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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 $a\parallel b⇔\left(\forall x\right)\phantom{\rule{0.166667em}{0ex}}x\perp a↔x\perp b$. 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 Ph. Balbiani, V. Goranko, R. Kellerman, 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)]. 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 W. Schwabhäuser and 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 $n\ge 4$. The limitations of line-orthogonality in Euclidean 3-dimensional space were the subject of R. Kramer [Geom. Dedicata 46, 207–210 (1993; Zbl 0778.51007)] and W. Benz and 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