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Absolute isotropic geometry

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Abstract

The term ‘absolute geometry’ was coined by János Bolyai to characterize the part of Euclidean geometry that does not depend on the parallel postulate. In the framework of Cayley–Klein geometries the parallel axiom characterizes three classical geometries, namely Euclidean, Galilean and Minkowskian planes. We study the part of Galilean geometry that does not depend on the parallel postulate (briefly called absolute isotropic geometry) and their models (isotropic planes). After an axiomatic foundation of absolute isotropic geometry, we develop the basic theory of isotropic planes, prove the theorem of Saccheri for the angle sum of a triangle, and construct models of the different types of planes (over fields and skew fields of characteristic \(\ne 2\)). Surprisingly, these models have counterparts in the theory of Hilbert planes. The article closes with a comparison between Hilbert planes and isotropic planes.

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Notes

  1. We consider in this article only plane geometries.

  2. We follow the reflection-geometric approach of Hjelmslev [16] and Bachmann [3] and formulate axioms in group-theoretical terms (see Sect. 2).

  3. This consensus has been reached by the end of the first half of the 20th century, based on the work of Skolem, Hilbert and Ackermann, Gödel, and Tarski.

  4. That is, a sentence \(\varphi \) is a theorem of \(\mathcal {T}\) if and only if the translation of \(\varphi \) is a theorem of \(\mathcal {T'}\), and vice versa.

  5. This holds under mild hypotheses with respect to the axioms of \(\mathcal {T}\), which all axiom systems of this article satisfy (see [32]).

  6. The (Witt) index of a quadratic space (V, f) is defined to be the maximal dimension of a totally isotropic subspace of V/RadV (see [3, p. 311]).

  7. A group is linearly orderable if and only if there exists a partition into cones; see [28, Theorem 3.3].

  8. The continuity axiom implies the Archimedean axiom which Hilbert used in [13]. Neither of these axioms can be stated in first-order logic.

  9. There is no commonly agreed naming of Hilbert planes. We follow Bachmann [3] and Hessenberg and Diller [12].

  10. In the literature division rings are also called ‘distributive Veblen–Wedderburn systems’ (see Hughes and Piper [17]).

  11. \(\angle (a, a, a^{+}, a^{-})\) is a zero angle (whose interior \( a^{+} \cap a^{-}\) is void). Any two zero angles are congruent.

  12. The neighborhood is defined with respect to the order topology.

  13. In [10] the field K and the metric constant k are denoted by M respectively \(\kappa \).

  14. The axiom was introduced in R. Struve [30] for a characterization of the part of Euclidean geometry which satisfies the principle of duality.

  15. In other words, if A is a point, which is not incident with a line b then \(n = 0\) or \(n = 1\) or \(n = 2\) according as there are 0, 1 or at least 2 lines through A which are ‘parallel’ to b (i.e., which have no common point with b).

  16. A non-Legendrean Hilbert plane is locally elliptic, since it is a locally complete subplane of the associated elliptic ideal plane (see Bachmann [3, § 18,3]).

  17. This holds also for plane absolute geometry [3] and for Hjelmslev Allgemeine Kongruenz-lehre [16].

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Authors and Affiliations

  1. Bochum, Germany

    Rolf Struve

  2. Institut für Mathematikdidaktik, Universität zu Köln, Gronewaldstraße 2, Köln, 50931, Germany

    Horst Struve

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  1. Rolf Struve
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Struve, R., Struve, H. Absolute isotropic geometry. J. Geom. 112, 1 (2021). https://doi.org/10.1007/s00022-020-00558-z

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