Clean, sweep, polish—the house is always spotless. At least, until you look too closely in the corners, where a little dust invariably remains. Looking away? You are probably telling yourself that this flaw in your housekeeping is so unobtrusive that your guests will barely notice it.
Mathematics has no wish to meddle in how you keep your home. It does, however, have theorems showing that this is not so, and that tiny specks of dust can prove far more substantial than they appear when viewed from the right angle. To illustrate this result, we shall draw a floor plan of your living room on a sheet of paper, identify both the room and its uncleaned area with subsets of the ordinary Euclidean plane, and finally reduce the problem to seemingly elementary geometric considerations.
A seemingly satisfactory sweep
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Your young niece has started computer science lessons and is enthralled: she talks about them constantly and programs whenever she gets the chance! On her computer, of course, but also on anything else that can be programmed... And so your little robot vacuum finds itself illustrating her lesson on recursion in a curious ballet whose successive movements recall the usual construction of the fractal Cantor set (see box).
Your living room is square, with side length c. To simplify the diagram and explanations, we shall take it more precisely to be the square whose vertices are the origin and the points with coordinates (c, 0), (c, c), and (0, c). In other words, we have equipped the plane with an orthonormal coordinate system centered at one corner of the room, with its axes running along the two walls that meet there. Initially—that is, before the cleaning begins—the whole room, or square, is covered in dust.
