Calculation of Fundamental Groups via Computational Paths

We address the question as to how to formalise the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. The intention is to demonstrate the use of a term rewriting system in performing computations with these computational paths, establishing equalities between equalities, and further higher equalities, in particular, in the calculation of fundamental groups of surfaces such as the circle, the torus and the real projective plane.

Pappus’s Hexagon Theorem in Real Projective PlaneThis work has been supported by the “Centre autonome de formation et de recherche en mathématiques et sciences avec assistants de preuve” ASBL (non-profit organization). Enterprise number: 0777.779.751. Belgium.

Summary. In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear” https://en.wikipedia.org/wiki/Pappus’s_hexagon_theorem . More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leończuk and Krzysztof Prażmowski. Eugeniusz Kusak and Wojciech Leończuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover9 https://www.cs.unm.edu/~mccune/prover9/ , the successor of the Otter prover and ott2miz by Josef Urban See its homepage https://github.com/JUrban/ott2miz [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].

Limit Polygons of Convex Domains in the Projective Plane

For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determine all Hausdorff limit domains that one can obtain after normalizing each member of the sequence by a projective transformation. We then show that the result can be applied to convex domains generated by projective triangular reflection groups.

A systolic inequality with remainder in the real projective plane

The first paper in systolic geometry was published by Loewner’s student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric in the real projective plane. We prove a stronger version of Pu’s systolic inequality with a remainder term.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Gilles Deleuze's Luminous Philosophy

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.

Philosophical Times and Spaces

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.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.