euclid's
2129:and the geometric theory of
1992:(Alhazen), Omar Khayyam and
1910:wrote his astronomical work
7:
10199:(UK ed.). Allen Lane.
9616:. Bentley Institute Press.
9264:Handbook of Convex Geometry
8633:Huybrechts, Daniel (2005).
8271:Martin D. Crossley (2011).
7283:. Basic Books. p. 14.
7134:. Oxford University Press.
6919:A Course in Metric Geometry
6727:. Berlin: Springer-Verlag.
6632:Encyclopedia of Mathematics
5997:Geometry: Euclid and Beyond
4823:. Birkhäuser. pp. 1–.
4413:Lists of mathematics topics
4337:
4263:Other fields of mathematics
4207:, and the use of symmetry.
3864:travelling salesman problem
3762:complex algebraic varieties
3524:
3128:, determines what geometry
3065:
2654:, angles are used to study
2395:
1914:
1727:In the 7th century BC, the
1491:, as fundamental concepts.
10:
11260:
11150:Films about mathematicians
10071:The History of Mathematics
10031:
9815:Cambridge University Press
7394:Brian J. Cantwell (2002).
5680:C. H. Edwards Jr. (2012).
4357:
4257:quantum information theory
4214:
4210:
4180:
4087:
3983:
3904:
3844:
3799:
3770:holomorphic vector bundles
3766:complex analytic varieties
3707:
3592:
3528:
3458:
3381:
3328:
3239:
3069:
2979:
2961:
2935:
2894:
2801:
2718:
2704:
2610:
2553:
2519:
2399:
2357:
2326:
2288:
2239:
2171:
2037:Giovanni Girolamo Saccheri
1769:incommensurable magnitudes
1631:
1627:
1431:
1414:
1397:
1285:Relationship with sciences
18:
11173:
11123:
11080:
10990:
10952:
10919:
10871:
10843:
10790:
10737:
10719:Philosophy of mathematics
10694:
10659:
10599:
10566:
10535:
10485:
10433:
10408:
10273:Unusual Geometry Problems
10236:Resources in your library
9577:. Birkhäuser. p. 6.
9532:Phyllis Gelineau (2011).
9466:. Springer. p. 107.
8701:10.1007/978-0-387-73892-5
8689:Wells, R. O. Jr. (2008).
8427:Robin Hartshorne (2013).
8349:George E. Martin (1996).
8238:. Springer. p. 185.
7850:George L. Hersey (2001).
7704:. Springer. p. 252.
7089:George E. Martin (2012).
7011:Shlomo Libeskind (2008).
6752:Steven A. Treese (2018).
5464:10.1007/978-94-017-3274-1
5322:p. 141: "No work, except
4817:Vincenzo De Risi (2015).
4298:could now be represented
3974:right angled Artin groups
3947:Geometrization conjecture
3935:finitely generated groups
3723:several complex variables
3683:Millennium Prize problems
3635:Hilbert's Nullstellensatz
3150:geometric transformations
3042:, to infinite dimension (
2891:Congruence and similarity
2606:
2515:
2353:
2322:
2235:
2156:
1519:
21:Geometry (disambiguation)
11155:Recreational mathematics
10329:"4000 Years of Geometry"
10046:A History of Mathematics
9963:Arturo Sangalli (2009).
9610:Helmut Pottmann (2007).
8926:Peter M. Gruber (2007).
8076:Harley Flanders (2012).
8037:Gerard Walschap (2015).
7895:. Elsevier. p. 23.
7050:Mark A. Freitag (2013).
6230:10.1305/ndjfl/1093870761
6213:"Individuals and Points"
5602:, Vol. 2, pp. 447–494 ,
5584:University of St Andrews
5556:University of St Andrews
5528:University of St Andrews
5342:University of St Andrews
5216:James R. Choike (1980).
4888:Walter A. Meyer (2006).
4486:non-Euclidean geometries
4463:
3970:Gromov-hyperbolic groups
3955:Millennium Prize Problem
3939:low-dimensional topology
3915:The Cayley graph of the
3633:, with a theorem called
3624:multivariate polynomials
3618:, and defined as common
3132:. Symmetry in classical
2958:Rotation and orientation
2572:where every point has a
2500:) that are assembled by
2284:
2209:non-Euclidean geometries
2095:non-Euclidean geometries
1555:non-Euclidean geometries
124:Non-Archimedean geometry
11040:Mathematical statistics
11030:Mathematical psychology
11000:Engineering mathematics
10934:Algebraic number theory
10586:List of geometry topics
10254:Encyclopædia Britannica
9378:Brad J. Guigar (2004).
9261:Gerard Meurant (2014).
9219:Daniel T. Wise (2012).
9105:. International Press.
8887:Chuanming Zong (2006).
8469:Jean Dieudonné (1985).
8316:. Elsevier. p. 1.
8082:. Courier Corporation.
7478:. Courier Corporation.
7316:Stakhov Alexey (2009).
7128:Mark Blacklock (2018).
6625:Sidorov, L.A. (2001) .
6603:; Nadis, Steve (2010).
6270:. Amsterdam: Elsevier.
5917:Victor J. Katz (2000).
5845:. Springer. p. 4.
5767:. Courier Corporation.
5647:. Courier Corporation.
5606:, London and New York:
5452:Rāshid, Rushdī (1994).
5384:10.1023/A:1004364417713
5338:"A history of calculus"
5110:"Neolithic Skywatchers"
5044:10.1126/science.aad8085
4927:Friberg, Jöran (1981).
4544:{\displaystyle (a,b,c)}
4440:three-dimensional space
4403:List of geometry topics
3563:transformation geometry
3017:three-dimensional space
2982:Dimension (mathematics)
2926:transformation geometry
2816:Visual checking of the
2586:differentiable manifold
2580:to Euclidean space. In
2384:, which are defined as
1968:) (836–901) dealt with
1844:Woman teaching geometry
1694:Babylonian clay tablets
230:Noncommutative geometry
11186:Mathematics portal
11035:Mathematical sociology
11015:Mathematical economics
11010:Mathematical chemistry
10939:Analytic number theory
10820:Differential equations
10322:The Mathematical Atlas
9613:Architectural geometry
9339:Kimberly Elam (2001).
7970:Geometry of Navigation
7560:Geometric Group Theory
7436:Geometry of Lie Groups
6316:OxfordDictionaries.com
6184:"Pointless Geometries"
5722:; Jeremy Gray (2012).
5625:
4757:
4704:
4651:
4598:
4545:
4304:infinitesimal calculus
4276:
4187:Architectural geometry
4099:
4083:
3931:Geometric group theory
3927:
3907:Geometric group theory
3901:Geometric group theory
3868:minimum spanning trees
3852:Computational geometry
3847:Computational geometry
3841:Computational geometry
3815:
3651:Alexander Grothendieck
3607:
3543:
3482:non-Euclidean geometry
3474:
3461:Non-Euclidean geometry
3454:Non-Euclidean geometry
3379:
3170:geometric group theory
3088:
3004:
2968:Orientation (geometry)
2833:
2622:
2535:
2481:
2279:Alfred North Whitehead
2169:
2117:, and introducing the
2031:(1288–1344), Alfonso,
2006:Saccheri quadrilateral
1972:operations applied to
1858:
1834:surfaces of revolution
1824:. He also studied the
1651:
1622:mathematical structure
1586:combinatorial geometry
1574:computational geometry
198:Discrete/Combinatorial
11165:Mathematics education
11095:Theory of computation
10815:Hypercomplex analysis
10558:Differential geometry
10347:The Geometry Junkyard
10069:Cooke, Roger (2005).
9458:Doris Schattschneider
8193:P.A.M. Dirac (2016).
8012:The American Educator
7967:Roy Williams (1998).
7892:Geodesy: The Concepts
6723:Jost, Jürgen (2002).
5878:Morris Kline (1990).
5607:
4758:
4705:
4652:
4599:
4546:
4270:
4247:geometry are used in
4097:
4030:isoperimetric problem
3914:
3891:computer-aided design
3809:
3657:, which allows using
3602:
3575:differential topology
3538:
3516:, which give rise to
3470:
3426:postulation that the
3389:Differential geometry
3384:Differential geometry
3372:Differential geometry
3370:
3363:Differential geometry
3231:Contemporary geometry
3079:
3001:topological dimension
2990:
2901:Similarity (geometry)
2897:Congruence (geometry)
2815:
2808:Measure (mathematics)
2681:, the angles between
2675:differential geometry
2620:
2582:differential geometry
2529:
2490:differential geometry
2409:
2402:Surface (mathematics)
2164:
2115:mathematical analysis
2002:Lambert quadrilateral
1915:Brāhmasphuṭasiddhānta
1890:Diophantine equations
1842:
1641:
1566:differential geometry
1508:elementary arithmetic
1504:Fermat's Last Theorem
181:Discrete differential
16:Branch of mathematics
11145:Informal mathematics
11025:Mathematical physics
11020:Mathematical finance
11005:Mathematical biology
10944:Diophantine geometry
10339:Finitism in Geometry
10313:Nature Precedings –
10119:. pp. 360–375.
10096:. pp. 118–130.
9852:. Elsevier Science.
9805:Bengtsson, Ingemar;
9417:Mario Livio (2008).
9267:. Elsevier Science.
8639:. Berlin: Springer.
8235:Information Geometry
7472:Peter Pesic (2007).
7361:. World Scientific.
7355:Werner Hahn (1998).
7196:Roger Temam (2013).
6114:Audun Holme (2010).
5800:Jeremy Gray (2011).
5761:C. R. Wylie (2011).
5571:Robertson, Edmund F.
5543:Robertson, Edmund F.
5515:Robertson, Edmund F.
4933:Historia Mathematica
4859:Tabak, John (2014).
4714:
4661:
4608:
4555:
4517:
4434:, a book written by
4425:Descriptive geometry
4391:Category:Topologists
4217:Mathematical physics
4003:discrete mathematics
3778:Calabi–Yau manifolds
3605:Calabi–Yau threefold
3539:A thickening of the
3518:kinematic geometries
3416:mathematical physics
2840:. For instance, the
2804:Metric (mathematics)
2798:Metrics and measures
2592:to Euclidean space.
2510:polynomial equations
2346:using techniques of
2231:Spaces and subspaces
2221:Carl Friedrich Gauss
2088:artistic perspective
1994:Nasir al-Din al-Tusi
1964:(known as Thebit in
1927:cyclic quadrilateral
1894:Bakhshali manuscript
1880:, p. 363), the
1765:method of exhaustion
1515:Carl Friedrich Gauss
1250:Discrete mathematics
19:For other uses, see
11160:Mathematics and art
11070:Operations research
10825:Functional analysis
10306:– Advanced Geometry
9691:Spherical Astronomy
9054:Franco P. Preparata
7680:Projective Geometry
7596:W-H. Steeb (1996).
