Comments on: Notes on Deferred Computation – the Pythagorean Spiral http://unlearningmath.com/2009/02/11/notes-on-deferred-computation-the-pythagorean-spiral/ math as a garden, friendly and always new Fri, 12 Mar 2010 22:55:25 +0000 http://wordpress.com/ hourly 1 By: Bert Speelpenning http://unlearningmath.com/2009/02/11/notes-on-deferred-computation-the-pythagorean-spiral/#comment-73 Bert Speelpenning Mon, 09 Mar 2009 03:41:54 +0000 http://bertspeelpenning.wordpress.com/?p=482#comment-73 I remember reading somewhere, decades ago, about a random walk and a square root. Best I can reconstruct it is that you imagine a drunk walking around, with no clue as to where he is going, and at any moment, the direction he proceeds in is completely random. That is, he is just as likely to go forward than to turn left, just as likely to turn right as to turn back the way he came - and so for each successive moment. The claim was that such a random walk would gradually move further away from the starting point, and the mean distance to the starting point would be proportional to the square root of the time walked. At the time I had no idea how such a thing could be analyzed, but having written the account of the Pythagorean spiral in terms of the distance from home versus the total distance traveled, it occurred to me there might be a direct link between the two. The Pythagorean spiral is done in discrete steps, but it isn't terribly hard to imagine a continuous version of it. Conversely, the drunk's random walk could be construed as done in discrete steps also, with the drunk always lurching a fixed distance in the new direction before lurching in a new (and independent) direction. Reasoning very crudely, you could say that "on average" the new direction is at right angles with the old direction: just as likely to be net forward as to be net backward. "On average", then, the distance from home for the drunk would be just like the distance of the person traveling the Pythagorean spiral. It would be interesting to pursue if this line of reasoning holds any water. I remember reading somewhere, decades ago, about a random walk and a square root. Best I can reconstruct it is that you imagine a drunk walking around, with no clue as to where he is going, and at any moment, the direction he proceeds in is completely random. That is, he is just as likely to go forward than to turn left, just as likely to turn right as to turn back the way he came – and so for each successive moment. The claim was that such a random walk would gradually move further away from the starting point, and the mean distance to the starting point would be proportional to the square root of the time walked.

At the time I had no idea how such a thing could be analyzed, but having written the account of the Pythagorean spiral in terms of the distance from home versus the total distance traveled, it occurred to me there might be a direct link between the two. The Pythagorean spiral is done in discrete steps, but it isn’t terribly hard to imagine a continuous version of it. Conversely, the drunk’s random walk could be construed as done in discrete steps also, with the drunk always lurching a fixed distance in the new direction before lurching in a new (and independent) direction.

Reasoning very crudely, you could say that “on average” the new direction is at right angles with the old direction: just as likely to be net forward as to be net backward. “On average”, then, the distance from home for the drunk would be just like the distance of the person traveling the Pythagorean spiral.

It would be interesting to pursue if this line of reasoning holds any water.

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