You might have come across real life accounts of how a seasoned hunter can read the pug-marks of animals and figure out surprising amount of information from them, quite Sherlock Holmes style. Now move this scenario to the concrete jungle and consider the case where you are given track marks of a bicycle wheels, can you figure out which way was the bicycle moving?
In the book “Genius at Play”, a biography of John Conway, Siobhan Roberts discusses an interesting problem. For one of the classes, Conway and his co-teachers rode bicycles on large rolls of paper and handed out the resulting tracks to the students to analyze and figure out which way the bicycle moved.
There is an easy algorithm to answer the question, which has to do with curvature of the tracks and drawing tangents and measuring distances along them.
Notice that the track made by the front wheel always has a larger curvature because when a turn is being made by the front wheel, the back-wheel is free to move only along the direction of the bicycle.
If you draw a tangent to a point P on the track made by the back-wheel of the bicycle, then the direction of the tangent shows the direction of the velocity of the back-wheel. Since this velocity is along the direction of the bicycle frame, if one moves along the tangent, there has to be a point on the outer (front-wheel) track which is exactly at the distance between the two contact points of the wheel. Thus the direction along the tangent in which the distance between the contact point P (on the inner curve) and the point on the outer curve Q stay the same, no matter which point P you choose, is the direction of motion of the bicycle.
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There are two kinds of people in the world, those who enjoy mathematics and those who don’t. What is interesting is that even those who enjoy mathematics don’t quite agree on how should mathematics be taught. On one side there are people like the famous mathematician John Conway who would carry fruits and vegetables to the classroom to teach students about geometry and curvature while on the other side there are people like the famous computer scientist Dijkstra who insist that right from the start, students should learn rigorous abstract concepts. My own belief is that majority of people develop appreciation for mathematics only through specific examples and that it takes a level of expertise before which one may start appreciating rigorous mathematics. I belong to the first category of people who think that playing games, looking at bicycle tracks and thinking of real life situations is a much better way of getting introduction to mathematics.
Saw on a T-shirt in the canteen today 🙂