6875:H. S. Bear (2002).
5569:O'Connor, John J.;
5541:O'Connor, John J.;
5513:O'Connor, John J.;
5442:, pp. 121–122)
5036:2016Sci...351..482O
4551:with the property:
4478:hyperbolic geometry
4436:Edwin Abbott Abbott
4330:, which is used in
4326:or, more recently,
4324:geometry of numbers
4241:Riemannian geometry
4193:projective geometry
4153:hyperbolic geometry
4108:projective geometry
4090:Mathematics and art
4015:functional analysis
3951:Poincaré conjecture
3872:hidden-line removal
3663:cohomology theories
3659:topological methods
3647:commutative algebra
3548:continuous mappings
3506:hyperbolic geometry
3391:uses techniques of
3189:projective geometry
3178:Riemannian geometry
3142:projective geometry
3025:configuration space
2964:Rotation (geometry)
2818:Pythagorean theorem
2765:formulas for volume
2753:Pythagorean theorem
2666:forms the basis of
2386:algebraic varieties
2336:topological surface
2143:classical mechanics
2072:projective geometry
2041:hyperbolic geometry
1886:Pythagorean triples
1867:Shatapatha Brahmana
1812:under the arc of a
1750:Pythagorean theorem
1681:(2000–1800 BC) and
1634:History of geometry
1590:projective geometry
1547:Riemannian geometry
1184:Part of a series on
448:Pythagorean theorem
11105:Numerical analysis
10714:Mathematical logic
10709:Information theory
10553:Algebraic geometry
10298:– College Geometry
9503:Drawing Course 101
9141:Clara Löh (2017).
8966:Satyan L. Devadoss
8430:Algebraic Geometry
8388:J. P. May (1999).
8274:Essential Topology
6945:General Relativity
6881:. Academic Press.
6182:Gerla, G. (1995).
5314:, Saunders, 1990,
5157:, Saunders, 1990,
4975:. pp. 71–96.
4973:Dover Publications
4753:
4700:
4647:
4594:
4541:
4500:to understand the
4498:spherical geometry
4458:Molecular geometry
4452:Other applications
4386:Category:Geometers
4352:Mathematics portal
4277:
4249:general relativity
4197:forced perspective
4100:
4058:Gaussian curvature
3928:
3919:on two generators
3876:linear programming
3816:
3786:superstring theory
3608:
3595:Algebraic geometry
3589:Algebraic geometry
3579:algebraic topology
3571:geometric topology
3544:
3502:parallel postulate
3490:Euclidean geometry
3475:
3424:general relativity
3380:
3247:Euclidean geometry
3242:Euclidean geometry
3236:Euclidean geometry
3136:is represented by
3134:Euclidean geometry
3089:
3056:algebraic geometry
3029:degrees of freedom
3005:
2870:general relativity
2866:Riemannian metrics
2862:special relativity
2834:
2820:for the (3, 4, 5)
2745:Euclidean geometry
2652:Euclidean geometry
2623:
2597:general relativity
2568:, a manifold is a
2536:
2482:
2461:or implicitly (by
2309:incidence geometry
2262:, which is itself
2248:synthetic geometry
2174:Euclidean geometry
2170:
2167:parallel postulate
2127:algebraic topology
2103:Erlangen programme
2010:parallel postulate
1988:. The theorems of
1954:algebraic geometry
1935:rational triangles
1859:
1779:mathematical rigor
1746:Pythagorean School
1718:mean speed theorem
1714:Oxford Calculators
1652:
1578:algebraic topology
1570:algebraic geometry
1559:general relativity
1551:parallel postulate
1527:Gaussian curvature
1461:Euclidean geometry
1351:Mathematics Portal
11231:
11230:
10830:Harmonic analysis
10617:
10616:
10222:Library resources
10206:978-0-7139-9634-0
10167:978-981-4556-70-5
10126:978-1-4051-3251-0
10103:978-0-8018-7396-6
10080:978-0-471-44459-6
10061:978-0-471-54397-8
10009:978-1-4612-1974-3
9982:978-0-691-04955-7
9937:978-1-4704-5016-8
9898:978-1-4641-7499-5
9891:. W. H. Freeman.
9859:978-1-4832-6240-6
9824:978-1-107-02625-4
9807:Życzkowski, Karol
9779:978-0-465-02266-3
9740:978-3-03719-051-7
9701:978-0-521-31779-5
9662:978-1-85669-371-4
9623:978-1-934493-04-5
9584:978-3-99043-371-3
9545:978-1-111-30126-2
9518:978-1-4027-0383-6
9473:978-3-540-28849-7
9430:978-0-307-48552-6
9391:978-1-4405-2305-2
9352:978-1-56898-249-6
9313:978-3-642-17286-1
9274:978-0-08-093439-6
9232:978-0-8218-8800-1
9193:978-0-8218-5201-9
9154:978-3-319-72254-2
9112:978-1-57146-171-1
9073:978-1-4612-1098-6
9058:Michael I. Shamos
9027:978-1-4419-0600-7
8985:978-1-4008-3898-1
8939:978-3-540-71133-9
8900:978-0-521-85535-8
8861:978-1-4613-0039-7
8787:978-0-19-154584-9
8607:978-3-540-79814-9
8568:978-3-319-63931-4
8549:Kristin E. Lauter
8547:Everett W. Howe;
8521:978-0-8218-3679-8
8482:978-0-412-99371-8
8440:978-1-4757-3849-0
8401:978-0-226-51183-2
8362:978-0-387-90636-2
8323:978-0-08-057085-3
8284:978-1-85233-782-7
8245:978-3-319-56478-4
8206:978-1-4008-8419-3
8167:978-1-118-09952-0
8128:978-0-521-65116-5
8089:978-0-486-13961-6
8050:978-3-11-036954-0
7980:978-1-898563-46-4
7941:978-1-107-05374-8
7902:978-1-4832-9079-9
7863:978-0-226-32783-9
7817:978-94-017-2742-6
7804:W. Abbot (2013).
7750:978-94-009-0959-5
7711:978-3-540-39437-2
7698:G. Gierz (2006).
7651:978-0-521-02139-5
7636:Charles W. Misner
7609:978-981-310-503-4
7570:978-1-4704-1227-2
7527:978-1-4684-0397-8
7485:978-0-486-45350-7
7446:978-1-4757-5325-7
7407:978-1-139-43171-2
7368:978-981-02-2363-2
7329:978-981-4472-57-9
7290:978-0-465-08237-7
7248:978-0-8218-5154-8
7209:978-1-4612-0645-3
7141:978-0-19-875548-7
7102:978-1-4612-5680-9
7063:978-0-618-61008-2
7024:978-0-7637-4366-6
6985:978-0-8218-6919-2
6955:978-0-226-87033-5
6917:, Sergei Ivanov,
6888:978-0-12-083971-1
6849:978-0-9614088-2-4
6807:978-1-4704-3714-5
6765:978-3-319-77577-7
6734:978-3-540-42627-1
6712:978-0-538-49790-9
6613:978-0-465-02023-2
6589:978-0-321-57056-7
6562:978-3-540-63293-1
6404:978-90-277-1243-1
6351:Munkres, James R.
6277:978-0-444-88355-1
6127:978-3-642-14441-7
6088:978-1-4612-6135-3
6046:978-1-351-97353-3
6007:978-0-387-22676-7
5974:978-0-465-03863-3
5930:978-0-88385-163-0
5891:978-0-19-506137-6
5852:978-3-319-74830-6
5813:978-0-85729-060-1
5774:978-0-486-14170-1
5735:978-1-4613-8692-6
5693:978-1-4612-6230-5
5654:978-0-486-15451-0
5473:978-0-7923-2565-9
5194:978-90-481-5850-8
5030:(6272): 482–484.
4982:978-0-486-22332-2
4901:978-0-08-047803-6
4874:978-0-8160-4953-0
4830:978-3-319-12102-4
4371:List of geometers
4245:pseudo-Riemannian
4123:ideal proportions
3819:Discrete geometry
3802:Discrete geometry
3796:Discrete geometry
3758:complex manifolds
3750:Jean-Pierre Serre
3510:elliptic geometry
3444:Riemannian metric
3325:Euclidean vectors
3319:analytic geometry
3110:Leonardo da Vinci
3021:higher dimensions
2997:fractal dimension
2850:hyperbolic metric
2781:Lebesgue integral
2761:formulas for area
2749:analytic geometry
2570:topological space
2301:analytic geometry
2131:dynamical systems
2125:, the founder of
1978:analytic geometry
1848:Euclid's Elements
1802: 287–212 BC
1733:Thales of Miletus
1582:discrete geometry
1521:Theorema Egregium
1385:
1384:
1340:
1339:
1179:
1178:
1144:
1143:
867:List of geometers
550:Three-dimensional
539:
538:
11251:
11219:
11218:
11207:
11206:
11195:
11194:
11184:
11183:
11115:Computer algebra
11090:Computer science
10810:Complex analysis
10644:
10637:
10630:
10621:
10620:
10607:
10606:
10395:
10388:
10381:
10372:
10371:
10362:Geometry classes
10258:
10250:
10248:"Geometry"
10210:
10193:Leonard Mlodinow
10188:
10179:
10130:
10107:
10084:
10065:
10049:
10026:
10025:
10023:
10021:
9993:
9987:
9986:
9970:
9960:
9954:
9953:
9951:
9949:
9921:
9915:
9914:
9912:
9910:
9882:
9876:
9875:
9873:
9871:
9843:
9837:
9836:
9813:(2nd ed.).
9802:
9796:
9795:
9793:
9791:
9763:
9757:
9756:
9754:
9752:
9724:
9718:
9717:
9715:
9713:
9685:
9679:
9678:
9676:
9674:
9646:
9640:
9639:
9637:
9635:
9607:
9601:
9600:
9598:
9596:
9568:
9562:
9561:
9559:
9557:
9529:
9523:
9522:
9506:
9496:
9490:
9489:
9487:
9485:
9453:
9447:
9446:
9444:
9442:
9414:
9408:
9407:
9405:
9403:
9375:
9369:
9368:
9366:
9364:
9336:
9330:
9329:
9327:
9325:
9297:
9291:
9290:
9288:
9286:
9258:
9249:
9248:
9246:
9244:
9216:
9210:
9209:
9207:
9205:
9177:
9171:
9170:
9168:
9166:
9138:
9129:
9128:
9126:
9124:
9096:
9090:
9089:
9087:
9085:
9050:
9044:
9043:
9041:
9039:
9008:
9002:
9001:
8999:
8997:
8962:
8956:
8955:
8953:
8951:
8923:
8917:
8916:
8914:
8912:
8884:
8878:
8877:
8875:
8873:
8842:
8836:
8830:
8824:
8818:
8812:
8811:
8809:
8807:
8777:Riemann surfaces
8772:Donaldson, S. K.
