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Axiomatizing perpendicularity and parallelism. (English) Zbl 1250.97008
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. Reviewer’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 {\it Ph. Balbiani}, {\it V. Goranko}, {\it R. Kellerman}, {\it 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)]. {\it 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 {\it W. Schwabhäuser} and {\it 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 {\it R. Kramer} [Geom. Dedicata 46, 207--210 (1993; Zbl 0778.51007)] and {\it W. Benz} and {\it E. M. Schröder} [Geom. Dedicata 21, 265--276 (1986; Zbl 0605.51003)].

MSC:
 97G99 Geometry (educational aspects) 51-01 Textbooks (geometry) 51F20 Congruence and orthogonality (geometry)
Keywords:
mathematics education
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