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Incidences and tilings

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arxiv 2305.07728 v2 pith:2W6YMYML submitted 2023-05-12 math.CO math.MG

classification math.COmath.MG
keywords theoremsincidencearbitrarycasesclassicalclosedcomplexdesargues
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We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, M\"obius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Tropical linear series and matroids

    math.AG 2025-08 conditional novelty 7.0 of 10

    Tropical linear series on metric graphs are locally Bergman fans of matroids, yielding an exact condition for canonical tropicalizations to fill the realizable locus.

  2. Shear coordinates and braid invariants

    math.GT 2025-05 conditional novelty 5.0 of 10

    The paper constructs a braid invariant from shear-coordinate transformations applied to the edges of Delaunay triangulations.

  3. Octagon and tropical octagon yield braid invariants

    math.GT 2024-11 reject novelty 5.0 of 10

    Braids in the projective plane act on labels of a dual graph via the Desargues flip, and isotopic braids produce identical label transformations.

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