8768:
8762:
8759:
8753:
8750:
8744:
8741:
8735:
8734:
8732:
8730:
8686:
8680:
8677:
8671:
8670:
8668:
8666:
8630:
8624:
8623:
8621:
8619:
8591:
8585:
8584:
8582:
8580:
8544:
8538:
8537:
8535:
8533:
8505:
8499:
8498:
8496:
8494:
8466:
8457:
8456:
8454:
8452:
8424:
8418:
8417:
8415:
8413:
8385:
8379:
8378:
8376:
8374:
8346:
8340:
8339:
8337:
8335:
8307:
8301:
8300:
8298:
8296:
8268:
8262:
8261:
8259:
8257:
8229:
8223:
8222:
8220:
8218:
8190:
8184:
8183:
8181:
8179:
8151:
8145:
8144:
8142:
8140:
8112:
8106:
8105:
8103:
8101:
8073:
8067:
8066:
8064:
8062:
8034:
8028:
8027:
8003:
7997:
7996:
7994:
7992:
7964:
7958:
7957:
7955:
7953:
7925:
7919:
7918:
7916:
7914:
7886:
7880:
7879:
7877:
7875:
7847:
7834:
7833:
7831:
7829:
7801:
7795:
7794:
7792:
7790:
7773:
7767:
7766:
7764:
7762:
7734:
7728:
7727:
7725:
7723:
7695:
7689:
7688:
7674:
7668:
7667:
7665:
7663:
7632:
7626:
7625:
7623:
7621:
7593:
7587:
7586:
7584:
7582:
7550:
7544:
7543:
7541:
7539:
7508:
7502:
7501:
7499:
7497:
7469:
7463:
7462:
7460:
7458:
7430:
7424:
7423:
7421:
7419:
7391:
7385:
7384:
7382:
7380:
7352:
7346:
7345:
7343:
7341:
7313:
7307:
7306:
7304:
7302:
7271:
7265:
7264:
7262:
7260:
7232:
7226:
7225:
7223:
7221:
7193:
7187:
7186:
7184:
7182:
7164:
7158:
7157:
7155:
7153:
7125:
7119:
7118:
7116:
7114:
7086:
7080:
7079:
7077:
7075:
7047:
7041:
7040:
7038:
7036:
7008:
7002:
7001:
6999:
6997:
6966:
6960:
6959:
6936:
6930:
6911:
6905:
6904:
6902:
6900:
6872:
6866:
6865:
6863:
6861:
6830:
6824:
6823:
6821:
6819:
6788:
6782:
6781:
6779:
6777:
6749:
6740:
6738:
6720:
6714:
6697:
6691:
6690:
6688:
6686:
6647:
6641:
6640:
6622:
6616:
6598:
6592:
6581:
6575:
6574:
6551:(2nd ed.).
6541:
6532:
6531:
6529:
6527:
6491:
6480:
6477:
6471:
6470:
6468:
6466:
6431:Ahlfors, Lars V.
6427:
6421:
6420:
6418:
6416:
6385:
6379:
6378:
6347:
6332:
6331:
6329:
6327:
6318:. Archived from
6308:
6302:
6301:
6299:
6297:
6261:
6255:
6254:
6241:
6235:
6234:
6232:
6208:
6202:
6201:
6200:on 17 July 2011.
6199:
6188:
6179:
6173:
6159:
6144:
6143:
6141:
6139:
6111:
6105:
6104:
6102:
6100:
6069:
6063:
6062:
6060:
6058:
6030:
6024:
6023:
6021:
6019:
5992:Robin Hartshorne
5988:
5979:
5978:
5966:
5953:
5947:
5946:
5944:
5942:
5914:
5908:
5907:
5905:
5903:
5875:
5869:
5868:
5866:
5864:
5836:
5830:
5829:
5827:
5825:
5797:
5791:
5790:
5788:
5786:
5758:
5752:
5751:
5749:
5747:
5716:
5710:
5709:
5707:
5705:
5677:
5671:
5670:
5668:
5666:
5635:
5626:
5617:Kitab al-Manazir
5594:
5588:
5587:
5566:
5560:
5559:
5538:
5532:
5531:
5510:
5504:
5492:
5486:
5485:
5449:
5443:
5436:
5427:
5420:
5414:
5402:
5396:
5395:
5378:(1–2): 105–127.
5364:
5358:
5357:
5355:
5353:
5344:. Archived from
5333:
5327:
5303:
5297:
5290:
5284:
5277:
5271:
5264:
5258:
5257:
5255:
5253:
5213:
5207:
5206:
5172:
5166:
5149:
5143:
5136:
5130:
5129:
5127:
5125:
5105:
5099:
5098:
5070:
5064:
5063:
5019:
5013:
5012:, "Egypt" p. 19)
5006:
5000:
4998:
4996:
4994:
4961:Neugebauer, Otto
4957:
4951:
4950:
4948:
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4913:
4885:
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4656:
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4603:
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4579:
4567:
4566:
4550:
4548:
4547:
4542:
4511:
4505:
4474:
4354:
4349:
4348:
4235:celestial sphere
4054:convex polytopes
3966:quasi-isometries
3945:'s proof of the
3943:Grigori Perelman
3887:image processing
3774:coherent sheaves
3742:Riemann surfaces
3740:in his study of
3738:Bernhard Riemann
3715:Complex geometry
3710:Complex geometry
3704:Complex geometry
3687:Hodge conjecture
3643:polynomial rings
3632:
3629:
3583:general topology
3440:smooth manifolds
3411:, among others.
3374:uses tools from
3331:Euclidean vector
3172:, the latter in
3122:Erlangen program
3101:regular polygons
3086:hyperbolic plane
3052:fractal geometry
3036:general topology
3011:or curve), 2 (a
2854:hyperbolic plane
2842:Euclidean metric
2830:Euclidean metric
2777:Riemann integral
2479:
2460:
2446:
2428:
2382:algebraic curves
2360:Curve (geometry)
2348:complex analysis
2277:, formulated by
2242:Point (geometry)
2139:complex analysis
2111:Bernhard Riemann
2076:Girard Desargues
2060:Pierre de Fermat
2058:(1596–1650) and
2026:
2022:
2019:
2000:, including the
1962:Thābit ibn Qurra
1917:
1876:. According to (
1856:
1853:
1803:
1800:
1783:axiomatic method
1763:) developed the
1762:
1759:
1744:established the
1738:Thales's theorem
1716:, including the
1691:
1688:
1524:
1523:
1442:
1435:
1425:
1418:
1408:
1401:
1377:
1370:
1363:
1349:
1213:
1212:
1181:
1180:
1171:
1164:
1157:
885:
884:
404:
403:
337:Zero-dimensional
42:
28:
27:
11259:
11258:
11254:
11253:
11252:
11250:
11249:
11248:
11234:
11233:
11232:
11227:
11178:
11169:
11119:
11076:
11055:Systems science
10986:
10982:Homotopy theory
10948:
10915:
10867:
10839:
10786:
10733:
10704:Category theory
10690:
10655:
10648:
10618:
10613:
10595:
10562:
10531:
10488:
10481:
10436:
10429:
10404:
10399:
10333:Gresham College
10290:– K–12 Geometry
10245:
10242:
10241:
10240:
10230:
10229:
10225:
10218:
10213:
10207:
10191:
10168:
10142:
10138:
10136:Further reading
10133:
10127:
10117:Basil Blackwell
10104:
10081:
10062:
10052:Uta C. Merzbach
10034:
10029:
10019:
10017:
10010:
9994:
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9908:
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9883:
9879:
9869:
9867:
9860:
9844:
9840:
9825:
9803:
9799:
9789:
9787:
9780:
9772:. Basic Books.
9764:
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9709:
9702:
9686:
9682:
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9555:
9553:
9546:
9530:
9526:
9519:
9497:
9493:
9483:
9481:
9474:
9456:Michele Emmer;
9454:
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9392:
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9372:
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9113:
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9081:
9074:
9051:
9047:
9037:
9035:
9028:
9009:
9005:
8995:
8993:
8986:
8970:Joseph O'Rourke
8963:
8959:
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8109:
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8074:
8070:
8060:
8058:
8051:
8035:
8031:
8004:
8000:
7990:
7988:
7981:
7973:. Horwood Pub.
7965:
7961:
7951:
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7883:
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7112:
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7103:
7087:
7083:
7073:
7071:
7064:
7048:
7044:
7034:
7032:
7025:
7009:
7005:
6995:
6993:
6986:
6967:
6963:
6956:
6940:Wald, Robert M.
6937:
6933:
6913:Dmitri Burago,
6912:
6908:
6898:
6896:
6889:
6873:
6869:
6859:
6857:
6850:
6831:
6827:
6817:
6815:
6808:
6792:James W. Cannon
6789:
6785:
6775:
6773:
6766:
6750:
6743:
6735:
6721:
6717:
6698:
6694:
6684:
6682:
6667:
6651:Gelʹfand, I. M.
6648:
6644:
6623:
6619:
6607:. Basic Books.
6601:Yau, Shing-Tung
6599:
6595:
6582:
6578:
6563:
6553:Springer-Verlag
6542:
6535:
6525:
6523:
6508:
6492:
6483:
6478:
6474:
6464:
6462:
6447:
6428:
6424:
6414:
6412:
6405:
6389:Szmielew, Wanda
6386:
6382:
6367:
6348:
6335:
6325:
6323:
6322:on 15 July 2016
6310:
6309:
6305:
6295:
6293:
6278:
6262:
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6238:
6209:
6205:
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6147:
6137:
6135:
6128:
6112:
6108:
6098:
6096:
6089:
6070:
6066:
6056:
6054:
6047:
6031:
6027:
6017:
6015:
6008:
5989:
5982:
5975:
5967:. Basic Books.
