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Journal articles on the topic 'Geometry non-Euclidean geometry'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Gardiner, Tony, and H. S. M. Coxeter. "Non-Euclidean Geometry." Mathematical Gazette 86, no. 506 (2002): 364. http://dx.doi.org/10.2307/3621907.

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2

Stewart, Ian. "Geometry: Non-euclidean kaleidoscopes." Nature 323, no. 6084 (1986): 114. http://dx.doi.org/10.1038/323114a0.

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3

Bellot, F., and Eugene E. Krause. "Taxicab Geometry: An Adventure in Non-Euclidean Geometry." Mathematical Gazette 72, no. 461 (1988): 255. http://dx.doi.org/10.2307/3618288.

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4

Posamentier, Alfred S. "Delving Deeper: Trisecting the Circle: A Case for Euclidean Geometry." Mathematics Teacher 99, no. 6 (2006): 414–18. http://dx.doi.org/10.5951/mt.99.6.0414.

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As an undergraduate mathematics major, a prospective teacher usually takes at least one geometry course. Typically, these courses focus on non–Euclidean geometry (sometimes presented as Modern Geometry), or vectors, transformations, or topology. Instead, we at the City College of New York offer a course on more advanced Euclidean geometry in which prospective teachers investigate a plethora of geometric theorems (or relationships) that enrich their understanding of Euclidean geometry and, consequently, their teaching of it.

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5

Sun, B. W., L. T. Jiang, H. Pan, and H. Zhu. "Realization on Fractal Interpolation of Non-Rule Geometry." Key Engineering Materials 392-394 (October 2008): 523–25. http://dx.doi.org/10.4028/www.scientific.net/kem.392-394.523.

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The traditional geometric modeling is generally described by means of Euclidean geometry, and objects for the geometric modeling are usually artificial work-pieces with smooth and regular contour. However in real world, there are so many irregular geometric objects(such as cavernous body, geological body, rough surface body and so on) with extremely complicated structure that the constructing method based on Euclidean geometry equation has been already helpless, while the process constructing method based on fractal geometry can. Taking rough surface body as examples, in order to explore a met

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6

Barreto, Mylane dos Santos, and Salvador Tavares. "From the myth of Euclidean Geometry to the teaching of Non-Euclidean Geometry." Revista Vértices 9, no. 1 (2007): 73–81. http://dx.doi.org/10.5935/1809-2667.20070007.

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7

Darke, Ian P., and Patrick J. Ryan. "Euclidean and Non-Euclidean Geometry: An Analytic Approach." Mathematical Gazette 71, no. 458 (1987): 349. http://dx.doi.org/10.2307/3617111.

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8

Sunada, T. "Euclidean versus Non-Euclidean Aspects in Spectral Geometry." Progress of Theoretical Physics Supplement 116 (May 16, 2013): 235–50. http://dx.doi.org/10.1143/ptp.116.235.

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9

Sunada, Toshikazu. "Euclidean versus Non-Euclidean Aspects in Spectral Geometry." Progress of Theoretical Physics Supplement 116 (1994): 235–50. http://dx.doi.org/10.1143/ptps.116.235.

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10

Popov, A. "Non-Euclidean geometry and differential equations." Banach Center Publications 33, no. 1 (1996): 297–308. http://dx.doi.org/10.4064/-33-1-297-308.

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11

Storer, W. O., and B. A. Rosenfeld. "The History of Non-Euclidean Geometry." Mathematical Gazette 74, no. 468 (1990): 203. http://dx.doi.org/10.2307/3619413.

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12

BEESON, MICHAEL, PIERRE BOUTRY, and JULIEN NARBOUX. "HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY." Bulletin of Symbolic Logic 21, no. 2 (2015): 111–22. http://dx.doi.org/10.1017/bsl.2015.6.

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AbstractWe use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.

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13

Hansen, Vagn Lundsgaard. "The dawn of non‐Euclidean geometry." International Journal of Mathematical Education in Science and Technology 28, no. 1 (1997): 3–23. http://dx.doi.org/10.1080/0020739970280102.

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14

Blum, Zoltan, Sven Lidin, and Ronnie Thomasson. "Zeolites: Coneyers of non-euclidean geometry." Journal of Solid State Chemistry 74, no. 2 (1988): 353–55. http://dx.doi.org/10.1016/0022-4596(88)90365-9.

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15

Suter, Beth. "Non-Euclidean Geometry Before Breakfast, and: Inheritance." Colorado Review 45, no. 2 (2018): 151–52. http://dx.doi.org/10.1353/col.2018.0070.

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16

Leichtweiss, Kurt. "Polar Curves in the Non-euclidean Geometry." Results in Mathematics 52, no. 1-2 (2008): 143–60. http://dx.doi.org/10.1007/s00025-007-0247-3.

