## No. 93: Unreachaball

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Strip by: Ian Boreham

{Zeno, at the left, throws a ball over to the right. One of his dogs is already halfway towards the ball, and starts running for it. His other dog is standing next to him, and also starts running towards it. Zeno continues to watch them as they get closer and closer to the ball. One dog is always halfway between the other dog and the ball. As they get closer, they get visually compressed in the direction they are running, until the pair of dogs becomes a vertical line right next to the ball, never quite reaching it. Meanwhile, Zeno sits, lies down, dies and becomes a skeleton, while waiting for the dogs to reach the ball.}

caption: Zeno's Pair o' Dogs.

The author writes:

I remember hearing about Zeno's paradoxes when I was a teenager, and deciding that they were obviously wrong given my daily experience; and then after learning more mathematics and physics, deciding that I could prove it. It appears, though, that Zeno was actually trying to do something more subtle than simply making an argument that looks logically sound, but fails to correspond to reality. He was trying to get people to think about what motion means, and how it could even make sense, and it seems that these aspects are still debated, even though the conclusions that the paradoxes draw are not considered valid. Anyway, see Wikipedia.

The dogs in the strip are actually exemplifying two of the paradoxes: Achilles and the Tortoise (Achilles gives the Tortoise a headstart in a race; by the time Achilles has run to where the Tortoise started, the Tortoise will have moved a little further; by the time Achilles runs to that point, the Tortoise will have run a little further again, etc; so Achilles can never catch up to the Tortoise) and the Dichotomy (in order to reach the end of a path, Homer must first walk half the distance; from that point, he must walk half the remaining distance, etc; so he will never reach the end of the path; or since he has to walk halfway to the halfway point first, etc, he can never even begin).

Of course, if Achilles can never catch the Tortoise because he has to keep reaching the Tortoise's previous position first, then he is moving in an infinite series of steps of decreasing length with a bounded sum, which take decreasing time with a bounded sum. So, once the race has started, we know that time itself can never progress beyond the point where he catches up to the Tortoise. If this effect is global, then the race would doom the universe. If it's local, it would be very useful as a mechanism for time-freezing a region, to prevent accidents, catch criminals, do physics experiments, win masses of money through speculation or gambling, etc. Perhaps you can get your "Achilles" to bail out after a certain distance, to allow the normal flow of time to resume.

This local time-freezing effect appears to be the situation depicted in the strip, since Zeno dies while waiting for the dogs, who are apparently still alive in their time-dilated zone. Since their position is measured by their centroids, compression in the direction of travel is required to prevent them from reaching the ball. The combination of time-dilation and compression in the direction of motion make me suspect that Zeno stumbled across some weird form of relativity, but his contemporaries were unable to grasp it.

While I was writing this annotation, Mezzacotta AC suggested this could be used as a colouring-in sheet for children, but it needs an additional instruction that they should colour in exactly half of the remaining uncoloured proportion each day until finished.

Drawn in Krita and Inkscape.