5957:David Berlinski
5954:
5950:
5940:
5938:
5931:
5915:
5911:
5901:
5899:
5892:
5876:
5872:
5862:
5860:
5853:
5837:
5833:
5823:
5821:
5814:
5798:
5794:
5784:
5782:
5775:
5759:
5755:
5745:
5743:
5736:
5720:Judith V. Field
5717:
5713:
5703:
5701:
5694:
5678:
5674:
5664:
5662:
5655:
5636:
5629:
5595:
5591:
5567:
5563:
5539:
5535:
5511:
5507:
5493:
5489:
5474:
5450:
5446:
5437:
5430:
5421:
5417:
5403:
5399:
5365:
5361:
5351:
5349:
5348:on 15 July 2007
5334:
5330:
5304:
5300:
5291:
5287:
5278:
5274:
5265:
5261:
5251:
5249:
5234:10.2307/3026893
5214:
5210:
5195:
5173:
5169:
5150:
5146:
5137:
5133:
5123:
5121:
5106:
5102:
5087:10.2307/3822211
5071:
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5016:
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5003:
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4990:
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4902:
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4611:
4609:
4606:
4605:
4588:
4584:
4575:
4571:
4562:
4558:
4556:
4553:
4552:
4518:
4515:
4514:
4512:
4508:
4475:
4471:
4466:
4438:about two- and
4362:
4358:Main category:
4350:
4343:
4340:
4273:incommensurable
4265:
4219:
4213:
4189:
4181:Main articles:
4179:
4092:
4086:
4078:
4007:convex analysis
3991:Convex geometry
3988:
3986:Convex geometry
3982:
3980:Convex geometry
3909:
3903:
3895:medical imaging
3883:computer vision
3860:implementations
3849:
3843:
3827:sphere packings
3823:convex geometry
3812:sphere packings
3804:
3798:
3731:mirror symmetry
3712:
3706:
3681:. One of seven
3630:
3597:
3591:
3533:
3527:
3522:
3521:
3514:planar algebras
3498:affine geometry
3494:metric geometry
3473:
3464:
3456:
3420:Albert Einstein
3386:
3365:
3333:
3327:
3259:crystallography
3244:
3238:
3233:
3156:and Klein, and
3105:platonic solids
3074:
3068:
3040:natural numbers
2999:=log4/log3 and
2985:
2978:
2970:
2962:Main articles:
2960:
2940:
2934:
2903:
2895:Main articles:
2893:
2846:Euclidean plane
2826:Zhoubi Suanjing
2810:
2802:Main articles:
2800:
2727:
2717:
2705:Main articles:
2703:
2645:angular measure
2615:
2609:
2558:
2552:
2532:Euclidean space
2524:
2518:
2502:diffeomorphisms
2462:
2447:
2429:
2411:
2404:
2398:
2362:
2356:
2331:
2329:Euclidean plane
2325:
2305:linear equation
2293:
2291:Line (geometry)
2287:
2244:
2238:
2233:
2180:
2159:
2151:
2119:Riemann surface
2024:
2020:
1986:cubic equations
1931:Heron's formula
1854:
1806:Syracuse, Italy
1801:
1760:
1722:ancient Nubians
1689:
1636:
1630:
1614:geometric space
1598:finite geometry
1594:affine geometry
1584:(also known as
1535:Euclidean space
1381:
1336:
1335:
1286:
1278:
1277:
1273:Decision theory
1221:
1175:
1146:
1145:
882:
881:
872:
871:
662:
661:
645:
644:
630:
629:
617:
616:
553:
552:
541:
540:
401:
400:
398:Two-dimensional
389:
388:
362:
361:
359:One-dimensional
350:
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11140:
11133:Mathematicians
11129:
11127:
11125:Related topics
11121:
11120:
11118:
11117:
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11107:
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11097:
11092:
11086:
11084:
11078:
11077:
11075:
11074:
11073:
11072:
11067:
11062:
11060:Control theory
11052:
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11027:
11022:
11017:
11012:
11007:
11002:
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10994:
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10868:
10866:
10865:
10860:
10855:
10849:
10847:
10841:
10840:
10838:
10837:
10835:Measure theory
10832:
10827:
10822:
10817:
10812:
10807:
10802:
10796:
10794:
10788:
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10533:
10532:
10530:
10529:
10524:
10519:
10514:
10509:
10504:
10499:
10493:
10491:
10487:Non-Euclidean
10483:
10482:
10480:
10479:
10477:Solid geometry
10474:
10473:
10472:
10467:
10460:Plane geometry
10457:
10452:
10447:
10441:
10439:
10431:
10430:
10428:
10427:
10422:
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10420:
10409:
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10368:
10359:
10354:
10349:
10344:
10343:
10342:
10326:
10318:
10310:
10309:
10308:
10304:The Math Forum
10300:
10296:The Math Forum
10292:
10288:The Math Forum
10280:The Math Forum
10276:
10269:
10259:
10239:
10238:
10232:
10231:
10220:
10219:
10217:
10216:External links
10214:
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8625:
8606:
8586:
8567:
8553:Judy L. Walker
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8068:
8049:
8043:. De Gruyter.
8029:
7998:
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7796:
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7588:
7569:
7555:Karen Vogtmann
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6906:
6887:
6867:
6848:
6834:Gilbert Strang
6825:
6806:
6783:
6764:
6741:
6733:
6715:
6700:Stewart, James
6692:
6665:
6642:
6617:
6593:
6576:
6561:
6545:Mumford, David
6533:
6506:
6481:
6472:
6445:
6422:
6403:
6380:
6365:
6333:
6303:
6276:
6256:
6236:
6203:
6174:
6145:
6126:
6106:
6087:
6064:
6045:
6025:
6006:
5980:
5973:
5948:
5929:
5909:
5890:
5870:
5851:
5831:
5812:
5792:
5773:
5753:
5734:
5711:
5692:
5672:
5653:
5627:
5612:Book of Optics
5589:
5574:"Omar Khayyam"
5561:
5533:
5505:
5487:
5472:
5444:
5428:
5426:, p. 371)
5415:
5397:
5359:
5328:
5298:
5285:
5272:
5259:
5228:(5): 312–316.
5208:
5193:
5167:
5151:Eves, Howard,
5144:
5131:
5100:
5065:
5014:
5001:
4981:
4971:(2 ed.).
4952:
4939:(3): 277–318.
4919:
4900:
4880:
4873:
4848:
4829:
4809:
4783:
4781:
4778:
4775:
4774:
4765:
4750:
4746:
4742:
4737:
4733:
4729:
4724:
4720:
4697:
4693:
4689:
4684:
4680:
4676:
4671:
4667:
4644:
4640:
4636:
4631:
4627:
4623:
4618:
4614:
4591:
4587:
4583:
4578:
4574:
4570:
4565:
4561:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4506:
4468:
4467:
4465:
4462:
4461:
4460:
4454:
4453:
4449:
4448:
4443:
4427:
4421:
4420:
4419:Related topics
4416:
4415:
4410:
4405:
4400:
4395:
4394:
4393:
4388:
4383:
4378:
4367:
4366:
4356:
4355:
4339:
4336:
4315:ancient Greece
4288:René Descartes
4264:
4261:
4215:Main article:
4212:
4209:
4201:conic sections
4178:
4175:
4088:Main article:
4085:
4082:
4077:
4074:
4036:. Archimedes,
3984:Main article:
3981:
3978:
3905:Main article:
3902:
3899:
3845:Main article:
3842:
3839:
3831:triangulations
3800:Main article:
3797:
3794:
3708:Main article:
3705:
3702:
3616:algebraic sets
3593:Main article:
3590:
3587:
3567:homeomorphisms
3529:Main article:
3526:
3523:
3471:
3465:
3457:
3455:
3452:
3409:bioinformatics
3397:linear algebra
3382:Main article:
3364:
3361:
3329:Main article:
3326:
3323:
3240:Main article:
3237:
3234:
3232:
3229:
3162:symmetry group
3070:Main article:
3067:
3064:
3044:Hilbert spaces
2993:Koch snowflake
2977:
2974:
2959:
2956:
2936:Main article:
2933:
2930:
2892:
2889:
2877:measure theory
2858:Lorentz metric
2799:
2796:
2775:, such as the
2702:
2699:
2611:Main article:
2608:
2605:
2554:Main article:
2551:
2548:
2522:Solid geometry
2520:Main article:
2517:
2514:
2506:homeomorphisms
2400:Main article:
2397:
2394:
2358:Main article:
2355:
2352:
2327:Main article:
2324:
2321:
2289:Main article:
2286:
2283:
2281:in 1919–1920.
