Axiomatizing perpendicularity and parallelism
Motivated by the desire to introduce axiomatic geometric reasoning in a high school and teacher education setting, the authors introduce a rudimentary axiom system \(\Sigma\) for Euclidean-style line-orthogonality \(\perp\), consisting of four axioms -- irreflexivity, symmetry, \(3\)-transitivity, and non-emptiness -- and study some of its consequences, as well as notice that a parallelism relation \(\parallel\) can be defined in terms of \(\perp\) by \(a \parallel b\Leftrightarrow (\forall x)\, x\perp a \leftrightarrow x\perp b\). Noticing that \(3\)-transitivity fails in higher dimensions, an \(n\)-ary notion of perpendicularity is suggested for line-perpendicularity in \(n\)-dimensional spaces, and for the case \(n=3\) four axioms are stated. Models of \(\Sigma\) that do not bear any resemblance to the intended interpretation, as well as possible exercises that one may want to use in class round off the paper.NEWLINENEWLINEReviewer's remark: The authors seem to be unaware of some of the literature relevant to their project. Their axiom system \(\Sigma\), as well as its associated parallelism can be found on page 408 of \textit{Ph. Balbiani}, \textit{V. Goranko}, \textit{R. Kellerman}, \textit{D. Vakarelov}, [``Logical theories for fragments of elementary geometry'', in: M. Aiello, I. Pratt-Hartmann, J. van Benthem, (eds.), Handbook of spatial logics, 343--428, Springer, Dordrecht (2007; Zbl 1172.03001)]. \textit{R. Kellerman} [Log. J. IGPL 15, No. 3, 255--270 (2007; Zbl 1129.03003)] is devoted in whole to the very subject of Euclidean-style line-orthogonality. That one cannot say very much with lines and line-orthogonality (in fact with lines and any set of binary relations on lines) in plane Euclidean geometry was proved in Th.\ 3.1 \textit{W. Schwabhäuser} and \textit{L. W. Szczerba} [Fundam. Math. 82, 347--355 (1974; Zbl 0296.50001)], where it is also shown that one can express in terms of line-orthogonality all of \(n\)-dimensional Euclidean geometry over Euclidean fields for \(n\geq 4\). The limitations of line-orthogonality in Euclidean \(3\)-dimensional space were the subject of \textit{R. Kramer} [Geom. Dedicata 46, 207--210 (1993; Zbl 0778.51007)] and \textit{W. Benz} and \textit{E. M. Schröder} [Geom. Dedicata 21, 265--276 (1986; Zbl 0605.51003)].
scientific article; zbMATH DE number 6086377