Numerical Dual Affine Spaces of Desargues


Authors : Jesús Castañeda Rivera

Volume/Issue : Volume 9 - 2024, Issue 12 - December

Google Scholar : https://tinyurl.com/2swvrfnd

Scribd : https://tinyurl.com/2pnhve3n

DOI : https://doi.org/10.5281/zenodo.14636684

Abstract : Dual affine spaces are geometries with points and lines, lines have three points and, at most, a line passes through two points. Furthermore, we have that its planes are the duals of the affine plane over the field of two elements. If the space is connected, numerical invariants are associated with it. Let n be the number of points in space and k be the number of points that, given a fixed point, are not collinear with it. In this research we characterize the geometric spaces that satisfy the Desargues property “Every pair of non-collinear points has exactly four collinear points.” represented by pairs of numbers (n, k) that satisfy certain algebraic properties studied by D. Higman (1964), H. Cárdenas (1999, 2001, 2002) and J. Castañeda (2011, 2020).

Keywords : Dual Affine Space, Desargues Configuration, Numerical Dual Affine Spaces, Desargues Spaces.

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Dual affine spaces are geometries with points and lines, lines have three points and, at most, a line passes through two points. Furthermore, we have that its planes are the duals of the affine plane over the field of two elements. If the space is connected, numerical invariants are associated with it. Let n be the number of points in space and k be the number of points that, given a fixed point, are not collinear with it. In this research we characterize the geometric spaces that satisfy the Desargues property “Every pair of non-collinear points has exactly four collinear points.” represented by pairs of numbers (n, k) that satisfy certain algebraic properties studied by D. Higman (1964), H. Cárdenas (1999, 2001, 2002) and J. Castañeda (2011, 2020).

Keywords : Dual Affine Space, Desargues Configuration, Numerical Dual Affine Spaces, Desargues Spaces.

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