2240:Main article:
2237:
2234:
2232:
2229:
2158:
2155:
2150:
2147:
2123:Henri Poincaré
2056:René Descartes
1998:quadrilaterals
1990:Ibn al-Haytham
1731:mathematician
1690: 1890 BC
1683:Moscow Papyrus
1632:Main article:
1629:
1626:
1616:, or simply a
1383:
1382:
1380:
1379:
1372:
1365:
1357:
1354:
1353:
1342:
1341:
1338:
1337:
1334:
1333:
1328:
1323:
1318:
1313:
1308:
1303:
1298:
1293:
1287:
1284:
1283:
1280:
1279:
1276:
1275:
1266:
1261:
1252:
1247:
1238:
1233:
1228:
1222:
1217:
1216:
1209:
1208:
1207:
1206:
1201:
1193:
1192:
1186:
1185:
1177:
1176:
1174:
1173:
1166:
1159:
1151:
1148:
1147:
1142:
1141:
1140:
1139:
1134:
1126:
1125:
1121:
1120:
1119:
1118:
1113:
1108:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1060:
1059:
1055:
1054:
1053:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1009:
1008:
1004:
1003:
1002:
1001:
996:
991:
986:
981:
976:
971:
966:
961:
956:
951:
946:
938:
937:
933:
932:
931:
930:
925:
920:
915:
910:
905:
900:
892:
891:
883:
879:
878:
877:
874:
873:
870:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
663:
659:
658:
657:
654:
653:
647:
646:
643:
642:
637:
631:
624:
623:
622:
619:
618:
615:
614:
609:
604:
602:Platonic Solid
599:
594:
589:
584:
579:
574:
573:
572:
561:
560:
554:
548:
547:
546:
543:
542:
537:
536:
535:
534:
529:
524:
516:
515:
509:
508:
507:
506:
501:
493:
492:
486:
485:
484:
483:
478:
473:
468:
460:
459:
453:
452:
451:
450:
445:
440:
432:
431:
425:
424:
423:
422:
417:
412:
402:
396:
395:
394:
391:
390:
387:
386:
381:
380:
379:
374:
363:
357:
356:
355:
352:
351:
348:
347:
341:
335:
334:
333:
330:
329:
326:
325:
320:
315:
309:
308:
303:
298:
288:
283:
278:
272:
271:
262:
258:
257:
254:
250:
249:
248:
247:
244:
243:
240:
239:
238:
237:
227:
222:
217:
212:
207:
206:
205:
195:
190:
185:
184:
183:
178:
173:
163:
162:
161:
156:
146:
141:
136:
131:
126:
121:
120:
119:
114:
113:
112:
97:
91:
85:
84:
83:
80:
79:
77:
76:
66:
60:
57:
56:
43:
35:
34:
15:
9:
6:
4:
3:
2:
11256:
11245:
11242:
11241:
11239:
11224:
11223:
11214:
11212:
11211:
11202:
11200:
11199:
11190:
11188:
11187:
11182:
11176:
11175:
11172:
11166:
11163:
11161:
11158:
11156:
11153:
11151:
11148:
11146:
11143:
11139:
11136:
11135:
11134:
11131:
11130:
11128:
11126:
11122:
11116:
11113:
11111:
11108:
11106:
11103:
11101:
11098:
11096:
11093:
11091:
11088:
11087:
11085:
11083:
11082:Computational
11079:
11071:
11068:
11066:
11063:
11061:
11058:
11057:
11056:
11053:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11031:
11028:
11026:
11023:
11021:
11018:
11016:
11013:
11011:
11008:
11006:
11003:
11001:
10998:
10997:
10995:
10993:
10989:
10983:
10980:
10978:
10975:
10973:
10970:
10968:
10965:
10963:
10960:
10959:
10957:
10955:
10951:
10945:
10942:
10940:
10937:
10935:
10932:
10930:
10927:
10926:
10924:
10922:
10921:Number theory
10918:
10912:
10909:
10907:
10904:
10902:
10899:
10897:
10894:
10892:
10889:
10887:
10884:
10882:
10879:
10878:
10876:
10874:
10870:
10864:
10861:
10859:
10856:
10854:
10853:Combinatorics
10851:
10850:
10848:
10846:
10842:
10836:
10833:
10831:
10828:
10826:
10823:
10821:
10818:
10816:
10813:
10811:
10808:
10806:
10805:Real analysis
10803:
10801:
10798:
10797:
10795:
10793:
10789:
10783:
10780:
10778:
10775:
10773:
10770:
10768:
10765:
10763:
10760:
10758:
10755:
10753:
10750:
10748:
10745:
10744:
10742:
10740:
10736:
10730:
10727:
10725:
10722:
10720:
10717:
10715:
10712:
10710:
10707:
10705:
10702:
10701:
10699:
10697:
10693:
10687:
10684:
10682:
10679:
10675:
10672:
10670:
10667:
10666:
10665:
10662:
10661:
10658:
10653:
10645:
10640:
10638:
10633:
10631:
10626:
10625:
10622:
10610:
10602:
10601:
10598:
10592:
10589:
10587:
10584:
10580:
10577:
10576:
10575:
10572:
10571:
10569:
10565:
10559:
10556:
10554:
10551:
10549:
10546:
10544:
10541:
10540:
10538:
10534:
10528:
10525:
10523:
10520:
10518:
10515:
10513:
10510:
10508:
10505:
10503:
10500:
10498:
10495:
10494:
10492:
10490:
10484:
10478:
10475:
10471:
10468:
10466:
10463:
10462:
10461:
10458:
10456:
10453:
10451:
10448:
10446:
10445:Combinatorial
10443:
10442:
10440:
10438:
10432:
10426:
10423:
10419:
10416:
10415:
10414:
10411:
10410:
10407:
10403:
10396:
10391:
10389:
10384:
10382:
10377:
10376:
10373:
10367:
10363:
10360:
10358:
10355:
10353:
10350:
10348:
10345:
10340:
10337:
10336:
10334:
10330:
10327:
10325:
10323:
10319:
10317:
10316:
10311:
10307:
10305:
10301:
10299:
10297:
10293:
10291:
10289:
10285:
10284:
10283:
10281:
10277:
10275:
10274:
10270:
10268:
10264:
10260:
10256:
10255:
10249:
10244:
10243:
10237:
10234:
10233:
10228:
10223:
10208:
10202:
10198:
10194:
10190:
10186:
10181:
10177:
10173:
10169:
10163:
10159:
10155:
10151:
10150:
10145:
10141:
10140:
10128:
10122:
10118:
10114:
10109:
10105:
10099:
10095:
10091:
10086:
10082:
10076:
10072:
10067:
10063:
10057:
10053:
10048:
10047:
10041:
10037:
10036:
10015:
10011:
10005:
10001:
10000:
9992:
9984:
9978:
9974:
9969:
9968:
9959:
9943:
9939:
9933:
9929:
9928:
9920:
9904:
9900:
9894:
9890:
9889:
9881:
9865:
9861:
9855:
9851:
9850:
9842:
9834:
9830:
9826:
9820:
9816:
9812:
9808:
9801:
9785:
9781:
9775:
9771:
9770:
9762:
9746:
9742:
9736:
9732:
9731:
9723:
9707:
9703:
9697:
9693:
9692:
9684:
9668:
9664:
9658:
9654:
9653:
9645:
9629:
9625:
9619:
9615:
9614:
9606:
9590:
9586:
9580:
9576:
9575:
9567:
9551:
9547:
9541:
9537:
9536:
9528:
9520:
9514:
9510:
9505:
9504:
9495:
9479:
9475:
9469:
9465:
9464:
9459:
9452:
9436:
9432:
9426:
9422:
9421:
9413:
9397:
9393:
9387:
9383:
9382:
9374:
9358:
9354:
9348:
9344:
9343:
9335:
9319:
9315:
9309:
9305:
9304:
9296:
9280:
9276:
9270:
9266:
9265:
9257:
9255:
9238:
9234:
9228:
9224:
9223:
9215:
9199:
9195:
9189:
9185:
9184:
9176:
9160:
9156:
9150:
9146:
9145:
9137:
9135:
9118:
9114:
9108:
9104:
9103:
9095:
9079:
9075:
9069:
9065:
9064:
9059:
9055:
9049:
9033:
9029:
9023:
9019:
9018:
9013:
9012:Károly Bezdek
9007:
8991:
8987:
8981:
8977:
8976:
8971:
8967:
8961:
8945:
8941:
8935:
8931:
8930:
8922:
8906:
8902:
8896:
8892:
8891:
8883:
8867:
8863:
8857:
8853:
8852:
8847:
8846:Jiří Matoušek
8841:
8834:
8829:
8822:
8817:
8801:
8797:
8793:
8789:
8783:
8779:
8778:
8773:
8767:
8758:
8749:
8740:
8724:
8720:
8716:
8712:
8710:9780387738918
8706:
8702:
8698:
8694:
8693:
8685:
8676:
8660:
8656:
8652:
8648:
8646:9783540266877
8642:
8638:
8637:
8629:
8613:
8609:
8603:
8599:
8598:
8590:
8574:
8570:
8564:
8560:
8559:
8554:
8550:
8543:
8527:
8523:
8517:
8513:
8512:
8504:
8488:
8484:
8478:
8474:
8473:
8465:
8463:
8446:
8442:
8436:
8432:
8431:
8423:
8407:
8403:
8397:
8393:
8392:
8384:
8368:
8364:
8358:
8354:
8353:
8345:
8329:
8325:
8319:
8315:
8314:
8306:
8290:
8286:
8280:
8276:
8275:
8267:
8251:
8247:
8241:
8237:
8236:
8228:
8212:
8208:
8202:
8198:
8197:
8189:
8173:
8169:
8163:
8159:
8158:
8150:
8134:
8130:
8124:
8120:
8119:
8111:
8095:
8091:
8085:
8081:
8080:
8072:
8056:
8052:
8046:
8042:
8041:
8033:
8025:
8021:
8017:
8013:
8009:
8002:
7986:
7982:
7976:
7972:
7971:
7963:
7947:
7943:
7937:
7933:
7932:
7924:
7908:
7904:
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7893:
7885:
7869:
7865:
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7854:
7846:
7844:
7842:
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7819:
7813:
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7707:
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7702:
7694:
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7673:
7657:
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7642:
7637:
7631:
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7605:
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7600:
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7507:
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7390:
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7360:
7359:
7351:
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7325:
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7320:
7312:
7296:
7292:
7286:
7282:
7281:
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7270:
7254:
7250:
7244:
7240:
7239:
7231:
7215:
7211:
7205:
7201:
7200:
7192:
7176:
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7171:
7163:
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7143:
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7133:
7132:
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7098:
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7069:
7065:
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7026:
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7016:
7015:
7007:
6991:
6987:
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6977:
6976:
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6965:
6957:
6951:
6947:
6946:
6941:
6935:
6928:
6927:0-8218-2129-6
6924:
6920:
6916:
6910:
6894:
6890:
6884:
6880:
6879:
6871:
6855:
6851:
6845:
6841:
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6835:
6829:
6813:
6809:
6803:
6799:
6798:
6793:
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6767:
6761:
6757:
6756:
6748:
6746:
6736:
6730:
6726:
6719:
6713:
6709:
6705:
6701:
6696:
6680:
6676:
6672:
6668:
6666:0-8176-3914-4
6662:
6658:
6657:
6652:
6646:
6638:
6634:
6633:
6628:
6621:
6614:
6610:
6606:
6602:
6597:
6590:
6586:
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6558:
6554:
6550:
6546:
6540:
6538:
6521:
6517:
6513:
6509:
6507:0-13-212589-7
6503:
6499:
6498:
6490:
6488:
6486:
6476:
6460:
6456:
6452:
6448:
6446:9780070006577
6442:
6438:
6437:
6432:
6426:
6410:
6406:
6400:
6396:
6395:
6390:
6384:
6376:
6372:
6368:
6366:0-13-181629-2
6362:
6358:
6357:
6352:
6346:
6344:
6342:
6340:
6338:
6321:
6317:
6313:
6307:
6291:
6287:
6283:
6279:
6273:
6269:
6268:
6260:
6252:
6251:
6246:
6240:
6231:
6226:
6222:
6218:
6214:
6207:
6196:
6192:
6185:
6178:
6171:
6170:1-888009-18-7
6167:
6163:
6158:
6156:
6154:
6152:
6150:
6133:
6129:
6123:
6119:
6118:
6110:
6094:
6090:
6084:
6080:
6079:
6074:
6068:
6052:
6048:
6042:
6038:
6037:
6029:
6013:
6009:
6003:
5999:
5998:
5993:
5987:
5985:
5976:
5970:
5965:
5964:
5958:
5952:
5936:
5932:
5926:
5922:
5921:
5913:
5897:
5893:
5887:
5883:
5882:
5874:
5858:
5854:
5848:
5844:
5843:
5835:
5819:
5815:
5809:
5805:
5804:
5796:
5780:
5776:
5770:
5766:
