The Real Projective Plane

Author(s):  
H. S. M. Coxeter ◽  
George Beck
Author(s):  
Hanjo Berressem

Providing a comprehensive reading of Deleuzian philosophy, Gilles Deleuze’s Luminous Philosophy argues that this philosophy’s most consistent conceptual spine and figure of thought is its inherent luminism. When Deleuze notes in Cinema 1 that ‘the plane of immanence is entirely made up of light’, he ties this philosophical luminism directly to the notion of the complementarity of the photon in its aspects of both particle and wave. Engaging, in chronological order, the whole body and range of Deleuze’s and Deleuze and Guattari’s writing, the book traces the ‘line of light’ that runs through Deleuze’s work, and it considers the implications of Deleuze’s luminism for the fields of literary studies, historical studies, the visual arts and cinema studies. It contours Deleuze’s luminism both against recent studies that promote a ‘dark Deleuze’ and against the prevalent view that Deleuzian philosophy is a philosophy of difference. Instead, it argues, it is a philosophy of the complementarity of difference and diversity, considered as two reciprocally determining fields that are, in Deleuze’s view, formally distinct but ontologically one. The book, which is the companion volume toFélix Guattari’s Schizoanalytic Ecology, argues that the ‘real projective plane’ is the ‘surface of thought’ of Deleuze’s philosophical luminism.


2016 ◽  
Vol 24 (4) ◽  
pp. 239-251 ◽  
Author(s):  
Roland Coghetto

Summary The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12]. Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.


1956 ◽  
Vol 40 (332) ◽  
pp. 153
Author(s):  
D. Pedoe ◽  
H. S. M. Coxeter

2010 ◽  
Vol 21 (12) ◽  
pp. 1605-1617
Author(s):  
JOËL ROUYER

It is proved in this article, that in the framework of Riemannian geometry, the existence of large sets of antipodes (i.e. farthest points) for diametral points of a smooth surface has very strong consequences on the topology and the metric of this surface. Roughly speaking, if the sets of antipodes of diametral points are closed curves, then the surface is nothing but the real projective plane.


1998 ◽  
Vol 20 (2) ◽  
pp. 155-161 ◽  
Author(s):  
D. Forge ◽  
J. L. Ramírez Alfonsín

Author(s):  
Hanjo Berressem

In luminist and cinematic registers, the chapter first explicates Deleuze’s complementary chronologics of Aion and Chronos. While durational Aion is the time of luminist glow, flickering Chronos is the time of strobe light: wave and particle. Shifting from time to space, the chapter then addresses Deleuze’s topologics. After introducing the notion of fractal space in terms of Deleuze’s notion of becoming-imperceptible, it delineates, via Leibniz’ images of the baroque house and the camera obscura, Deleuze’s transformation of the spatial dualism of light surface and dark depth into the luminous space of a fractal chiaroscuro, and it shows how Deleuze’s luminous philosophy resonates with Leibniz’ proposition that monads, as points or centres of light, have a luminous nature. After explicating the mathematical concept of the ‘real projective plane,’ the chapter argues that Deleuze’s shift from a Cartesian to a projective topology of thought is fundamental for an understanding of his philosophy.


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