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**Thrackle Conjecture Part I**

September 17, 2018

Â Thrackle Conjecture, Pencil on Bristol, 11 x 17 inches, 2018

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These things are "thrackles," drawings where every lineÂ touches every other line once and only once (by endpoint or traverse). Everyone draws thrackles all the time - a five pointed star and a triangle are both thrackles. This concept isÂ remarkably simple which is interesting because the thrackle conjecture is one of the great unsolved geometry problems in mathematics. The goalÂ is to draw a thrackle that has more lines that points. Since the problem was proposed byÂ mathematician George Horton Conway in the 1960's not a single thrackle has been discovered that fits these constraints. Conway himself accepts drawings of thrackles by snail mail at his office at Princeton, if you can draw one he'll mail you $1,000. Above are my thrackle studies, all correct thrackles but none of them are a counterexample to the Conway Conjecture.

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Star Thrackles, Pencil on Bristol, 11 x 17 inches, 2018

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Two more complicated thrackles. The left one is a 18-vertex enneagram thrackle and the right is a 14-vertex heptagon thrackle. My current goal is to draw every conceivable known thrackle in search of the counterexample. The likelihood that I, or anyone, will solve it is exceptionally low but I like the idea of an artist solving a problem within pure mathematics. One can dream.

Â Cycloids,Â Pencil on Bristol, 11 x 17 inches, 2017

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This is an engineering drawing from last year that I forgot to post. These are cycloids, they determine the point on a circle as it rolls along a path. This is important in engineering because it gives you the ability to determine the location and spacing of gear teeth. The upper right drawing, for example, shows the path of a point on a gear as it rotates inside a circle. Computers take care of this kind of work today but as a drafting historian I was eager to draw them - beautifully simple but really difficult to draw!

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