Abstract
An old question regarding the world we live in concerns what is real regarding points and lines: if two distinct lines intersect, is their intersection a unique point? In this paper, we take the approach that the answer is no, that all the points in the intersection are somehow close to one another (neighbourly) and that two non-neighbourly points determine a unique line. These are the Affine Klingenberg spaces (AK-spaces). How does one put a logical structure on points and lines that reflect the preceding view of reality? History has shown that such a structure is based upon the concept of coordinatization, which leads naturally to algebraic structures that allow a faithful representation of incidence, which in turn reflects the existence of relations between points and lines that recognise incidence. The preceding view of reality is not new, and the history of this subject is of approaches that are too general (there are conditions on neighbourly points). Our approach is novel in that it is based upon a minimum number of assumptions that yield the existence of dilatations that are translations: the corner stones of coordinatization.
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T. Bisztriczky was supported by a NSERC Discovery Grant.
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Bisztriczky, T., Lorimer, J.W. Translations in affine Klingenberg spaces. J. Geom. 99, 15–42 (2010). https://doi.org/10.1007/s00022-011-0067-1
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DOI: https://doi.org/10.1007/s00022-011-0067-1