geometry

 

  • This enlargement of the scope of geometry led to a change of meaning of the word “space”, which originally referred to the three-dimensional space of the physical world and
    its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined.

  • 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.

  • For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,[46] but in a more abstract
    setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

  • [43] In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described.

  • In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.

  • [71] One example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the system’s degrees of freedom.

  • 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—differential
    geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment
    of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

  • [62] 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.

  • [73] In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.

  • For example, methods of algebraic geometry are fundamental in Wiles’s proof of Fermat’s Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained
    unsolved for several centuries.

  • [80] Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,[81][82] the latter in Lie theory and Riemannian
    geometry.

  • [85] A similar and closely related form of duality exists between a vector space and its dual space.

  • The characteristic feature of Euclid’s approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.

  • 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[15] as well as a theory of ratios that avoided the problem
    of incommensurable magnitudes, which enabled subsequent geometers to make significant advances.

  • Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,[16] introduced mathematical
    rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof.

  • [66] In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where
    the measures follow rules similar to those of classical area and volume.

  • Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron’s formula), as well as a complete description of rational triangles (i.e.

  • Geometry (from Ancient Greek (geōmetría) ‘land measurement’; from (gê) ‘earth, land’ and (métron) ‘a measure’)[1] is a branch of mathematics concerned with properties of space
    such as the distance, shape, size, and relative position of figures.

  • In modern mathematics, they are generally defined as elements of a set called space, which is itself axiomatically defined.

  • [3][36] Axioms An illustration of Euclid’s parallel postulate See also: Euclidean geometry and Axiom Euclid took an abstract approach to geometry in his Elements,[37] one
    of the most influential books ever written.

  • For instance, planes can be studied as a topological surface without reference to distances or angles;[49] it can be studied as an affine space, where collinearity and ratios
    can be studied but not distances;[50] it can be studied as the complex plane using techniques of complex analysis;[51] and so on.

  • [3] Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related
    to graphics.

  • [61] Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects.

  • [78] Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations,
    geometric transformations that take straight lines into straight lines.

  • As a consequence of these major changes in the conception of geometry, the concept of “space” became something rich and varied, and the natural background for theories as
    different as complex analysis and classical mechanics.

  • [5][6] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need
    in surveying, construction, astronomy, and various crafts.

  • [75] Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers[76] and were investigated in detail before
    the time of Euclid.

  • Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry,[a] which includes the notions of point, line, plane, distance, angle, surface, and curve,
    as fundamental concepts.

  • The Koch snowflake, with fractal dimension and topological dimension=1 Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient
    world conceived of as three-dimensional space).

  • [32] Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.

  • One of the oldest such discoveries is Carl Friedrich Gauss’s Theorema Egregium (“remarkable theorem”) that asserts roughly that the Gaussian curvature of a surface is independent
    from any specific embedding in a Euclidean space.

  • However, there are modern geometries in which points are not primitive objects, or even without points.

  • [79] However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein’s idea to ‘define a geometry via its symmetry group’
    found its inspiration.

  • [72] In general topology, the concept of dimension has been extended from natural numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in
    fractal geometry).

  • [59][60] Measures: length, area, and volume Main articles: Length, Area, and Volume See also: Area § List of formulas, and Volume § Volume formulas Length, area, and volume
    describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.

  • Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.

  • [43] In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

  • They may be defined by the properties that they must have, as in Euclid’s definition as “that which has no part”,[43] or in synthetic geometry.

  • [4] Geometry also has applications in areas of mathematics that are apparently unrelated.

  • Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.

  • [52] In topology, a curve is defined by a function from an interval of the real numbers to another space.

  • [68] 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.

  • [33] Two developments in geometry in the 19th century changed the way it had been studied previously.

  • [35] Main concepts The following are some of the most important concepts in geometry.

  • Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between
    complex geometry and algebraic geometry.

  • [110][111][112] Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and has found applications
    to string theory and mirror symmetry.

  • Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi–Yau manifolds, and these spaces find uses in string theory.

  • [147] Since the 19th century, geometry has been used for solving problems in number theory, for example through the geometry of numbers or, more recently, scheme theory, which
    is used in Wiles’s proof of Fermat’s Last Theorem.

  • [edit] Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms
    closely related to those that specify Euclidean geometry.

  • [30] For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such
    as plane curves could now be represented analytically in the form of functions and equations.

  • [106] From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme
    theory, which allows using topological methods, including cohomology theories in a purely algebraic context.

  • Wiles’ proof of Fermat’s Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory.

  • [113] Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.

  • [137][138] Applications of geometry to architecture include the use of projective geometry to create forced perspective,[139] the use of conic sections in constructing domes
    and similar objects,[90] the use of tessellations,[90] and the use of symmetry.

