*(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 $a\parallel b\iff (\forall x)\phantom{\rule{0.166667em}{0ex}}x\perp a\leftrightarrow x\perp b$. 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 *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) |