Mathematical Impressions: Bicycle Tracks

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Bicycle Diagram

A nice mathematical puzzle, with a solution anyone can understand, is to determine the direction a bicycle went when you come upon its tracks. The answer involves thinking about tangent lines, geometric constraints and the bicycle’s steering mechanism. Once you learn the trick, you’ll find yourself using it every time you happen upon a bike trail.

The question goes back to the early days of the bicycle age and a 1903 Sherlock Holmes story by Sir Arthur Conan Doyle called “The Adventure of the Priory School.” Surprisingly, Holmes did not analyze the tangents as we do in the video and his reasoning in the story was incorrect. A geometric approach to the question was included in a 1990s “Geometry and the Imagination” course that John Conway, Peter Doyle, Jane Gilman and William Thurston taught at Princeton and the Geometry Center at the University of Minnesota. Holmes’ error is discussed in the book “Which Way Did the Bicycle Go?” by Joseph D. E. Konhauser, Dan Velleman and Stan Wagon.

 

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More videos from the Mathematical Impressions series.

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  • The viewers might be interested in current research on “bicycle mathematics”:
    http://www.tphys.uni-heidelberg.de/~wegner/Fl2mvs/Movies.html,
    (where one finds other examples, besides the concentric circles); and
    http://www.math.psu.edu/tabachni/prints/bike7.pdf or math/0405445
    (Israel J. Math., 151 (2006), 1-28)
    http://www.math.psu.edu/tabachni/prints/bicycle9.pdf or arXiv:0801.4396
    (Experimental Math., 18 (2009), 173-186),
    http://www.math.psu.edu/tabachni/prints/FLT8.pdf or arXiv:1207.0834
    (March 2013 special issue of the American Mathematical Monthly),
    http://www.math.psu.edu/tabachni/prints/Discrbike3.pdf or arXiv:1211.2345
    (preprint)

  • This still does not solve the problem fully. There are bikes that go in reverse gear when you pedal backwards, and in that case one cannot say which way the bike came from because the tracks are exactly the same for the two bikes. It would be analogous to riding a bike on which the back wheel stirs. To complicate the problem further, when the vertical axis of the bike is tilted, the bike starts to turn towards the tilting side, which is a result of a torque on tire-ground intersection, generated be the gravity. On a bike that only turns by that mechanism (operated by a very skilled biker!), it would be again, impossible to say which way the bike came from.

  • Fascinating, as always. Note that a complicating (but surmountable) detail is glossed over in the video. Consider the red and blue tire tracks left by the model bicycle. The solid tangent lines drawn by Prof. Hart cross the front-wheel track *multiple* times if extended indefinitely. Prof. Hart sometimes chooses the first crossing, and at other times the second. In practice, one would presumably ignore crossings that don’t correspond to a plausible bicycle wheelbase.

    The New Annotated Sherlock Holmes devotes a few pages to the history of the debate over the error Holmes makes, and possible explanations (including a suggestion that Dr. Watson simplified the explanation in his recounting of the tale!). In his memoir, Sir. Arthur Conan Doyle admits to the error, and offers an alternate solution:

    ‘There are some questions concerned with particular stories which turn up periodically from every quarter of the globe. In “The Adventure of the Priory School” Holmes remarks in his offhand way that by looking at a bicycle track on a damp moor one can say which way it was heading. I had so many remonstrances upon this point, varying from pity to anger, that I took out my bicycle and tried. I had imagined that the observations of the way in which the track of the hind wheel overlaid the track of the front one when the machine was not running dead straight would show the direction. I found that my correspondents were right and I was wrong, for this would be the same whichever way the cycle was moving. On the other hand the real solution was much simpler, for on an undulating moor the wheels make a much deeper impression uphill and a more shallow one downhill, so Holmes was justified of his wisdom after all.’ http://gutenberg.net.au/ebooks14/1400681h.html#ch-11

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