Graph Theoretical Models of Closed n-Dimensional Manifolds: Digital Models of a Moebius Strip, a Projective Plane a Klein Bottle and n-Dimensional Spheres

Alexander V. Evako^1,

¹“Dianet”, Laboratory of Digital Technologies, Moscow, Russia

Cite this paper:
Alexander V. Evako. Graph Theoretical Models of Closed n-Dimensional Manifolds: Digital Models of a Moebius Strip, a Projective Plane a Klein Bottle and n-Dimensional Spheres. Applied Mathematics and Physics. 2017; 5(1):19-27. doi: 10.12691/amp-5-1-3

Abstract

This paper presents discretization schemes for building graph theoretical models of n-dimensional continuous objects with the same topological properties as their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. The digital model of a continuous n-dimensional object is the intersection graph of an LCL cover of the object. We prove that the digital model of a continuous closed n-dimensional manifold is a digital closed n-dimensional manifold. It is shown that the digital model of a continuous n-dimensional sphere is a digital n-sphere with at least 2n+2 points, the digital model of a continuous projective plane is a digital projective plane with at least eleven points and the digital model of a continuous Klein bottle is the digital Klein bottle with at least sixteen points.

Keywords:
graph manifold digital space sphere Klein bottle projective plane Moebius band

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