Abstract
The principle of duality is well established in projective geometry but can hardly be found in the literature on Euclidean geometry where it is “more a principle of analogy than a scientific principle with a logical foundation” (cp. Sommerville, The Elements of Non-Euclidean Geometry. The Open Court, London, 1919). We close this gap and develop a theory of duality in Euclidean geometry. Following Hilbert’s Grundlagen der Geometrie we consider the incidence, order and metric structure of a Euclidean plane and show (a) that there is a large class of theorems of Euclidean incidence geometry which allow a dualization (b) that Hilbert’s order structure can be introduced in a Euclidean plane in a self-dual way and (c) that appropriate definitions of metric notions (e.g., of an angle, a segment or a circle) lead to Euclidean theorems with meaningful dual versions. This shows that duality in Euclidean geometry is not a collection of isolated phenomena but corresponds to a rich and coherent theory.
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Notes
As in deVilliers (2009) where mainly fragments of duality are studied and “not a true duality like that in projective geometry”.
We consider in this article only projective planes which can be coordinatized over fields of characteristic \(\ne 2\), i.e., which satisfy the axioms of Pappus and Fano.
If all assumptions \(A_{i}\) are positive literals then (4.1) is the normal form of sentences of ‘geometric theories’ which have the property that a theorem, which can be derived within classical logic, can also be derived within intuitionistic logic (see Pambuccian 2017 for a discussion within the context of a direct proof of the Steiner–Lehmus theorem and Dyckhoff and Negri (2015) for the so-called geometrisation of first-order logic).
Or, equivalently, by ‘If two of the lines a, b and c are incident with E then E is incident with all lines a, b and c’.
The set of points on a line a is called a row of collinear points. The set of lines through a point A is called a pencil of concurrent lines.
The theorem of Pappus implies the theorem of Desargues (see Coxeter 1961, §14.3).
i.e., if all flags are conjugated or, equivalently, if any two lines with a common point have a bisector.
In the sense of Definition 4.6.
Lines a, b, c lie in a pencil if \(abc \in S\) (see Bachmann 1973, §3,5).
For the notion of a quadratic set of points of a projective plane we refer to Buekenhout (1969) who introduced this notion for a characterization of projective conics.
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Struve, R. A theory of duality in Euclidean geometry. Beitr Algebra Geom 59, 221–246 (2018). https://doi.org/10.1007/s13366-017-0370-6
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DOI: https://doi.org/10.1007/s13366-017-0370-6