5765:
5757:
5741:
5737:
5731:
5727:
5726:
5721:
5715:
5699:
5695:
5689:
5685:
5684:
5676:
5660:
5656:
5650:
5646:
5645:
5640:
5639:Carl B. Boyer
5634:
5632:
5624:
5622:
5618:
5614:
5613:
5605:
5601:
5600:
5593:
5585:
5581:
5580:
5575:
5572:
5565:
5557:
5553:
5552:
5547:
5544:
5537:
5529:
5525:
5524:
5519:
5516:
5509:
5501:
5497:
5491:
5483:
5479:
5475:
5469:
5465:
5461:
5457:
5456:
5448:
5441:
5435:
5433:
5425:
5419:
5411:
5407:
5401:
5393:
5389:
5385:
5381:
5377:
5373:
5369:
5363:
5347:
5343:
5339:
5332:
5325:
5321:
5320:0-03-029558-0
5317:
5313:
5312:
5307:
5302:
5295:
5289:
5282:
5276:
5269:
5263:
5247:
5243:
5239:
5235:
5231:
5227:
5223:
5219:
5212:
5204:
5200:
5196:
5190:
5186:
5182:
5178:
5171:
5164:
5163:0-03-029558-0
5160:
5156:
5155:
5148:
5141:
5135:
5119:
5115:
5111:
5104:
5096:
5092:
5088:
5084:
5080:
5076:
5069:
5061:
5057:
5053:
5049:
5045:
5041:
5037:
5033:
5029:
5025:
5018:
5011:
5005:
4988:
4984:
4978:
4974:
4970:
4966:
4962:
4956:
4947:
4942:
4938:
4934:
4930:
4923:
4907:
4903:
4897:
4893:
4892:
4884:
4876:
4870:
4865:
4864:
4855:
4853:
4836:
4832:
4826:
4822:
4821:
4813:
4798:
4794:
4788:
4784:
4769:
4748:
4744:
4740:
4735:
4731:
4727:
4722:
4718:
4695:
4691:
4687:
4682:
4678:
4674:
4669:
4665:
4642:
4638:
4634:
4629:
4625:
4621:
4616:
4612:
4589:
4585:
4581:
4576:
4572:
4568:
4563:
4559:
4535:
4532:
4529:
4526:
4523:
4510:
4503:
4502:Earth geodesy
4499:
4495:
4491:
4487:
4483:
4479:
4473:
4469:
4459:
4456:
4455:
4451:
4450:
4447:
4444:
4441:
4437:
4433:
4432:
4428:
4426:
4423:
4422:
4418:
4417:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4373:
4372:
4369:
4368:
4364:
4363:
4361:
4353:
4347:
4342:
4335:
4333:
4329:
4328:scheme theory
4325:
4320:
4316:
4312:
4311:number theory
4307:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4274:
4269:
4260:
4258:
4254:
4253:String theory
4250:
4246:
4242:
4238:
4236:
4232:
4228:
4224:
4221:The field of
4218:
4208:
4206:
4205:tessellations
4202:
4199:, the use of
4198:
4194:
4188:
4184:
4174:
4172:
4168:
4164:
4160:
4156:
4154:
4150:
4146:
4142:
4138:
4137:this legend.
4135:
4130:
4128:
4124:
4120:
4116:
4111:
4109:
4105:
4096:
4091:
4081:
4073:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4022:
4020:
4019:number theory
4016:
4012:
4008:
4004:
4000:
3999:real analysis
3996:
3993:investigates
3992:
3987:
3977:
3975:
3971:
3967:
3963:
3958:
3956:
3952:
3948:
3944:
3941:, such as in
3940:
3936:
3932:
3926:
3922:
3918:
3913:
3908:
3898:
3896:
3892:
3888:
3884:
3879:
3877:
3873:
3869:
3865:
3861:
3857:
3853:
3848:
3838:
3836:
3835:combinatorics
3832:
3828:
3824:
3820:
3813:
3808:
3803:
3793:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3734:
3732:
3728:
3727:string theory
3724:
3720:
3719:complex plane
3716:
3711:
3701:
3699:
3698:string theory
3695:
3690:
3688:
3684:
3680:
3676:
3675:number theory
3672:
3668:
3667:number theory
3664:
3660:
3656:
3655:scheme theory
3652:
3648:
3644:
3640:
3636:
3625:
3621:
3617:
3613:
3606:
3601:
3596:
3586:
3584:
3580:
3576:
3572:
3568:
3564:
3559:
3557:
3553:
3552:connectedness
3549:
3542:
3537:
3532:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3479:
3469:
3462:
3451:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3417:
3412:
3410:
3406:
3402:
3398:
3394:
3390:
3385:
3377:
3373:
3369:
3360:
3358:
3354:
3350:
3346:
3342:
3338:
3332:
3322:
3320:
3316:
3312:
3311:solid figures
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3243:
3228:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3181:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3146:collineations
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3106:
3102:
3098:
3094:
3091:The theme of
3087:
3083:
3078:
3073:
3063:
3061:
3057:
3053:
3049:
3045:
3041:
3037:
3032:
3030:
3026:
3022:
3018:
3014:
3010:
3002:
2998:
2994:
2989:
2983:
2973:
2969:
2965:
2955:
2953:
2949:
2945:
2939:
2929:
2927:
2922:
2920:
2916:
2911:
2907:
2902:
2898:
2888:
2886:
2882:
2878:
2873:
2871:
2867:
2864:and the semi-
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2831:
2827:
2823:
2819:
2814:
2809:
2805:
2795:
2793:
2789:
2784:
2782:
2778:
2774:
2770:
2766:
2762:
2756:
2754:
2750:
2746:
2741:
2739:
2735:
2731:
2726:
2722:
2716:
2712:
2708:
2698:
2696:
2692:
2688:
2684:
2680:
2676:
2671:
2669:
2665:
2661:
2657:
2653:
2648:
2646:
2642:
2641:
2636:
2633:, called the
2632:
2627:
2619:
2614:
2604:
2602:
2601:string theory
2598:
2593:
2591:
2590:diffeomorphic
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2557:
2547:
2545:
2541:
2533:
2528:
2523:
2513:
2511:
2507:
2503:
2499:
2498:neighborhoods
2495:
2491:
2487:
2477:
2473:
2469:
2465:
2458:
2454:
2450:
2444:
2440:
2436:
2432:
2426:
2422:
2418:
2414:
2408:
2403:
2393:
2391:
2387:
2383:
2377:
2375:
2371:
2367:
2361:
2351:
2350:; and so on.
2349:
2345:
2344:complex plane
2341:
2337:
2330:
2320:
2318:
2317:curved spaces
2314:
2310:
2306:
2302:
2297:
2292:
2282:
2280:
2276:
2271:
2267:
2265:
2264:axiomatically
2261:
2257:
2253:
2249:
2243:
2228:
2226:
2225:David Hilbert
2222:
2219:(1802–1860),
2218:
2215:(1792–1856),
2214:
2210:
2206:
2205:
2200:
2196:
2192:
2188:
2184:
2179:
2175:
2168:
2163:
2154:
2149:Main concepts
2146:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2091:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2044:
2042:
2038:
2034:
2030:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1942:
1940:
1936:
1932:
1928:
1923:
1919:
1918:
1916:
1909:
1905:
1904:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1874:
1869:
1868:
1863:
1849:
1845:
1841:
1837:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1796:
1792:
1788:
1784:
1780:
1776:
1775:
1770:
1766:
1761: 355 BC
1755:
1751:
1747:
1743:
1739:
1734:
1730:
1725:
1723:
1719:
1715:
1711:
1707:
1703:
1699:
1695:
1684:
1680:
1679:Rhind Papyrus
1677:
1673:
1669:
1665:
1661:
1657:
1649:
1645:
1640:
1635:
1625:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1567:
1562:
1560:
1556:
1552:
1548:
1544:
1540:
1539:intrinsically
1536:
1532:
1528:
1522:
1516:
1511:
1509:
1505:
1501:
1500:Wiles's proof
1497:
1492:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1457:
1452:
1448:
1444:
1441:
1434:
1430:
1427:
1424:
1417:
1413:
1410:
1407:
1400:
1396:
1393:
1392:Ancient Greek
1389:
1378:
1373:
1371:
1366:
1364:
1359:
1358:
1356:
1355:
1352:
1348:
1344:
1343:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1294:
1292:
1289:
1288:
1282:
1281:
1274:
1270:
1267:
1265:
1262:
1260:
1256:
1253:
1251:
1248:
1246:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1226:Number theory
1224:
1223:
1220:
1215:
1214:
1211:
1210:
1205:
1202:
1200:
1197:
1196:
1195:
1194:
1191:
1188:
1187:
1183:
1182:
1172:
1167:
1165:
1160:
1158:
1153:
1152:
1150:
1149:
1138:
1135:
1133:
1130:
1129:
1128:
1127:
1123:
1122:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1063:
1062:
1061:
1057:
1056:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1012:
1011:
1010:
1006:
1005:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
941:
940:
939:
935:
934:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
895:
894:
893:
890:
887:
886:
876:
875:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
664:
656:
655:
652:
649:
648:
641:
638:
636:
633:
632:
627:
621:
620:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
571:
568:
567:
566:
563:
562:
559:
556:
555:
551:
545:
544:
533:
530:
528:
527:Circumference
525:
523:
520:
519:
518:
517:
514:
511:
510:
505:
502:
500:
497:
496:
495:
494:
491:
490:Quadrilateral
488:
487:
482:
479:
477:
474:
472:
469:
467:
464:
463:
462:
461:
458:
457:Parallelogram
455:
454:
449:
446:
444:
441:
439:
436:
435:
434:
433:
430:
427:
426:
421:
418:
416:
413:
411:
408:
407:
406:
405:
399:
393:
392:
385:
382:
378:
375:
373:
370:
369:
368:
365:
364:
360:
354:
353:
346:
343:
342:
338:
332:
331:
324:
321:
319:
316:
314:
311:
310:
307:
304:
302:
299:
296:
295:Perpendicular
292:
291:Orthogonality
289:
287:
284:
282:
279:
277:
274:
273:
270:
267:
266:
265:
255:
252:
251:
246:
245:
236:
233:
232:
231:
228:
226:
223:
221:
218:
216:
215:Computational
213:
211:
208:
204:
201:
200:
199:
196:
194:
191:
189:
186:
182:
179:
177:
174:
172:
169:
168:
167:
164:
160:
157:
155:
152:
151:
150:
147:
145:
142:
140:
137:
135:
132:
130:
127:
125:
122:
118:
115:
111:
108:
107:
106:
103:
102:
101:
100:Non-Euclidean
98:
96:
93:
92:
88:
82:
81:
74:
70:
67:
65:
62:
61:
59:
58:
54:
50:
46:
41:
37:
36:
33:
30:
29:
26:
22:
11220:
11208:
11196:
11177:
11110:Optimization
10972:Differential
10896:Differential
10872:
10863:Order theory
10858:Graph theory
10762:Group theory
10543:Trigonometry
10401:
10366:Khan Academy
10321:
10314:
10303:
10295:
10287:
10279:
10272:
10265:course from
10252:
10226:
10196:
10184:
10158:10.1142/8952
10148:
10144:Jay Kappraff
10112:
10089:
10070:
10045:
10020:25 September
10018:. Retrieved
9998:
9991:
9966:
9958:
9948:25 September
9946:. Retrieved
9926:
9919:
9909:25 September
9907:. Retrieved
9887:
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9870:25 September
9868:. Retrieved
9848:
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9800:
9790:25 September
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9768:
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9749:. Retrieved
9729:
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9710:. Retrieved
9690:
9683:
9673:25 September
9671:. Retrieved
9651:
9644:
9634:25 September
9632:. Retrieved
9612:
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9593:. Retrieved
9573:
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9554:. Retrieved
9534:
9527:
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9494:
9484:25 September
9482:. Retrieved
9462:
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9419:
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9380:
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9341:
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9322:. Retrieved
9302:
9295:
9285:24 September
9283:. Retrieved
9263:
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9241:. Retrieved
9221:
9214:
9204:25 September
9202:. Retrieved
9182:
9175:
9165:25 September
9163:. Retrieved
9147:. Springer.
