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: Flammarion. Fihman G. (1999). “Bergson, Zénon d’Élée et le cinéma /Bergson, Deleuze und das Kino.” Der Film bei Deleuze / Le cinéma selon Deleuze. Eds. O. Fahle and L. Engell. Weimar: Verlag der Bauhaus-Universität Weimar. 62-85. Günzel S. (2007). “Raum - Topographie - Topologie.” Topologie. Zur Raumbeschreibung in den Kultur- und Medienwissenschaften. Ed. S. Günzel. Bielefeld: Transcript Verlag. 13-29. Heidegger, M. (1954). Vom Wessen der Wahrheit. Frankfurt am Main: Vittorio Klostermann GmbH. Henderson, L.D. (1983). The Fourth Dimension and Non-Euclidean Geometry in

, Vol. I, Macmillan, London, 254–340. Conrey, B. (2003), “The Riemann Hypothesis”, Notices of AMS , March, 341–353. Coxeter, H.S.M. (1998), Non-Euclidean Geometry (sixth ed.), The Mathematical Association of America. De Jong, W.R. (1997), “Kant’s Theory of Geometrical Reasoning and the Analytic-Synthetic Distinction. On Hintikka’s Interpretation of Kant’s Philosophy of Mathematics”, Studies in History and Philosophy of Science , 28 (1), 141–166. Desanti, J.-T. (1983), Les idéalités mathématiques , Seuil, Paris. Donaldson, S.K. (1983), “An application of gauge

Urban Compression Patterns: Fractals and Non-Euclidean Geometries - Inventory and Prospect

Urban growth and fractality is a topic that opens an entrance for a range of radical ideas: from the theoretical to the practical, and back again. We begin with a brief inventory of related ideas from the past, and proceed to one specific application of fractals in the non-Euclidean geometry of Manhattan space. We initialize our discussion by inventorying selected existing knowledge about fractals and urban areas, and then presenting empirical evidence about the geometry of and movement in physical urban space.

Selected empirical analyses of minimum path distances between places in urban space indicate that its metric is best described by a general Minkowskian one whose parameters are between those for Manhattan and Euclidean space. Separate analyses relate these results to the fractal dimensions of the underlying physical spaces. One principal implication is that theoretical, as well as applied, ideas based upon fractals and the Manhattan distance metric should be illuminating in a variety of contexts. These specific analyses are the focus of this paper, leading a reader through analytical approaches to fractal metrics in Manhattan geometry. Consequently, they suggest metrics for evaluating urban network densities as these represent compression of human activity. Because geodesics are not unique in Manhattan geometry, that geometry offers a better fit to human activity than do Euclidean tools with their unique geodesic activities: human activity often moves along different paths to get from one place to another.

Real-world evidence motivates our specific application, although an interested reader may find the subsequent "prospect" section of value in suggesting a variety of future research topics that are currently in progress. Does "network science" embrace tools such as these for network compression as it might link to urban function and form? Stay tuned for forthcoming work in Geographical Analysis.

References [1] O.Y. Bodnar, The golden section and non-Euclidean geometry in nature and art , Publishing House ”Svit”, Lvov, 1994(Russian). [2] W.K. Clifford, Preliminary sketch of bi-quaternions , Proc. London Math. Soc. 4 (1873), 381-395. [3] J. Cockle, On systems of algebra involving more than one imaginary , Philos. Mag. 35 (series 3) (1849), 434–436. [4] H.S.M. Coxeter, S.L. Greitzer, Geometry revisited The Mathematical Association of America (nc.), International and Pan American Conventions, Washington:1967. [5] R.A. Dunlap, The golden ratio and

. R¨oschel, O. - Die Geometrie des Galileischen Raumes, [The geometry of Galilei space] Berichte [Reports], 256, Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 1985. 17. Yaglom, I.M. - A Simple Non-Euclidean Geometry and its Physical Basis, Springer- Verlag, New York-Heidelberg, 1979.

. Geom. 33(1988), 53-57. [6] M. J. Greenberg - Euclidean and Non-Euclidean geometries, Development and history, Third edition, W. H. Freeman & Co., 1993. [7] R. Hartshorne - Non-Euclidean III.36 Amer. Math. Monthly 110(2003), 495-502. [8] D. Hilbert - The Foundations of Geometry, Open Court Publishing Com- pany, La Salle, 19G2 (reprint of the translation by E. J. Townsend, 1902). [9] R. A. Johnson - A Circle Theorem, Amer. Math. Monthly, 23(1916), 161-162. [10] B. Kerekjarto Les fondements de la geometrie, vol. 1, Akad. Kiado, Bu- dapest, 1955. [11] T. Lalescu

R eferences [1] Norbert A’Campo and Athanase Papadopoulos. On Klein’s so-called non-Euclidean geometry. arXiv preprint arXiv:1406.7309 , 2014. [2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics , volume 9150 of Lecture Notes in Computer Science , pages 261–279. Springer International Publishing, 2015. ISBN 978

translation), Vol. 7 , 224–228. Einstein, A., 1922, Four lectures on the theory of relativity, held at Princeton University on May 1921. In: CPAE (English translation), Vol. 7 , 261–368. Einstein, A., 1923, Fundamental ideas and problems of the theory of relativity. In: CPAE (English translation), Vol. 14 , 74–81. Einstein, A., 1924, Review of Albert C. Elsbach, Kant und Einstein. In: CPAE (English translation), Vol. 14 , 322–327. Einstein, A., 1925, Non-Euclidean geometry and physics. In: CPAE (English translation), Vol. 14 , 215–218. Einstein, A., 1949a


, demonstrated by Gödel, as another one. Non-Euclidean geometries shattered the idea that necessary unique truths could be naturally formulated. Moreover and less obviously, mathe- ISBN 978-83-7431-480-0 ISSN 0860-150X 7 Stanisław Krajewski, Kazimierz Trzęsicki matical ideologies, like Intuitionism or Platonism, function in some way as distinct “religions”, religions that compete – and “conversions” are possible. Von Neumann said that the shift from one to another, and back, can be felt as humiliation. One can also maintain that these ideologies do not compete but just