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Thrackle Conjecture

Drawings constructed with hard tools investigating the Thrackle Conjecture, a pure geometry problem proposed by the mathematician John Horton Conway (England, 1937-2020) in the 1960's. A Thrackle is a drawing with lines and endpoints where each line touches every other line once and only once, either by touching it's endpoint or moving across it. By defenition: "A thrackle is an embedding of a graph in the plane in which each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, they must cross at their intersection point: the intersection must be transverse." (Lovász, L.; Pach, J.; Szegedy, M. (1997), "On Conway's thrackle conjecture", Discrete and Computational Geometry, 18 (4): 369–376). Conway conjectured that, in any thrackle, the number of edges is at most equal to the number of vertices. Conway himself used the terminology paths and spots (for edges and vertices respectively), so Conway's thrackle conjecture was originally stated in the form every thrackle has at least as many spots as paths.

Conway's conjecture has never been disproven - no one has ever made a drawing of this kind that has more paths than spots. Below is a gallery of my drawings displaying the myriad directions the problem has taken me over the years. Descriptions are written on the drawings themselves, all 11 x 17 inches, graphite on bristol.

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