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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

References

[1]Baker, H. F., Principles of Geometry (Cambridge University Press, London, 1922, Second Edition 1929), Vol. I, Foundations; Vol. III, Solid Geometry.Google Scholar
[2]Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system I: the user language. Computational algebra and number theory’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
[3]Brown, K. S., Min-Energy Configuration of Electrons on a Sphere, http://www.mathpages.com/home/kmath005/kmath005.htm, (1994–2008).Google Scholar
[4]Brylawski, T. H. and Lucas, T. D., Uniquely Representable Combinatorial Geometries, Atti dei Convegni Lincei, 17 (Accademia Nazionale dei Lincei, Rome, 1976), pp. 83–104 (in Proc. Vol. I of ‘Colloquio Internazionale sulle Teorie Combinatorie’, Roma, 1973).Google Scholar
[5]Carver, W. B., ‘On the Cayley–Veronese class of configurations’, Trans. Amer. Math. Soc. 6 (1905), 534–545.CrossRefGoogle Scholar
[6]Cayley, A., ‘Sur quelques théorèmes de la géométrie de position’, J. Reine Angew. Math. [Crelle’s J.] 31 (1846) (Collected Papers, vol. 1, p. 317).Google Scholar
[7]Coxeter, H. S. M., Introduction to Geometry, 2nd edn (John Wiley and Sons, New York, 1969).Google Scholar
[8]Dembowski, P., Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 (Springer, Berlin, 1968).CrossRefGoogle Scholar
[9]Fishback, W. T., Projective and Euclidean Geometry (John Wiley and Sons, New York, 1962).Google Scholar
[10]Glynn, D. G., ‘A note on Nk configurations and theorems in projective space’, Bull. Aust. Math. Soc. 76 (2007), 15–31.CrossRefGoogle Scholar
[11]Hilbert, D., Foundations of Geometry, 2nd edn. Translated from the 10th German edition by Leo Unger, Open Court, LaSalle, Ill. 1971.Google Scholar
[12]Hilbert, D. and Cohn-Vossen, S., Anschauliche Geometrie, 2nd edn (Springer, Berlin, 1996) (translated as Geometry and the Imagination).CrossRefGoogle Scholar
[13]Levi, F. W., Finite Geometrical Systems, Public Lecture Series, University of Calcutta, 1940 (University of Calcutta, Calcutta, 1942).Google Scholar
[14]McKay, B., NAUTY, The Graph Package (Australian National University, Canberra, 1984–2008), http://cs.anu.edu.au/∼bdm/nauty/.Google Scholar
[15]Möbius, A. F., ‘Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heissen?’, J. Reine Angew. Math. 3 (1828), 273–278 (plus figures 1 and 2 in Tafel IV of Heft 3) (Gesammelte Werke, Vol. 1 (1885), 437–443).Google Scholar
[16]Strambach, K. and Plaumann (eds), P., Geometry—Von Staudt’s point of view, Proceedings of the NATO Advanced Study Institute, Bad Windsheim, 1980(D. Reidel, Dordrecht, 1981).Google Scholar
[17]Tutte, W. T., ‘A homotopy theorem for matroids. I, II’, Trans. Amer. Math. Soc. 88 (1958), 144–174.Google Scholar
[18]Tutte, W. T., ‘Matroids and graphs’, Trans. Amer. Math. Soc. 90 (1959), 527–552.CrossRefGoogle Scholar
[19]Veronese, A., ‘Behandlung der projectivischen Verhältnisse der Räume von verschiedenen Dimensionen durch das Princip des Projicirens und Schneidens’ (Treatment of the projective relationships of spaces of different dimensions using the principle of projections and intersection), Math. Ann. 19 (1882), 161–234.CrossRefGoogle Scholar
[20]White, N., ‘The bracket ring of a combinatorial geometry. I’, Trans. Amer. Math. Soc. 202 (1975), 79–95; ‘The bracket ring of a combinatorial geometry. II: unimodular geometries’, Trans. Amer. Math. Soc. 214 (1975), 233–248.CrossRefGoogle Scholar
[21]Wilson, R. J., Introduction to Graph Theory, 3rd edn (Longman, New York, 1985).Google Scholar
[22]von Staudt, K. G. C., Geometrie der Lage (Bauer und Raspe, Nürnberg, 1847).Google Scholar