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THEOREMS OF POINTS AND PLANES IN THREE-DIMENSIONAL PROJECTIVE SPACE

Part of: Linear incidence geometry Geometry Designs and configurations

Published online by Cambridge University Press:  02 February 2010

DAVID G. GLYNN
DAVID G. GLYNN*
Affiliation:
School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
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Abstract

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We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

Keywords

geometryconfigurationprojective spacetheoremminimal energy electronsforbidden minorFanoPappusDesarguesMöbius

MSC classification

Secondary: 51A20: Configuration theorems 05B25: Finite geometries 05B35: Matroids, geometric lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 1 , February 2010 , pp. 75 - 92
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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