9143:
9123:25 September
9121:. Retrieved
9101:
9094:
9084:25 September
9082:. Retrieved
9062:
9048:
9038:25 September
9036:. Retrieved
9016:
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8996:25 September
8994:. Retrieved
8974:
8960:
8950:25 September
8948:. Retrieved
8928:
8921:
8911:25 September
8909:. Retrieved
8889:
8882:
8872:25 September
8870:. Retrieved
8850:
8840:
8833:Serre, J. P.
8828:
8821:Serre, J. P.
8816:
8804:. Retrieved
8776:
8766:
8757:
8748:
8739:
8727:. Retrieved
8691:
8684:
8675:
8665:10 September
8663:. Retrieved
8635:
8628:
8618:24 September
8616:. Retrieved
8600:. Springer.
8596:
8589:
8579:24 September
8577:. Retrieved
8561:. Springer.
8557:
8542:
8532:24 September
8530:. Retrieved
8510:
8503:
8493:24 September
8491:. Retrieved
8471:
8451:24 September
8449:. Retrieved
8429:
8422:
8412:24 September
8410:. Retrieved
8390:
8383:
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8371:. Retrieved
8351:
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8332:. Retrieved
8312:
8305:
8295:24 September
8293:. Retrieved
8273:
8266:
8256:23 September
8254:. Retrieved
8234:
8227:
8217:23 September
8215:. Retrieved
8195:
8188:
8178:23 September
8176:. Retrieved
8156:
8149:
8139:23 September
8137:. Retrieved
8117:
8110:
8100:23 September
8098:. Retrieved
8078:
8071:
8061:23 September
8059:. Retrieved
8039:
8032:
8018:(2): 10–26.
8015:
8011:
8001:
7991:20 September
7989:. Retrieved
7969:
7962:
7952:20 September
7950:. Retrieved
7930:
7923:
7913:20 September
7911:. Retrieved
7891:
7884:
7874:20 September
7872:. Retrieved
7852:
7828:20 September
7826:. Retrieved
7806:
7799:
7789:20 September
7787:. Retrieved
7777:
7771:
7761:20 September
7759:. Retrieved
7739:
7732:
7722:23 September
7720:. Retrieved
7700:
7693:
7679:
7672:
7662:23 September
7660:. Retrieved
7640:
7630:
7620:23 September
7618:. Retrieved
7598:
7591:
7581:23 September
7579:. Retrieved
7559:
7548:
7538:23 September
7536:. Retrieved
7516:
7506:
7496:23 September
7494:. Retrieved
7474:
7467:
7457:23 September
7455:. Retrieved
7435:
7428:
7418:23 September
7416:. Retrieved
7396:
7389:
7379:23 September
7377:. Retrieved
7357:
7350:
7340:23 September
7338:. Retrieved
7318:
7311:
7301:23 September
7299:. Retrieved
7279:
7269:
7259:18 September
7257:. Retrieved
7237:
7230:
7220:18 September
7218:. Retrieved
7198:
7191:
7181:18 September
7179:. Retrieved
7169:
7162:
7152:18 September
7150:. Retrieved
7130:
7123:
7113:25 September
7111:. Retrieved
7091:
7084:
7074:25 September
7072:. Retrieved
7052:
7045:
7035:25 September
7033:. Retrieved
7013:
7006:
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6994:. Retrieved
6974:
6964:
6944:
6934:
6918:
6909:
6899:25 September
6897:. Retrieved
6877:
6870:
6860:25 September
6858:. Retrieved
6838:
6828:
6818:25 September
6816:. Retrieved
6796:
6786:
6776:25 September
6774:. Retrieved
6754:
6724:
6718:
6703:
6695:
6685:10 September
6683:. Retrieved
6656:Trigonometry
6655:
6645:
6630:
6620:
6604:
6596:
6579:
6548:
6524:. Retrieved
6496:
6475:
6463:. Retrieved
6435:
6425:
6413:. Retrieved
6397:. Springer.
6393:
6383:
6355:
6324:. Retrieved
6320:the original
6306:
6294:. Retrieved
6266:
6259:
6249:
6239:
6223:(1): 61–75.
6220:
6216:
6206:
6195:the original
6190:
6177:
6161:
6138:14 September
6136:. Retrieved
6116:
6109:
6099:14 September
6097:. Retrieved
6077:
6073:I. M. Yaglom
6067:
6057:14 September
6055:. Retrieved
6035:
6028:
6018:14 September
6016:. Retrieved
5996:
5962:
5951:
5941:14 September
5939:. Retrieved
5919:
5912:
5902:14 September
5900:. Retrieved
5880:
5873:
5863:18 September
5861:. Retrieved
5841:
5834:
5824:18 September
5822:. Retrieved
5802:
5795:
5785:18 September
5783:. Retrieved
5763:
5756:
5746:18 September
5744:. Retrieved
5724:
5714:
5704:18 September
5702:. Retrieved
5682:
5675:
5665:18 September
5663:. Retrieved
5643:
5620:
5616:
5610:
5608:
5597:
5592:
5577:
5564:
5549:
5536:
5521:
5508:
5499:
5490:
5454:
5447:
5440:Hayashi 2003
5424:Hayashi 2005
5418:
5410:Śulva Sūtras
5409:
5400:
5375:
5371:
5368:Staal, Frits
5362:
5350:. Retrieved
5346:the original
5331:
5309:
5301:
5288:
5275:
5262:
5250:. Retrieved
5225:
5221:
5211:
5176:
5170:
5152:
5147:
5134:
5122:. Retrieved
5113:
5103:
5078:
5074:
5068:
5027:
5023:
5017:
5004:
4991:. Retrieved
4968:
4955:
4936:
4932:
4922:
4912:14 September
4910:. Retrieved
4894:. Elsevier.
4890:
4883:
4862:
4841:14 September
4839:. Retrieved
4819:
4812:
4800:. Retrieved
4796:
4787:
4768:
4509:
4472:
4429:
4319:Pythagoreans
4308:
4300:analytically
4296:plane curves
4278:
4239:
4220:
4190:
4177:Architecture
4157:
4149:M. C. Escher
4139:
4134:golden ratio
4131:
4127:Michelangelo
4122:
4112:
4101:
4079:
4076:Applications
4052:all studied
4044:, and later
4023:
4011:optimization
3989:
3962:Cayley graph
3959:
3929:
3924:
3920:
3880:
3850:
3817:
3735:
3713:
3694:cryptography
3691:
3679:stack theory
3661:, including
3609:
3560:
3545:
3541:trefoil knot
3447:
3435:
3413:
3405:econometrics
3387:
3353:acceleration
3341:displacement
3334:
3275:aerodynamics
3267:architecture
3245:
3221:vector space
3216:
3212:
3208:
3204:
3200:
3196:
3182:
3129:
3114:M. C. Escher
3090:
3048:real numbers
3033:
3006:
2971:
2948:straightedge
2941:
2923:
2904:
2880:
2874:
2848:, while the
2835:
2785:
2757:
2742:
2728:
2687:space curves
2683:plane curves
2672:
2668:trigonometry
2649:
2638:
2634:
2624:
2594:
2578:homeomorphic
2574:neighborhood
2559:
2537:
2483:
2475:
2471:
2467:
2463:
2456:
2452:
2448:
2442:
2438:
2434:
2430:
2424:
2420:
2416:
2412:
2378:
2374:space curves
2370:plane curves
2363:
2340:affine space
2332:
2294:
2272:
2268:
2245:
2217:János Bolyai
2202:
2198:
2181:
2152:
2092:
2045:
1982:Omar Khayyam
1943:
1938:
1911:
1901:
1882:Śulba Sūtras
1881:
1878:Hayashi 2005
1873:Sulba Sutras
1871:
1865:
1860:
1843:
1790:
1786:
1781:through the
1772:
1726:
1698:Plimpton 322
1668:construction
1653:
1617:
1613:
1585:
1563:
1538:
1512:
1496:architecture
1493:
1454:
1439:
1436:
1429:
1422:
1419:
1412:
1405:
1402:
1395:
1387:
1386:
1230:
999:Parameshvara
812:Parameshvara
582:Dodecahedron
166:Differential
31:
25:
11222:WikiProject
11065:Game theory
11045:Probability
10782:Homological
10772:Multilinear
10752:Commutative
10729:Type theory
10696:Foundations
10652:mathematics
10267:Wikiversity
10185:Pangeometry
10040:Boyer, C.B.
8806:9 September
8729:9 September
7512:Michio Kaku
7275:Ian Stewart
6970:Terence Tao
6915:Yu D Burago
6526:9 September
6465:9 September
6415:9 September
6296:9 September
5518:"Al-Mahani"
5306:Howard Eves
5252:9 September
5081:: 171–180.