  • [107] Algebraic geometry has applications in many areas, including cryptography[108] and string theory.

  • [125] Geometric group theory Main article: Geometric group theory The Cayley graph of the free group on two generators a and b Geometric group theory uses large-scale geometric
    techniques to study finitely generated groups.

  • [90] Physics Main article: Mathematical physics The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and
    describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.

  • [109] Complex geometry Main article: Complex geometry Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane.

  • [114][115][116] Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s.

  • [99] Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric,
    which determines how distances are measured near each point) or extrinsic (where the object under study is a part of some ambient flat Euclidean space).

  • Discrete geometry Main article: Discrete geometry Discrete geometry includes the study of various sphere packings.

  • [129] It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.

  • [141] String theory makes use of several variants of geometry,[142] as does quantum information theory.

  • Applications Geometry has found applications in many fields, some of which are described below.

  • This is still used in art theory today, although the exact list of shapes varies from author to author.

  • It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles.

 

Works Cited

[‘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 hyperbolic geometry by Lobachevsky and other non-Euclidean geometries
by Gauss and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of Desargues in the 17th century, all the way back to the implicit use of spherical geometry to understand the Earth
geodesy and to navigate the oceans since antiquity.
2. ^ Pythagorean triples are triples of integers with the property: . Thus, , , etc.
3. ^ The ancient Greeks had some constructions using other instruments.
4. Boyer, C.B. (1991) [1989]. A
History of Mathematics (Second edition, revised by Uta C. Merzbach ed.). New York: Wiley. ISBN 978-0-471-54397-8.
5. Cooke, Roger (2005). The History of Mathematics. New York: Wiley-Interscience. ISBN 978-0-471-44459-6.
6. Hayashi, Takao (2003).
“Indian Mathematics”. In Grattan-Guinness, Ivor (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. Baltimore, MD: The Johns Hopkins University Press. pp. 118–130. ISBN 978-0-8018-7396-6.
7. Hayashi,
Takao (2005). “Indian Mathematics”. In Flood, Gavin (ed.). The Blackwell Companion to Hinduism. Oxford: Basil Blackwell. pp. 360–375. ISBN 978-1-4051-3251-0.
8. “Geometry – Formulas, Examples | Plane and Solid Geometry”. Cuemath. Retrieved 31 August
2023.
9. ^ Vincenzo De Risi (2015). Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age. Birkhäuser. pp. 1–. ISBN 978-3-319-12102-4. Archived from the original on 20 February 2021. Retrieved 14 September 2019.
10. ^
Jump up to:a b Tabak, John (2014). Geometry: the language of space and form. Infobase Publishing. p. xiv. ISBN 978-0-8160-4953-0.
11. ^ Walter A. Meyer (2006). Geometry and Its Applications. Elsevier. ISBN 978-0-08-047803-6. Archived from the original
on 1 September 2021. Retrieved 14 September 2019.
12. ^ Friberg, Jöran (1981). “Methods and traditions of Babylonian mathematics”. Historia Mathematica. 8 (3): 277–318. doi:10.1016/0315-0860(81)90069-0.
13. ^ Neugebauer, Otto (1969) [1957]. “Chap.
IV Egyptian Mathematics and Astronomy”. The Exact Sciences in Antiquity (2 ed.). Dover Publications. pp. 71–96. ISBN 978-0-486-22332-2. Archived from the original on 14 August 2020. Retrieved 27 February 2021..
14. ^ (Boyer 1991, “Egypt” p. 19)
15. ^
Ossendrijver, Mathieu (29 January 2016). “Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph”. Science. 351 (6272): 482–484. Bibcode:2016Sci…351..482O. doi:10.1126/science.aad8085. PMID 26823423.
S2CID 206644971.
16. ^ Depuydt, Leo (1 January 1998). “Gnomons at Meroë and Early Trigonometry”. The Journal of Egyptian Archaeology. 84: 171–180. doi:10.2307/3822211. JSTOR 3822211.
17. ^ Slayman, Andrew (27 May 1998). “Neolithic Skywatchers”.
Archaeology Magazine Archive. Archived from the original on 5 June 2011. Retrieved 17 April 2011.
