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Pascal’s Triangle of Configurations

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)

Abstract

We introduce an infinite class of configurations which we call Desargues–Cayley–Danzer configurations. The term is motivated by the fact that they generalize the classical \((10_3)\) Desargues configuration and Danzer’s \((35_4)\) configuration; moreover, their construction goes back to Cayley. We show that these configurations can be arranged in a triangular array which resembles the classical Pascal triangle also in the sense that it can be recursively generated. As an interesting consequence, we show that all these configurations are connected to incidence theorems, like in the classical case of Desargues. We also show that these configurations can be represented not only by points and lines, but points and circles, too.

Keywords

Combinatorial configuration Desargues–Cayley–Danzer configuration Geometric configuration Incidence sum Incidence theorem Point-circle configuration 

Notes

Acknowledgements

The author is grateful to the (anonymous) reviewers for their helpful comments. Special thanks go to Reviewer #1 for the insightful remarks and valuable suggestions that improved the exposition, results and proofs of this paper. This research is supported by the OTKA grant NN-114614.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

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