Pappus configurations in finite Hall affine planes
Date
2020
Authors
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Publisher
University of Delaware
Abstract
In the classical projective planes, both Desargues's theorem and Pappus's theorem hold. According to a result of Ostrom, the Desargues configuration can also be found in every finite projective plane on at least twenty-one points, classical or not. In fact, Ostrom's argument shows that the number of Desargues configurations in every finite plane is actually quite large. The result is also true in the finite projective plane on thirteen points. The existence of Pappus configurations in every non-classical finite affine or projective plane is unknown. We study whether the Pappus configuration is present in such planes. ☐ In particular, we endeavor to prove that in finite Hall affine planes, the following strong version for the existence of Pappus configurations holds: For every pair of lines ℓ1, ℓ2 and every triple of points on ℓ1 and every choice of a single point on ℓ2, a pair of points on ℓ2 can be found to complete a Pappus configuration. This statement is not proven in every case. When it is not, weaker versions for existence are shown. Hall planes are not Pappian, yet this work implies that the number of Pappus configurations in Hall planes is actually quite large.
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Keywords
Affine planes, Configurations, Finite planes, Hall planes, Pappus configurations