18. ^ (Boyer 1991, “Ionia and the Pythagoreans” p. 43)
19. ^ Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0.
20. ^
Kurt Von Fritz (1945). “The Discovery of Incommensurability by Hippasus of Metapontum”. Classics in the History of Greek Mathematics. Annals of Mathematics; Boston Studies in the Philosophy of Science. Vol. 240. pp. 211–231. doi:10.1007/978-1-4020-2640-9_11.
ISBN 978-90-481-5850-8. JSTOR 1969021.
21. ^ James R. Choike (1980). “The Pentagram and the Discovery of an Irrational Number”. The Two-Year College Mathematics Journal. 11 (5): 312–316. doi:10.2307/3026893. JSTOR 3026893. Archived from the original
on 9 September 2022. Retrieved 9 September 2022.
22. ^ (Boyer 1991, “The Age of Plato and Aristotle” p. 92)
23. ^ (Boyer 1991, “Euclid of Alexandria” p. 119)
24. ^ (Boyer 1991, “Euclid of Alexandria” p. 104)
25. ^ Howard Eves, An Introduction
to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: “No work, except The Bible, has been more widely used….”
26. ^ O’Connor, J.J.; Robertson, E.F. (February 1996). “A history of calculus”. University of St Andrews. Archived
from the original on 15 July 2007. Retrieved 7 August 2007.
27. ^ Staal, Frits (1999). “Greek and Vedic Geometry”. Journal of Indian Philosophy. 27 (1–2): 105–127. doi:10.1023/A:1004364417713. S2CID 170894641.
28. ^ (Cooke 2005, p. 198): “The
arithmetic content of the Śulva Sūtras 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 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.”
29. ^ (Hayashi 2005, p. 371)
30. ^ Jump up to:a b (Hayashi 2003, pp. 121–122)
31. ^ Rāshid, Rushdī
(1994). The development of Arabic mathematics : between arithmetic and algebra. Boston Studies in the Philosophy of Science. Vol. 156. p. 35. doi:10.1007/978-94-017-3274-1. ISBN 978-0-7923-2565-9. OCLC 29181926.
32. ^ (Boyer 1991, “The Arabic Hegemony”
pp. 241–242) “Omar Khayyam (c. 1050–1123), the “tent-maker,” wrote an Algebra 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 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.””.
33. ^ O’Connor, John J.; Robertson, Edmund F. “Al-Mahani”. MacTutor History of Mathematics Archive. University of St Andrews.
34. ^ O’Connor,
John J.; Robertson, Edmund F. “Al-Sabi Thabit ibn Qurra al-Harrani”. MacTutor History of Mathematics Archive. University of St Andrews.
35. ^ O’Connor, John J.; Robertson, Edmund F. “Omar Khayyam”. MacTutor History of Mathematics Archive. University
of St Andrews.
36. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), “Geometry”, in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447–494 [470], Routledge, London and New York:
“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 Book of
Optics (Kitab al-Manazir)—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 Pseudo-Tusi’s Exposition of Euclid had stimulated both J. Wallis’s and G. Saccheri’s studies of the theory of parallel lines.”
37. ^ Jump up to:a b Carl B. Boyer (2012). History
of Analytic Geometry. Courier Corporation. ISBN 978-0-486-15451-0. Archived from the original on 26 December 2019. Retrieved 18 September 2019.
38. ^ C. H. Edwards Jr. (2012). The Historical Development of the Calculus. Springer Science & Business
Media. p. 95. ISBN 978-1-4612-6230-5. Archived from the original on 29 December 2019. Retrieved 18 September 2019.
39. ^ Judith V. Field; Jeremy Gray (2012). The Geometrical Work of Girard Desargues. Springer Science & Business Media. p. 43. ISBN
978-1-4613-8692-6. Archived from the original on 27 December 2019. Retrieved 18 September 2019.
40. ^ C. R. Wylie (2011). Introduction to Projective Geometry. Courier Corporation. ISBN 978-0-486-14170-1. Archived from the original on 28 December
2019. Retrieved 18 September 2019.
41. ^ Jeremy Gray (2011). Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. Springer Science & Business Media. ISBN 978-0-85729-060-1. Archived from the original on 7 December 2019.
Retrieved 18 September 2019.
42. ^ Eduardo Bayro-Corrochano (2018). Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing. Springer. p. 4. ISBN 978-3-319-74830-6. Archived from the original on 28 December 2019. Retrieved
18 September 2019.
43. ^ Morris Kline (1990). Mathematical Thought From Ancient to Modern Times: Volume 3. US: Oxford University Press. pp. 1010–. ISBN 978-0-19-506137-6. Archived from the original on 1 September 2021. Retrieved 14 September 2019.
