Journal ArticleDOI
The Thomsen–Bachmann correspondence in metric geometry II
Rolf Struve,Horst Struve +1 more
TLDR
Struve and R. Struve as mentioned in this paper showed that the Thomsen-Bachmann correspondence between metric geometries and groups can be precisely stated in a framework of first-order logic.Abstract:
We continue the investigations of the Thomsen–Bachmann correspondence between metric geometries and groups, which is often summarized by the phrase ‘Geometry can be formulated in the group of motions’. In the first part (H. Struve and R. Struve in J Geom, 2019. https://doi.org/10.1007/s00022-018-0465-8
) of this paper it was shown that the Thomsen–Bachmann correspondence can be precisely stated in a framework of first-order logic. We now prove that the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley–Klein geometries, isotropic and equiform geometries, and that these geometries and the theory of their group of motions are mutually faithfully interpretable (and bi-interpretable, but not definitionally equivalent). Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method.read more
Citations
More filters
Journal ArticleDOI
Cyclic order: a geometric analysis
TL;DR: In this article, the role of cyclic order in the foundations of geometry has been discussed and a characterization of cyclically ordered groups by cyclic cones and the introduction of the notion of a cyclically-ordered field has been presented.
Journal ArticleDOI
Absolute isotropic geometry
Rolf Struve,Horst Struve +1 more
TL;DR: In this paper, the authors study the part of absolute isotropic geometry that does not depend on the parallel postulate and their models (isotropic planes) and prove the theorem of Saccheri for the angle sum of a triangle, and construct models of the different types of planes.
References
More filters
Journal ArticleDOI
On the Calculus of Relations
TL;DR: The logical theory which is called the calculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development.
BookDOI
Aufbau der Geometrie aus dem Spiegelungsbegriff
TL;DR: In this article, the authors show that metrischen Ebenen, die with den axiomatisch gegebenen Gruppen definiert sind, sind daher, with einem Ausdruck von J. BOLYAI, a also ebene absolute Geometrie genannt.
Related Papers (5)
Constructing partial geometries from overlarge sets of Steiner systems
Sven Reichard,Andrew J. Woldar +1 more