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17

Dobbs, David E. "A proof of the arithmetic-geometric mean inequality using non-Euclidean geometry." International Journal of Mathematical Education in Science and Technology 32, no. 5 (2001): 778–82. http://dx.doi.org/10.1080/002073901753124655.

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18

Bashir, Asma, Benjamin Koch, and Muhammad Abdul Wasay. "Geometric description of Schrödinger equation in Finsler and Funk geometry." International Journal of Geometric Methods in Modern Physics 16, no. 07 (2019): 1950098. http://dx.doi.org/10.1142/s0219887819500981.

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For a system of [Formula: see text] non-relativistic spinless bosons, we show by using a set of suitable matching conditions that the quantum equations in the pilot-wave limit can be translated into a geometric language for a Finslerian manifold. We further link these equations to Euclidean time-like relative Funk geometry and show that the two different metrics in both of these geometric frameworks lead to the same coupling.

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19

Bellone, T., F. Fiermonte, and L. Mussio. "THE COMMON EVOLUTION OF GEOMETRY AND ARCHITECTURE FROM A GEODETIC POINT OF VIEW." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-5/W1 (May 16, 2017): 623–30. http://dx.doi.org/10.5194/isprs-archives-xlii-5-w1-623-2017.

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Throughout history the link between geometry and architecture has been strong and while architects have used mathematics to construct their buildings, geometry has always been the essential tool allowing them to choose spatial shapes which are aesthetically appropriate. Sometimes it is geometry which drives architectural choices, but at other times it is architectural innovation which facilitates the emergence of new ideas in geometry. <br><br> Among the best known types of geometry (Euclidean, projective, analytical, Topology, descriptive, fractal,…) those most frequently employed

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20

Afgoustidis, Alexandre. "Orientation Maps in V1 and Non-Euclidean Geometry." Journal of Mathematical Neuroscience 5, no. 1 (2015): 12. http://dx.doi.org/10.1186/s13408-015-0024-7.

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21

Bruss, I. R., and G. M. Grason. "Non-Euclidean geometry of twisted filament bundle packing." Proceedings of the National Academy of Sciences 109, no. 27 (2012): 10781–86. http://dx.doi.org/10.1073/pnas.1205606109.

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22

Griffin, Nicholas. "Non-Euclidean geometry: Still some problems for Kant." Studies in History and Philosophy of Science Part A 22, no. 4 (1991): 661–63. http://dx.doi.org/10.1016/0039-3681(91)90038-t.

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23

Wu, Wei, Guangmin Hu, and Fucai Yu. "An Unsupervised Learning Method for Attributed Network Based on Non-Euclidean Geometry." Symmetry 13, no. 5 (2021): 905. http://dx.doi.org/10.3390/sym13050905.

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Many real-world networks can be modeled as attributed networks, where nodes are affiliated with attributes. When we implement attributed network embedding, we need to face two types of heterogeneous information, namely, structural information and attribute information. The structural information of undirected networks is usually expressed as a symmetric adjacency matrix. Network embedding learning is to utilize the above information to learn the vector representations of nodes in the network. How to integrate these two types of heterogeneous information to improve the performance of network em

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24

Shillor, Irith. "Gifted Mathematicians Constructing Their Own Geometries — Changes in Knowledge and Attitudes." Gifted Education International 12, no. 2 (1997): 102–5. http://dx.doi.org/10.1177/026142949701200210.

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A previous paper (1) looked at how gifted mathematicians in primary school respond to the study of Non-Euclidean geometries. In this study children in the secondary school are asked to construct their own Non-Euclidean Geometries. Using Taxi-Cab Geometry (2) as the starting point, children are asked to focus on the non-Euclidean elements of this geometry, and consider the differences between Euclidean and Non-Euclidean geometries. They are then asked to construct their own geometry, and consider the non-Euclidean elements within it. Young children find this task quite complex, but slightly old

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25

Loiola, Carlos Augusto Gomes, and Chrsitine Sertã Costa. "AS CÔNICAS NA GEOMETRIA DO TÁXI." Ciência e Natura 37 (August 7, 2015): 179. http://dx.doi.org/10.5902/2179460x14596.

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http://dx.doi.org/10.5902/2179460X14596This paper aims to present conics when defined in a non-Euclidean geometry: the Taxicab geometry. The choice of this geometry was due to the simplicity of its definitions enabling diverse applications in Basic Education. It differs from Euclidean geometry by its metric and presents interesting and surprising results that enable the development of a more critical and meaningful learning.