44. ^
Victor J. Katz (2000). Using History to Teach Mathematics: An International Perspective. Cambridge University Press. pp. 45–. ISBN 978-0-88385-163-0. Archived from the original on 1 September 2021. Retrieved 14 September 2019.
45. ^ David Berlinski
(2014). The King of Infinite Space: Euclid and His Elements. Basic Books. ISBN 978-0-465-03863-3.
46. ^ Jump up to:a b Robin Hartshorne (2013). Geometry: Euclid and Beyond. Springer Science & Business Media. pp. 29–. ISBN 978-0-387-22676-7. Archived
from the original on 1 September 2021. Retrieved 14 September 2019.
47. ^ Pat Herbst; Taro Fujita; Stefan Halverscheid; Michael Weiss (2017). The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective. Taylor & Francis. pp.
20–. ISBN 978-1-351-97353-3. Archived from the original on 1 September 2021. Retrieved 14 September 2019.
48. ^ I. M. Yaglom (2012). A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean
Principle of Relativity. Springer Science & Business Media. pp. 6–. ISBN 978-1-4612-6135-3. Archived from the original on 1 September 2021. Retrieved 14 September 2019.
49. ^ Audun Holme (2010). Geometry: Our Cultural Heritage. Springer Science
& Business Media. pp. 254–. ISBN 978-3-642-14441-7. Archived from the original on 1 September 2021. Retrieved 14 September 2019.
50. ^ Jump up to:a b c d e Euclid’s Elements – All thirteen books in one volume, Based on Heath’s translation, Green
Lion Press ISBN 1-888009-18-7.
51. ^ Gerla, G. (1995). “Pointless Geometries” (PDF). In Buekenhout, F.; Kantor, W. (eds.). Handbook of incidence geometry: buildings and foundations. North-Holland. pp. 1015–1031. Archived from the original (PDF)
on 17 July 2011.
52. ^ Clark, Bowman L. (January 1985). “Individuals and Points”. Notre Dame Journal of Formal Logic. 26 (1): 61–75. doi:10.1305/ndjfl/1093870761.
53. ^ John Casey (1885). Analytic Geometry of the Point, Line, Circle, and Conic
Sections.
54. ^ Francis Buekenhout, ed. (1995). Handbook of incidence geometry : buildings and foundations. Amsterdam: Elsevier. ISBN 978-0-444-88355-1. OCLC 162589397. Archived from the original on 1 March 2023. Retrieved 9 September 2022.
55. ^
“geodesic – definition of geodesic in English from the Oxford dictionary”. OxfordDictionaries.com. Archived from the original on 15 July 2016. Retrieved 20 January 2016.
56. ^ Jump up to:a b c d e Munkres, James R. (2000). Topology. Vol. 2 (2nd
ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 0-13-181629-2. OCLC 42683260.
57. ^ Szmielew, Wanda (1983). From Affine to Euclidean Geometry. Springer. ISBN 978-90-277-1243-1. Archived from the original on 1 March 2023. Retrieved 9 September
2022.
58. ^ Ahlfors, Lars V. (1979). Complex analysis : an introduction to the theory of analytic functions of one complex variable (3rd ed.). New York: McGraw-Hill. ISBN 9780070006577. OCLC 4036464. Archived from the original on 1 March 2023. Retrieved
9 September 2022.
59. ^ Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.
60. ^ Jump up to:a b c Carmo, Manfredo Perdigão do (1976). Differential geometry of curves and surfaces. Vol. 2. Englewood Cliffs, N.J.: Prentice-Hall.
ISBN 0-13-212589-7. OCLC 1529515. Archived from the original on 1 March 2023. Retrieved 9 September 2022.
61. ^ Jump up to:a b Mumford, David (1999). The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians
(2nd ed.). Springer-Verlag. ISBN 978-3-540-63293-1. Zbl 0945.14001.
62. ^ Briggs, William L., and Lyle Cochran Calculus. “Early Transcendentals.” ISBN 978-0-321-57056-7.
63. ^ Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String
Theory and the Geometry of the Universe’s Hidden Dimensions. Basic Books. ISBN 978-0-465-02023-2.
64. ^ Sidorov, L.A. (2001) [1994]. “Angle”. Encyclopedia of Mathematics. EMS Press.
65. ^ Gelʹfand, I. M. (2001). Trigonometry. Mark E. Saul. Boston:
Birkhäuser. pp. 1–20. ISBN 0-8176-3914-4. OCLC 41355833. Archived from the original on 1 March 2023. Retrieved 10 September 2022.
66. ^ Stewart, James (2012). Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning. ISBN 978-0-538-49790-9
67. ^
Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 978-3-540-42627-1..
68. ^ Jump up to:a b Steven A. Treese (2018). History and Measurement of the Base and Derived Units. Springer International Publishing.
