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.
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Notes
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Not every incidence statement involves merely a single incidence in this sense. For example, the author presented a conjecture in [9] which is connected with a \((100_4)\) configuration; the conjecture essentially states that if 350 (suitable) incidences in this configuration are satisfied, then the remaining 50 are also satisfied. (We note that the same \((100_4)\) configuration also occurs in [17], in two different versions.).
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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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Gévay, G. (2018). Pascal’s Triangle of Configurations. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_10
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