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26

Zhao, Yuzhou, Chenyu Zhang, Daniel D. Kohler, et al. "Supertwisted spirals of layered materials enabled by growth on non-Euclidean surfaces." Science 370, no. 6515 (2020): 442–45. http://dx.doi.org/10.1126/science.abc4284.

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Euclidean geometry is the fundamental mathematical framework of classical crystallography. Traditionally, layered materials are grown on flat substrates; growing Euclidean crystals on non-Euclidean surfaces has rarely been studied. We present a general model describing the growth of layered materials with screw-dislocation spirals on non-Euclidean surfaces and show that it leads to continuously twisted multilayer superstructures. This model is experimentally demonstrated by growing supertwisted spirals of tungsten disulfide (WS2) and tungsten diselenide (WSe2) draped over nanoparticles near th

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27

Jovanovic, Radmila. "Three views on epistemological status of geometry." Theoria, Beograd 55, no. 4 (2012): 21–38. http://dx.doi.org/10.2298/theo1204021j.

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The subject of this paper is the epistemological status of geometry. This problem appeared in the beginning of XIX century, after the birth of non-euclidean geometries. This scientific discovery had a big influence on philosophy of science. We will deal with different systems of geometry, interpreted by means of physics. We will discuss several positions regarding their epistemological status. The key question is about the nature of geometry as a science- is there one real geometry of our world and which one? Both euclidean and non-euclidean geometries can be seen as to be in accordance with o

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28

Sitenko, Yurii, and Volodymyr Gorkavenko. "Non-Euclidean Geometry, Nontrivial Topology and Quantum Vacuum Effects." Universe 4, no. 2 (2018): 23. http://dx.doi.org/10.3390/universe4020023.

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29

Kisil, Vladimir V. "MoebInv: C++ libraries for manipulations in non-Euclidean geometry." SoftwareX 11 (January 2020): 100385. http://dx.doi.org/10.1016/j.softx.2019.100385.

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30

Leichtweiss, K. "Curves of constant width in the non-euclidean geometry." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 75, no. 1 (2005): 257–84. http://dx.doi.org/10.1007/bf02942046.

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31

Leichtweiss, Kurt. "Linear combinations of convex hypersurfaces in non-Euclidean geometry." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 53, no. 1 (2011): 77–88. http://dx.doi.org/10.1007/s13366-011-0080-4.

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32

Noro, Shuta, Masahiko Okumura, Satoshi Hongo, et al. "Langevin simulations of protoplasmic streaming in non-Euclidean geometry." Journal of Physics: Conference Series 1730, no. 1 (2021): 012037. http://dx.doi.org/10.1088/1742-6596/1730/1/012037.

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33

Gray, Jeremy. "A history of non-euclidean geometry: Evolution of the concept of a geometric space." Historia Mathematica 18, no. 4 (1991): 373–74. http://dx.doi.org/10.1016/0315-0860(91)90380-g.

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34

Coxeter, H. S. M., and Jan van de Craats. "Philon lines in non-Euclidean planes." Journal of Geometry 48, no. 1-2 (1993): 26–55. http://dx.doi.org/10.1007/bf01226799.

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35

El Khaldi, Khaldoun, and Elias G. Saleeby. "On the density of lines and Santalo’s formula for computing geometric size measures." Monte Carlo Methods and Applications 26, no. 4 (2020): 315–23. http://dx.doi.org/10.1515/mcma-2020-2071.

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AbstractMethods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established

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36

Blok, Johan. "Toward a Formal Interpretation of Kant's Analogies of Experience." Hegel Bulletin 28, no. 1-2 (2007): 107–20. http://dx.doi.org/10.1017/s0263523200000665.

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Very often, the rise of non-Euclidean geometry and Einstein's theory of relativity are seen as the decisive defeat of Kant's theoretical philosophy. Scientific progress seems to render Kant's philosophy obsolete. This view became dominant during the first decades of the twentieth century, when the movement of logical positivism arose. Despite extensive criticism of basic tenets of this movement later in the twentieth century, its view of Kant's philosophy is still common. Although it is not my intention to defend Kant infinitely, I think that this view is rather unsatisfactory and even mislead

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37

Сафиулина, Yu Safiulina, Шмурнов, and V. Shmurnov. "Graphical Proof of the Main Theorem of Non-Euclidean Geometry." Geometry & Graphics 3, no. 3 (2015): 18–23. http://dx.doi.org/10.12737/14416.

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The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recogn

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38

Shiguo, Yang, and Wang Jia. "Some geometric inequalities in non-euclidean space." Journal of Geometry 56, no. 1-2 (1996): 196–201. http://dx.doi.org/10.1007/bf01222696.

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39

Struve, Horst, and Rolf Struve. "Non-euclidean geometries: the Cayley-Klein approach." Journal of Geometry 98, no. 1-2 (2010): 151–70. http://dx.doi.org/10.1007/s00022-010-0053-z.