pp. 101–. ISBN 978-3-319-77577-7. Archived from the original on 30 December 2019. Retrieved 25 September 2019.
69. ^ James W. Cannon (2017). Geometry of Lengths, Areas, and Volumes. American Mathematical Soc. p. 11. ISBN 978-1-4704-3714-5. Archived
from the original on 31 December 2019. Retrieved 25 September 2019.
70. ^ Gilbert Strang (1991). Calculus. SIAM. ISBN 978-0-9614088-2-4. Archived from the original on 24 December 2019. Retrieved 25 September 2019.
71. ^ H. S. Bear (2002). A Primer
of Lebesgue Integration. Academic Press. ISBN 978-0-12-083971-1. Archived from the original on 25 December 2019. Retrieved 25 September 2019.
72. ^ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society,
2001, ISBN 0-8218-2129-6.
73. ^ Wald, Robert M. (1984). General Relativity. University of Chicago Press. ISBN 978-0-226-87033-5.
74. ^ Terence Tao (2011). An Introduction to Measure Theory. American Mathematical Soc. ISBN 978-0-8218-6919-2. Archived
from the original on 27 December 2019. Retrieved 25 September 2019.
75. ^ Shlomo Libeskind (2008). Euclidean and Transformational Geometry: A Deductive Inquiry. Jones & Bartlett Learning. p. 255. ISBN 978-0-7637-4366-6. Archived from the original
on 25 December 2019. Retrieved 25 September 2019.
76. ^ Mark A. Freitag (2013). Mathematics for Elementary School Teachers: A Process Approach. Cengage Learning. p. 614. ISBN 978-0-618-61008-2. Archived from the original on 28 December 2019. Retrieved
25 September 2019.
77. ^ George E. Martin (2012). Transformation Geometry: An Introduction to Symmetry. Springer Science & Business Media. ISBN 978-1-4612-5680-9. Archived from the original on 7 December 2019. Retrieved 25 September 2019.
78. ^
Mark Blacklock (2018). The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle. Oxford University Press. ISBN 978-0-19-875548-7. Archived from the original on 27 December 2019. Retrieved 18 September 2019.
79. ^ Charles
Jasper Joly (1895). Papers. The Academy. pp. 62–. Archived from the original on 27 December 2019. Retrieved 18 September 2019.
80. ^ Roger Temam (2013). Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer Science & Business
Media. p. 367. ISBN 978-1-4612-0645-3. Archived from the original on 24 December 2019. Retrieved 18 September 2019.
81. ^ Bill Jacob; Tsit-Yuen Lam (1994). Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD
Year, Berkeley, 1990–1991. American Mathematical Soc. p. 111. ISBN 978-0-8218-5154-8. Archived from the original on 28 December 2019. Retrieved 18 September 2019.
82. ^ Ian Stewart (2008). Why Beauty Is Truth: A History of Symmetry. Basic Books.
p. 14. ISBN 978-0-465-08237-7. Archived from the original on 25 December 2019. Retrieved 23 September 2019.
83. ^ Stakhov Alexey (2009). Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science. World Scientific. p. 144.
ISBN 978-981-4472-57-9. Archived from the original on 29 December 2019. Retrieved 23 September 2019.
84. ^ Werner Hahn (1998). Symmetry as a Developmental Principle in Nature and Art. World Scientific. ISBN 978-981-02-2363-2. Archived from the original
on 1 January 2020. Retrieved 23 September 2019.
85. ^ Brian J. Cantwell (2002). Introduction to Symmetry Analysis. Cambridge University Press. p. 34. ISBN 978-1-139-43171-2. Archived from the original on 27 December 2019. Retrieved 23 September
2019.
86. ^ B. Rosenfeld; Bill Wiebe (2013). Geometry of Lie Groups. Springer Science & Business Media. pp. 158ff. ISBN 978-1-4757-5325-7. Archived from the original on 24 December 2019. Retrieved 23 September 2019.
87. ^ Peter Pesic (2007). Beyond
Geometry: Classic Papers from Riemann to Einstein. Courier Corporation. ISBN 978-0-486-45350-7. Archived from the original on 1 September 2021. Retrieved 23 September 2019.
88. ^ Michio Kaku (2012). Strings, Conformal Fields, and Topology: An Introduction.
Springer Science & Business Media. p. 151. ISBN 978-1-4684-0397-8. Archived from the original on 24 December 2019. Retrieved 23 September 2019.
89. ^ Mladen Bestvina; Michah Sageev; Karen Vogtmann (2014). Geometric Group Theory. American Mathematical
Soc. p. 132. ISBN 978-1-4704-1227-2. Archived from the original on 29 December 2019. Retrieved 23 September 2019.
90. ^ W-H. Steeb (1996). Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra. World Scientific Publishing
