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BRUCK NETS AND PARTIAL SHERK PLANES

Part of: Metric geometry Finite geometry and special incidence structures

Published online by Cambridge University Press:  19 June 2017

JOHN BAMBERG ,
JOANNA B. FAWCETT  and
JESSE LANSDOWN Open the ORCID record for JESSE LANSDOWN [Opens in a new window]
JOHN BAMBERG
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email John.Bamberg@uwa.edu.au
JOANNA B. FAWCETT
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email j.b.fawcett@dpmms.cam.ac.uk
JESSE LANSDOWN*
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email Jesse.Lansdown@research.uwa.edu.au
*
✉ Jesse.Lansdown@research.uwa.edu.au
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Abstract

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In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

Keywords

Bruck netaffine planefinite geometrymetric plane

MSC classification

Primary: 51E14: Finite partial geometries (general), nets, partial spreads
Secondary: 51E05: General block designs 51E15: Affine and projective planes 51F05: Absolute planes

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 1 , February 2018 , pp. 1 - 12
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author acknowledges the support of the Australian Research Council (ARC) Future Fellowship FT120100036. The second author acknowledges the support of the ARC Discovery Grant DP130100106. The third author acknowledges the support of the ARC Discovery Grant DP0984540.

†

Current address: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

References

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Dembowski, P., ‘Finite geometries’, in: Classics in Mathematics (Springer, Berlin, 1997), reprint of the 1968 original.Google Scholar
Sherk, F. A., ‘Finite incidence structures with orthogonality’, Canad. J. Math. 19 (1967), 1078–1083.CrossRefGoogle Scholar