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40

Berrett, Joshua, Virginia Marquardt, and Linda Dalrymple Henderson. "The Fourth Dimension and Non-Euclidean Geometry in Modern Art." Technology and Culture 26, no. 4 (1985): 879. http://dx.doi.org/10.2307/3105651.

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41

Henderson, Andrea. "Math for Math's Sake: Non-Euclidean Geometry, Aestheticism, and Flatland." PMLA/Publications of the Modern Language Association of America 124, no. 2 (2009): 455–71. http://dx.doi.org/10.1632/pmla.2009.124.2.455.

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This essay argues that Edwin Abbott's Flatland brings into focus the wide-ranging implications of the dethroning of what Victorians regarded as the preeminent representational system: Euclidean geometry. The contemporary debate surrounding the challenge to Euclid, conducted not just in mathematical but also in psychological, philosophical, and aesthetic terms, turned on an anxiety that signs might not have the capacity to bridge subjective and objective worlds, and Flatland seeks solace for this uncertainty by granting even empty signs unprecedented virtues.

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42

Kalimuthu, S. "Two Findings for the Origin of Third Non Euclidean Geometry." National Academy Science Letters 36, no. 6 (2013): 621–23. http://dx.doi.org/10.1007/s40009-013-0179-2.

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43

Leichtweiss, Kurt. "On Steiner’s Symmetrization of Convex Bodies in Non-Euclidean Geometry." Results in Mathematics 52, no. 3-4 (2008): 339–46. http://dx.doi.org/10.1007/s00025-008-0315-3.

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44

Trzesowski, Andrzej. "Geometry of crystal structure with defects. II. Non-Euclidean picture." International Journal of Theoretical Physics 26, no. 4 (1987): 335–55. http://dx.doi.org/10.1007/bf00672243.

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45

Gudmundsson, Sigmundur. "On the geometry of harmonic morphisms." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (1990): 461–66. http://dx.doi.org/10.1017/s0305004100069358.

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AbstractLet π:M→B be a horizontally conformal submersion. We give necessary curvature conditions on the manifolds M and B, which lead to non-existence results for certain horizontally conformal maps, and harmonic morphisms. We then classify all such maps between open subsets of Euclidean spaces, which additionally have totally geodesic fibres and are horizontally homothetic. They are orthogonal projections on each connected component, followed by a homothety.

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46

Khrennikov, Andrei. "Bell Could Become the Copernicus of Probability." Open Systems & Information Dynamics 23, no. 02 (2016): 1650008. http://dx.doi.org/10.1142/s1230161216500086.

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Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as

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47

Tolchelnikova, S. A., and K. N. Naumov. "On the matter of proving Euclidean fifth postulate and the origin of non-Euclidean geometries." Geodesy and Cartography 950, no. 8 (2019): 2–11. http://dx.doi.org/10.22389/0016-7126-2019-950-8-2-11.

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The Euclidean geometry was developed as a mathematical system due to generalizing thousands years of measurements on the plane and spherical surfaces. The development of celestial mechanics and stellar astronomy confirmed its validity as mathematical principles of natural philosophy, in particular for studying the Solar System bodies’ and Galaxy stars motions. In the non-Euclidean geometries by Lobachevsky and Riemann, the third axiom of modern geometry manuals is substituted. We show that the third axiom of these manuals is a corollary of the Fifth Euclidean postulate. The idea of spherical,

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48

KORCHEMSKY, G. P. "QUANTUM GEOMETRY OF DIRAC FERMIONS." International Journal of Modern Physics A 07, no. 02 (1992): 339–80. http://dx.doi.org/10.1142/s0217751x9200020x.

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A bosonic path integral formalism is developed for Dirac fermions interacting with a non-Abelian gauge field in D-dimensional Euclidean space–time. The representation for the effective action and correlation functions of interacting fermions as sums over all bosonic paths on the complex projective space CP2d−1 (2d = 2[D/2]) is derived where the whole of the spinor structure is absorbed by the one-dimensional Wess–Zumino term. It is the Wess–Zumino term that ensures all necessary properties of Dirac fermions under quantization, i.e. quantized values of the spin, Dirac equation, and Fermi statis

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49

EDER, GÜNTHER, and GEORG SCHIEMER. "HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY." Review of Symbolic Logic 11, no. 1 (2017): 48–86. http://dx.doi.org/10.1017/s1755020317000260.

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AbstractThe article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular,

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50

Tamm, Martin. "Minimizing Curvature in Euclidean and Lorentz Geometry." Symmetry 13, no. 8 (2021): 1433. http://dx.doi.org/10.3390/sym13081433.

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In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in R3, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However,

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