4993:27 February
4482:Lobachevsky
4284:coordinates
4145:Islamic art
4117:in design.
4104:perspective
3854:deals with
3782:worldsheets
3631: 1900
3556:compactness
3478:mathematics
3345:deformation
3263:engineering
3195:, exchange
3138:congruences
3118:Felix Klein
2792:compactness
2664:unit circle
2107:Felix Klein
2080:projections
2048:coordinates
2033:John Wallis
2025: 1314
2021: 1230
1946:Middle Ages
1908:Brahmagupta
1903:Aryabhatiya
1855: 1310
1692:), and the
1656:Mesopotamia
1600:that omits
1447:mathematics
1316:Linguistics
1306:Computation
1301:Geosciences
1264:Probability
1190:Mathematics
1124:Present day
1071:Lobachevsky
1058:1700s–1900s
1015:Jyeṣṭhadeva
1007:1400s–1700s
959:Brahmagupta
782:Lobachevsky
762:Jyeṣṭhadeva
712:Brahmagupta
640:Hypersphere
612:Tetrahedron
587:Icosahedron
159:Diophantine
11050:Statistics
10929:Arithmetic
10891:Arithmetic
10757:Elementary
10724:Set theory
10527:Riemannian
10522:Projective
10507:Symplectic
10502:Hyperbolic
10435:Euclidean
10282:– Geometry
10176:1364.00004
10115:. Oxford:
9833:1004572791
6571:0945.14001
6326:20 January
6245:John Casey
5496:Boyer 1991
5406:Cooke 2005
5294:Boyer 1991
5281:Boyer 1991
5268:Boyer 1991
5140:Boyer 1991
5010:Boyer 1991
4780:References
4484:and other
4195:to create
4169:, and the
4115:proportion
4062:algorithms
4026:Archimedes
3917:free group
3858:and their
3856:algorithms
3307:similarity
3303:congruence
3279:navigation
3225:dual space
3174:Lie theory
3158:Sophus Lie
2910:similarity
2906:Congruence
2824:as in the
2719:See also:
2695:derivative
2195:postulates
2172:See also:
2029:Gersonides
2023: – c.
1970:arithmetic
1795:Archimedes
1742:Pythagoras
1696:, such as
1602:continuity
1451:arithmetic
1390:(from
1326:Philosophy
1269:Statistics
1259:Set theory
984:al-Yasamin
928:Apollonius
923:Archimedes
913:Pythagoras
903:Baudhayana
857:al-Yasamin
807:Pythagoras
702:Baudhayana
692:Archimedes
687:Apollonius
592:Octahedron
443:Hypotenuse
318:Similarity
313:Congruence
225:Incidence
176:Symplectic
171:Riemannian
154:Arithmetic
129:Projective
117:Hyperbolic
45:Projecting
10977:Geometric
10967:Algebraic
10906:Euclidean
10881:Algebraic
10777:Universal
10548:Lie group
10512:Spherical
10042:(1991) .
8796:861200296
8719:233971394
8655:209857590
8024:118964353
6637:EMS Press
6286:162589397
5604:Routledge
5392:170894641
5324:The Bible
5060:206644971
4963:(1969) .
4802:31 August
4494:Desargues
4223:astronomy
4119:Vitruvius
4034:Zenodorus
3790:spacetime
3612:algebraic
3448:extrinsic
3436:intrinsic
3299:triangles
3255:astronomy
3251:mechanics
2976:Dimension
2788:curvature
2773:integrals
2660:triangles
2550:Manifolds
2390:dimension
2266:defined.
2204:synthetic
2199:axiomatic
2052:equations
1958:Al-Mahani
1898:Aryabhata
1892:. In the
1816:with the
1706:trapezoid
1672:astronomy
1664:surveying
1543:manifolds
1531:embedding
1406:geōmetría
1399:γεωμετρία
1331:Education
1321:Economics
1296:Chemistry
1101:Minkowski
1020:Descartes
954:Aryabhata
949:Kātyāyana
880:by period
792:Minkowski
767:Kātyāyana
727:Descartes
672:Aryabhata
651:Geometers
635:Tesseract
499:Trapezoid
471:Rectangle
264:Dimension
149:Algebraic
139:Synthetic
110:Spherical
95:Euclidean
11244:Geometry
11238:Category
11198:Category
10954:Topology
10901:Discrete
10886:Analytic
10873:Geometry
10845:Discrete
10800:Calculus
10792:Analysis
10747:Abstract
10686:Glossary
10669:Timeline
10609:Category
10497:Elliptic
10489:geometry
10470:Polyform
10455:Discrete
10437:geometry
10418:Timeline
10402:Geometry
10263:geometry
10227:Geometry
10195:(2002).
10146:(2014).
10014:Archived
9942:Archived
9903:Archived
9888:Calculus
9864:Archived
9809:(2017).
9784:Archived
9745:Archived
9706:Archived
9667:Archived
9628:Archived
9589:Archived
9550:Archived
9478:Archived
9460:(2007).
9435:Archived
9396:Archived
9357:Archived
9318:Archived
9279:Archived
9237:Archived
9198:Archived
9159:Archived
9117:Archived
9078:Archived
9060:(2012).
9032:Archived
9014:(2010).
8990:Archived
8972:(2011).
8944:Archived
8905:Archived
8866:Archived
8848:(2013).
8800:Archived
8774:(2011).
8723:Archived
8659:Archived
8612:Archived
8573:Archived
8555:(2017).
8526:Archived
8487:Archived
8445:Archived
8406:Archived
8367:Archived
8328:Archived
8289:Archived
8250:Archived
8211:Archived
8172:Archived
8133:Archived
8094:Archived
8055:Archived
7985:Archived
7946:Archived
7907:Archived
7868:Archived
7822:Archived
7783:Archived
7755:Archived
7716:Archived
7656:Archived
7638:(2005).
7614:Archived
7575:Archived
7557:(2014).
7532:Archived
7514:(2012).
7490:Archived
7451:Archived
7412:Archived
7373:Archived
7334:Archived
7295:Archived
7277:(2008).
7253:Archived
7214:Archived
7175:Archived
7146:Archived
7107:Archived
7068:Archived
7029:Archived
6990:Archived
6972:(2011).
6942:(1984).
6893:Archived
6854:Archived
6842:. SIAM.
6839:Calculus
6836:(1991).
6812:Archived
6794:(2017).
6770:Archived
6702:(2012).
6679:Archived
6675:41355833
6653:(2001).
6547:(1999).
6520:Archived
6459:Archived
6433:(1979).
6409:Archived
6391:(1983).
6375:42683260
6356:Topology
6353:(2000).
6290:Archived
6247:(1885).
6132:Archived
6093:Archived
6075:(2012).
6051:Archived
6012:Archived
5994:(2013).
5959:(2014).
5935:Archived
5896:Archived
5857:Archived
5818:Archived
5779:Archived
5740:Archived
5698:Archived
5659:Archived
5641:(2012).
5482:29181926
5352:7 August
5246:Archived
5124:17 April
5118:Archived
5052:26823423
4987:Archived
4906:Archived
4835:Archived
4604:. Thus,
4431:Flatland
4360:Geometry
4338:See also
4280:Calculus
4275:lengths.
4171:cylinder
4070:lattices
3603:Quintic
3531:Topology
3525:Topology
3428:universe
3393:calculus
3376:calculus
3349:velocity
3337:position
3223:and its
3217:contains
3166:topology
3154:Clifford
3093:symmetry
3072:Symmetry
3066:Symmetry
2822:triangle
2769:calculus
2691:surfaces
2679:calculus
2656:polygons
2576:that is
2566:topology
2562:manifold
2556:Manifold
2494:topology
2396:Surfaces
2313:geodesic
2252:elements
2187:Elements
2099:symmetry
2084:sections
2064:calculus
1922:Sanskrit
1814:parabola
1791:Elements
1787:Elements
1774:Elements
1676:Egyptian
1644:European
1477:distance
1456:geometer
1388:Geometry
1245:Analysis
1241:Calculus
1231:Geometry
1091:Poincaré
1035:Minggatu
994:Yang Hui
964:Virasena
852:Yang Hui
847:Virasena
817:Poincaré
797:Minggatu
577:Cylinder
522:Diameter
481:Rhomboid
438:Altitude
429:Triangle
323:Symmetry
301:Parallel
286:Diagonal
256:Features
253:Concepts
144:Analytic
105:Elliptic
87:Branches
73:Timeline
32:Geometry
11210:Commons
10992:Applied
10962:General
10739:Algebra
10664:History
10465:Polygon
10413:History
10032:Sources
7778:Science
6627:"Angle"
6516:1529515
6455:4036464
5500:Algebra
5242:3026893
5203:1969021
5095:3822211
5032:Bibcode
5024:Science
4797:Cuemath
4292:algebra
4233:on the
4231:planets
4211:Physics
4159:Cézanne
4141:Tilings
4066:tilings
4050:Coxeter
3897:, etc.
3754:sheaves
3418:due to
3401:physics
3359:, etc.
3315:circles
3271:geodesy
3213:lies in
3193:theorem
3185:duality
3084:of the
2995:, with
2944:compass
2915:Hilbert
2881:measure
2838:metrics
2779:or the
2486:surface
2258:called
2068:physics
2014:Vitello
1944:In the
1830:volumes
1754:Eudoxus
1702:frustum
1646:and an
1628:History
1485:surface
1311:Biology
1291:Physics
1236:Algebra
1199:History
1116:Coxeter
1096:Hilbert
1081:Riemann
1030:Huygens
989:al-Tusi
979:Khayyám
969:Alhazen
936:1–1400s
837:al-Tusi
822:Riemann
772:Khayyám
757:Huygens
752:Hilbert
722:Coxeter
682:Alhazen
660:by name
597:Pyramid
476:Rhombus
420:Polygon
372:segment
220:Fractal
203:Digital
188:Complex
69:History
64:Outline
10911:Finite
10767:Linear
10674:Future
10650:Major
10517:Affine
10450:Convex
10224:about
10203:
10174:
10164:
10123:
10100:
10077:
10058:
10006:
9979:
9934:
9895:
9856:
9831:
9821:
9776:
9737:
9698:
9659:
9620:
9581:
9542:
9515:
9470:
9427:
9388:
9349:
9310:
9271:
9229:
9190:
9151:
9109:
9070:
9024:
8982:
8936:
8897:
8858:
8794:
8784:
8717:
8707:
8653:
8643:
8604:
8565:
8518:
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