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Books on the topic 'Non-Euclidean spaces'

Author: Grafiati

Published: 28 June 2021

Last updated: 1 February 2022

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1

Borwein, Jonathan M. Convex functions: Constructions, characterizations and counterexamples. Cambridge: Cambridge University Press, 2010.

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2

Jeremy, Gray. Ideas of space: Euclidean, non-Euclidean, and relativistic. 2nd ed. Oxford: Clarendon Press, 1989.

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3

Outer billiards on kites. Princeton, N.J: Princeton University Press, 2009.

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4

Aravinda, C. S. Geometry, groups and dynamics: ICTS program, groups, geometry and dynamics, December 3-16, 2012, CEMS, Kumaun University, Almora, India. Providence, Rhode Island: American Mathematical Society, 2015.

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5

Bolyai, János. Appendix, the theory of space. Amsterdam: North-Holland, 1987.

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6

Appendix, the theory of space. Budapest: Akadémiai Kiadó, 1987.

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7

Rozenfelʹd, Boris Abramovich. A history of non-Euclidean geometry: Evolution of the concept of a geometric space. New York: Springer-Verlag, 1988.

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8

Jeremy, Gray. János Bolyai, non-Euclidean geometry, and the nature of space. Cambridge, Mass: Burndy Library, 2004.

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9

Ah istory of non-euclidean geometry: Evolution of the concept of a geometric space. New York: Springer-Verlag, 1987.

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10

Tazzioli, Rossana. Beltrami e i matematici "relativisti": La meccanica in spazi curvi nella seconda metà dell'Ottocento. Bologna: Pitagora, 2000.

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11

Lʹvova, L. V. Interpretat︠s︡ii︠a︡ kvaziėllipticheskogo prostranstva: Monografii︠a︡. Barnaul: Barnaulʹskiĭ gos. pedagogicheskiĭ universitet, 2005.

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12

Christensen, Jens Gerlach. Trends in harmonic analysis and its applications: AMS special session on harmonic analysis and its applications : March 29-30, 2014, University of Maryland, Baltimore County, Baltimore, MD. Providence, Rhode Island: American Mathematical Society, 2015.

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13

Small Universal Cellular Automata In Hyperbolic Spaces A Collection Of Jewels. Springer-Verlag Berlin and Heidelberg GmbH &, 2013.

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14

Path Integrals, Hyperbolic Spaces and Selberg Trace Formulae. World Scientific Publishing Co Pte Ltd, 2013.

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15

Ideas of Space, Euclidean, Non-Euclidean. Oxford University Press, 1995.

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16

Hellman, Geoffrey, and Stewart Shapiro. Non-Euclidean Extensions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198712749.003.0006.

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This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.

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17

Blacklock, Mark. The Emergence of the Fourth Dimension. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198755487.001.0001.

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The idea of the fourth dimension of space has been of sustained interest to nineteenth-century and Modernist studies since the publication of Linda Dalrymple Henderson’s The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983). An idea from mathematics that was appropriated by occultist thought, it emerged in the fin de siècle as a staple of genre fiction and grew to become an informing idea for a number of important Modernist writers and artists. Describing the post-Euclidean intellectual landscape of the late nineteenth century, The Emergence of the Fourth Dimension works with the concepts derived from the mathematical possibilities of n-dimensional geometry—co-presence, bi-location, and interpenetration; the experiences of two consciousnesses sharing the same space, one consciousness being in two spaces, and objects and consciousness pervading each other—to examine how a crucially transformative idea in the cultural history of space was thought and to consider the forms in which such thought was anchored. It identifies a corpus of higher-dimensional fictions by Conrad and Ford, H.G. Wells, Henry James, H.P. Lovecraft, and others and reads these closely to understand how fin de siècle and early twentieth-century literature shaped and were in turn shaped by the reconfiguration of imaginative space occasioned by the n-dimensional turn. In so doing it traces the intellectual history of higher-dimensional thought into diverse terrains, describing spiritualist experiments and how an extended abstract space functioned as an analogue for global space in occult groupings such as the Theosophical Society.

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18

Rosenfeld, Boris A., Abe Shenitzer, and Hardy Grant. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer, 2012.

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19

Rosenfeld, Boris A., Abe Shenitzer, and Hardy Grant. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer, 2012.

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20

Henderson, Andrea. Geometry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198809982.003.0002.

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Edwin Abbott’s Flatland dramatizes the implications of dethroning what Victorians regarded as the preeminent representational system: Euclidean geometry. The displacement of the singular Euclidean account of space with a multiplicity of non-referential spatial regimes did more than introduce the possibility of varying perspectives on the world; the challenge to the “sacredness” of Euclid met with resistance partly because it suggested the ideal of a transparent representational system was inherently untenable. Flatland explores the repercussions of this problem for the novel, shifting emphasis from the revelation of the content of character to focus on the vagaries of point of view. The characters are Euclidean figures shown the limitations of their constructions of the world, and epistemic certainty is unavailable because all representational systems are contingent. Abbott finds consolation for this loss of certainty in the formalist, aesthetic character of projective geometry, insisting on the beauty of signs in and of themselves.

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21

McMorran, Ciaran. Joyce and Geometry. University Press of Florida, 2020. http://dx.doi.org/10.5744/florida/9780813066288.001.0001.

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Following the development of non-Euclidean geometries from the mid-nineteenth century onwards, Euclid’s system came to be re-conceived as a language for describing reality rather than a set of transcendental laws. As Henri Poincaré famously put it, “[i]f several geometries are possible, is it certain that our geometry [...] is true?” By examining James Joyce’s linguistic play and conceptual engagement with ground-breaking geometric constructs in Ulysses and Finnegans Wake, this book explores how his topographical writing of place encapsulates a common crisis between geometric and linguistic modes of representation within the context of modernity. More specifically, it investigates how Joyce presents Euclidean geometry and its topographical applications as languages, rather than ideally objective systems, for describing the visible world; and how, conversely, he employs language figuratively to emulate the systems by which the world is commonly visualized. With reference to his early readings of Giordano Bruno, Henri Poincaré, and other critics of the Euclidean tradition, it examines how Joyce’s obsession with measuring and mapping space throughout his works enters into his more developed reflections on the codification of visual signs in Finnegans Wake. In particular, this book sheds new light on Joyce’s fascination with the “geometry of language” practiced by Bruno, whose massive influence on Joyce is often assumed to exist in Joyce studies yet is rarely explored in any detail.

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22

Rosenfeld, Boris A. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space (Studies in the History of Mathematics and Physical Sciences). Springer, 1988.

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23

Behrens, Stefan, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray, eds. The Disc Embedding Theorem. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198841319.001.0001.

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The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

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24

Continuity and Infinity between Science and Philosophy. Alexandria, Egypt: Al Maaref Establishment Press, 1998.

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