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What Can You Draw?

For his research project during the summer of 2019, Fei Peng raised the following deceptively simple question: What can you draw?

Your canvas is the plane 2 — colored white to begin with — and you are given a pencil, which produces a black unit disk wherever it meets the canvas, and an eraser, which produces a white unit disk. There are no further restrictions on your artistic freedom. You may raise the tool off the canvas, that is, there is no continuity requirement for the centers of disks you draw and you can switch tools as many times as desired. Peng and his research mentor Florian Frick showed the main result is that drawability cannot be characterized by local obstructions. A bounded set can be locally drawable, while not being drawable. The figure presented here is an example of such a set; the boundary has curvature less than one, but the set is not drawable in the large.