Company. ISBN 978-981-310-503-4. Archived from the original on 26 December 2019. Retrieved 23 September 2019.
91. ^ Charles W. Misner (2005). Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland:
Papers in Honor of Charles Misner. Cambridge University Press. p. 272. ISBN 978-0-521-02139-5. Archived from the original on 26 December 2019. Retrieved 23 September 2019.
92. ^ Linnaeus Wayland Dowling (1917). Projective Geometry. McGraw-Hill book
Company, Incorporated. p. 10.
93. ^ G. Gierz (2006). Bundles of Topological Vector Spaces and Their Duality. Springer. p. 252. ISBN 978-3-540-39437-2. Archived from the original on 27 December 2019. Retrieved 23 September 2019.
94. ^ Robert E.
Butts; J.R. Brown (2012). Constructivism and Science: Essays in Recent German Philosophy. Springer Science & Business Media. pp. 127–. ISBN 978-94-009-0959-5. Archived from the original on 1 September 2021. Retrieved 20 September 2019.
95. ^ Science.
Moses King. 1886. pp. 181–. Archived from the original on 27 December 2019. Retrieved 20 September 2019.
96. ^ W. Abbot (2013). Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students. Springer Science & Business
Media. pp. 6–. ISBN 978-94-017-2742-6. Archived from the original on 25 December 2019. Retrieved 20 September 2019.
97. ^ Jump up to:a b c d George L. Hersey (2001). Architecture and Geometry in the Age of the Baroque. University of Chicago Press.
ISBN 978-0-226-32783-9. Archived from the original on 25 December 2019. Retrieved 20 September 2019.
98. ^ P. Vanícek; E.J. Krakiwsky (2015). Geodesy: The Concepts. Elsevier. p. 23. ISBN 978-1-4832-9079-9. Archived from the original on 31 December
2019. Retrieved 20 September 2019.
99. ^ Russell M. Cummings; Scott A. Morton; William H. Mason; David R. McDaniel (2015). Applied Computational Aerodynamics. Cambridge University Press. p. 449. ISBN 978-1-107-05374-8. Archived from the original
on 1 September 2021. Retrieved 20 September 2019.
100. ^ Roy Williams (1998). Geometry of Navigation. Horwood Pub. ISBN 978-1-898563-46-4. Archived from the original on 7 December 2019. Retrieved 20 September 2019.
101. ^ Schmidt, W.; Houang,
R.; Cogan, Leland S. (2002). “A Coherent Curriculum: The Case of Mathematics”. The American Educator. 26 (2): 10–26. S2CID 118964353.
102. ^ Gerard Walschap (2015). Multivariable Calculus and Differential Geometry. De Gruyter. ISBN 978-3-11-036954-0.
Archived from the original on 27 December 2019. Retrieved 23 September 2019.
103. ^ Harley Flanders (2012). Differential Forms with Applications to the Physical Sciences. Courier Corporation. ISBN 978-0-486-13961-6. Archived from the original on
1 September 2021. Retrieved 23 September 2019.
104. ^ Paul Marriott; Mark Salmon (2000). Applications of Differential Geometry to Econometrics. Cambridge University Press. ISBN 978-0-521-65116-5. Archived from the original on 1 September 2021. Retrieved
23 September 2019.
105. ^ Matthew He; Sergey Petoukhov (2011). Mathematics of Bioinformatics: Theory, Methods and Applications. John Wiley & Sons. p. 106. ISBN 978-1-118-09952-0. Archived from the original on 27 December 2019. Retrieved 23 September
2019.
106. ^ P.A.M. Dirac (2016). General Theory of Relativity. Princeton University Press. ISBN 978-1-4008-8419-3. Archived from the original on 26 December 2019. Retrieved 23 September 2019.
107. ^ Nihat Ay; Jürgen Jost; Hông Vân Lê; Lorenz
Schwachhöfer (2017). Information Geometry. Springer. p. 185. ISBN 978-3-319-56478-4. Archived from the original on 24 December 2019. Retrieved 23 September 2019.
108. ^ Martin D. Crossley (2011). Essential Topology. Springer Science & Business Media.
ISBN 978-1-85233-782-7. Archived from the original on 28 December 2019. Retrieved 24 September 2019.
109. ^ Charles Nash; Siddhartha Sen (1988). Topology and Geometry for Physicists. Elsevier. p. 1. ISBN 978-0-08-057085-3. Archived from the original
on 26 December 2019. Retrieved 24 September 2019.
110. ^ George E. Martin (1996). Transformation Geometry: An Introduction to Symmetry. Springer Science & Business Media. ISBN 978-0-387-90636-2. Archived from the original on 22 December 2019. Retrieved
24 September 2019.
111. ^ J. P. May (1999). A Concise Course in Algebraic Topology. University of Chicago Press. ISBN 978-0-226-51183-2. Archived from the original on 23 December 2019. Retrieved 24 September 2019.
