Doodling with Fractals and Persistent Worms.

Suppose we have a rubber line of length 1 m and a worm at the other end. The worm moves 10 cm in a minute and its goal is to reach the other end moving along the rubber line. However every one minute the line is being stretched to be ten times longer than it was. So after one minute the line is 10 m, after two minutes it is 100 m and so on. The rubber grows homogeneously, i.e. the distance both in front and behind the worm grow ten times bigger every minute.

Does the worm ever reach the other end of the rubber line?

There is a video by Vi Hart somehow related to this kind of recreational thinking, in my opinion awesome:

Link to the vieo

About Vadim Kulikov

For details see this
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1 Response to Doodling with Fractals and Persistent Worms.

  1. Tjl says:

    No. Consider the fraction f of the line that the worm has traversed. f(0) = 0, f(1) = 0.1, f(2) = 0.11, f(3) = 0.111
    (this is assuming that the lengthening would happen in a single moment after the minute has elapsed but that won’t alter the result). This series converges to 1/9.

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