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The Ministry of Silly Walks

June2

We have been posting about directions which brings us to a rather different idea. John Cleese (see his website here) was hilarious in the Monty Python’s “Ministry of Silly Walks” (click on image to link to the video at YouTube) but, to the young growing mathematician, such fun has a more serious site.

The diagram below relates the seemingly random flow of rivers to that of human journeys on foot. The author, Brian Hayes, in his American Scientist mathematical article, mentions that, “Trajectories of random walks meandering across a plane may hold clues to the nature of river meanders. A walker departs from point a at an initial bearing of 110 degrees (counterclockwise from the positive x axis) and takes steps of unit length; at each step the walker’s direction rotates by an angle selected at random from a normal distribution with a mean of zero and a standard deviation of about 17 degrees. After exactly 40 steps, if the walker is within one unit of point b, it moves directly to b; all other walks are discarded. A million trials yielded 259 success- ful walks. The average trajectory (yellow) is calculated by averaging the x and y coordinates of the accepted walks at each step; it traces the movement of the center of mass of the population of walkers. The average path resembles a sine-generated curve, but the individual paths are highly variable; some even intersect themselves, as shown by the example highlighted in red.

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–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
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[I think that the left had side was a mistake and should have read (x+4)?]

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