112. ^ Robin Hartshorne (2013).
Algebraic Geometry. Springer Science & Business Media. ISBN 978-1-4757-3849-0. Archived from the original on 27 December 2019. Retrieved 24 September 2019.
113. ^ Jump up to:a b Jean Dieudonné (1985). History of Algebraic Geometry. Translated by
Judith D. Sally. CRC Press. ISBN 978-0-412-99371-8. Archived from the original on 25 December 2019. Retrieved 24 September 2019.
114. ^ James Carlson; James A. Carlson; Arthur Jaffe; Andrew Wiles (2006). The Millennium Prize Problems. American Mathematical
Soc. ISBN 978-0-8218-3679-8. Archived from the original on 30 May 2016. Retrieved 24 September 2019.
115. ^ Everett W. Howe; Kristin E. Lauter; Judy L. Walker (2017). Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA,
February 2016. Springer. ISBN 978-3-319-63931-4. Archived from the original on 27 December 2019. Retrieved 24 September 2019.
116. ^ Marcos Marino; Michael Thaddeus; Ravi Vakil (2008). 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. Springer. ISBN 978-3-540-79814-9. Archived from the original on 27 December 2019. Retrieved 24 September 2019.
117. ^ Huybrechts, Daniel (2005). Complex geometry
: an introduction. Berlin: Springer. ISBN 9783540266877. OCLC 209857590. Archived from the original on 1 March 2023. Retrieved 10 September 2022.
118. ^ Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.
119. ^
Wells, R. O. Jr. (2008). Differential analysis on complex manifolds. Graduate Texts in Mathematics. Vol. 65. O. García-Prada (3rd ed.). New York: Springer-Verlag. doi:10.1007/978-0-387-73892-5. ISBN 9780387738918. OCLC 233971394. Archived from the
original on 1 March 2023. Retrieved 9 September 2022.
120. ^ Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., … & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.
121. ^ Forster, O. (2012). Lectures
on Riemann surfaces (Vol. 81). Springer Science & Business Media.
122. ^ Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.
123. ^ Donaldson, S. K. (2011). Riemann surfaces. Oxford: Oxford University
Press. ISBN 978-0-19-154584-9. OCLC 861200296. Archived from the original on 1 March 2023. Retrieved 9 September 2022.
124. ^ Serre, J. P. (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.
125. ^ Serre, J. P. (1956). Géométrie
algébrique et géométrie analytique. In Annales de l’Institut Fourier (vol. 6, pp. 1–42).
126. ^ Jiří Matoušek (2013). Lectures on Discrete Geometry. Springer Science & Business Media. ISBN 978-1-4613-0039-7. Archived from the original on 27 December
2019. Retrieved 25 September 2019.
127. ^ Chuanming Zong (2006). The Cube – A Window to Convex and Discrete Geometry. Cambridge University Press. ISBN 978-0-521-85535-8. Archived from the original on 23 December 2019. Retrieved 25 September 2019.
128. ^
Peter M. Gruber (2007). Convex and Discrete Geometry. Springer Science & Business Media. ISBN 978-3-540-71133-9. Archived from the original on 24 December 2019. Retrieved 25 September 2019.
129. ^ Satyan L. Devadoss; Joseph O’Rourke (2011). Discrete
and Computational Geometry. Princeton University Press. ISBN 978-1-4008-3898-1. Archived from the original on 27 December 2019. Retrieved 25 September 2019.
130. ^ Károly Bezdek (2010). Classical Topics in Discrete Geometry. Springer Science & Business
Media. ISBN 978-1-4419-0600-7. Archived from the original on 28 December 2019. Retrieved 25 September 2019.
131. ^ Franco P. Preparata; Michael I. Shamos (2012). Computational Geometry: An Introduction. Springer Science & Business Media. ISBN 978-1-4612-1098-6.
Archived from the original on 28 December 2019. Retrieved 25 September 2019.
132. ^ Xianfeng David Gu; Shing-Tung Yau (2008). Computational Conformal Geometry. International Press. ISBN 978-1-57146-171-1. Archived from the original on 24 December
2019. Retrieved 25 September 2019.
133. ^ Jump up to:a b Clara Löh (2017). Geometric Group Theory: An Introduction. Springer. ISBN 978-3-319-72254-2. Archived from the original on 29 December 2019. Retrieved 25 September 2019.
134. ^ John Morgan;
Gang Tian (2014). The Geometrization Conjecture. American Mathematical Soc. ISBN 978-0-8218-5201-9. Archived from the original on 24 December 2019. Retrieved 25 September 2019.
135. ^ Daniel T. Wise (2012). From Riches to Raags: 3-Manifolds, Right-Angled
Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry. American Mathematical Soc. ISBN 978-0-8218-8800-1. Archived from the original on 28 December 2019. Retrieved 25 September 2019.
136. ^ Jump up to:a
b Gerard Meurant (2014). Handbook of Convex Geometry. Elsevier Science. ISBN 978-0-08-093439-6. Archived from the original on 1 September 2021. Retrieved 24 September 2019.
137. ^ Jürgen Richter-Gebert (2011). Perspectives on Projective Geometry:
A Guided Tour Through Real and Complex Geometry. Springer Science & Business Media. ISBN 978-3-642-17286-1. Archived from the original on 29 December 2019. Retrieved 25 September 2019.
138. ^ Kimberly Elam (2001). Geometry of Design: Studies in
Proportion and Composition. Princeton Architectural Press. ISBN 978-1-56898-249-6. Archived from the original on 31 December 2019. Retrieved 25 September 2019.
139. ^ Brad J. Guigar (2004). The Everything Cartooning Book: Create Unique And Inspired
Cartoons For Fun And Profit. Adams Media. pp. 82–. ISBN 978-1-4405-2305-2. Archived from the original on 27 December 2019. Retrieved 25 September 2019.
140. ^ Mario Livio (2008). The Golden Ratio: The Story of PHI, the World’s Most Astonishing
Number. Crown/Archetype. p. 166. ISBN 978-0-307-48552-6. Archived from the original on 30 December 2019. Retrieved 25 September 2019.
141. ^ Michele Emmer; Doris Schattschneider (2007). M. C. Escher’s Legacy: A Centennial Celebration. Springer.
p. 107. ISBN 978-3-540-28849-7. Archived from the original on 22 December 2019. Retrieved 25 September 2019.
142. ^ Robert Capitolo; Ken Schwab (2004). Drawing Course 101. Sterling Publishing Company, Inc. p. 22. ISBN 978-1-4027-0383-6.
143. ^
Phyllis Gelineau (2011). Integrating the Arts Across the Elementary School Curriculum. Cengage Learning. p. 55. ISBN 978-1-111-30126-2. Archived from the original on 7 December 2019. Retrieved 25 September 2019.
144. ^ Cristiano Ceccato; Lars Hesselgren;
Mark Pauly; Helmut Pottmann, Johannes Wallner (2016). Advances in Architectural Geometry 2010. Birkhäuser. p. 6. ISBN 978-3-99043-371-3. Archived from the original on 25 December 2019. Retrieved 25 September 2019.
145. ^ Helmut Pottmann (2007).
Architectural geometry. Bentley Institute Press. ISBN 978-1-934493-04-5. Archived from the original on 24 December 2019. Retrieved 25 September 2019.
146. ^ Marian Moffett; Michael W. Fazio; Lawrence Wodehouse (2003). A World History of Architecture.
Laurence King Publishing. p. 371. ISBN 978-1-85669-371-4. Archived from the original on 27 December 2019. Retrieved 25 September 2019.
147. ^ Robin M. Green; Robin Michael Green (1985). Spherical Astronomy. Cambridge University Press. p. 1. ISBN
978-0-521-31779-5. Archived from the original on 21 December 2019. Retrieved 25 September 2019.
148. ^ Dmitriĭ Vladimirovich Alekseevskiĭ (2008). Recent Developments in Pseudo-Riemannian Geometry. European Mathematical Society. ISBN 978-3-03719-051-7.
Archived from the original on 28 December 2019. Retrieved 25 September 2019.
149. ^ Shing-Tung Yau; Steve Nadis (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions. Basic Books. ISBN 978-0-465-02266-3.
Archived from the original on 24 December 2019. Retrieved 25 September 2019.
150. ^ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement (2nd ed.). Cambridge University Press. ISBN 978-1-107-02625-4.
OCLC 1004572791.
151. ^ Harley Flanders; Justin J. Price (2014). Calculus with Analytic Geometry. Elsevier Science. ISBN 978-1-4832-6240-6. Archived from the original on 24 December 2019. Retrieved 25 September 2019.
152. ^ Jon Rogawski; Colin
Adams (2015). Calculus. W. H. Freeman. ISBN 978-1-4641-7499-5. Archived from the original on 1 January 2020. Retrieved 25 September 2019.
153. ^ Álvaro Lozano-Robledo (2019). Number Theory and Geometry: An Introduction to Arithmetic Geometry. American
Mathematical Soc. ISBN 978-1-4704-5016-8. Archived from the original on 27 December 2019. Retrieved 25 September 2019.
154. ^ Arturo Sangalli (2009). Pythagoras’ Revenge: A Mathematical Mystery. Princeton University Press. p. 57. ISBN 978-0-691-04955-7.
155. ^
Gary Cornell; Joseph H. Silverman; Glenn Stevens (2013). Modular Forms and Fermat’s Last Theorem. Springer Science & Business Media. ISBN 9 Photo credit: https://www.flickr.com/photos/joceykinghorn/13231224664/’]