Wednesday, June 18, 2014

Daily 13: Mathematical Paradoxes

Aristotle's Wheel

Aristotle's Wheel is a paradox found in the Mechanica and consists of two wheels, one within another, that roll along a surface. It is constructed so, like in the image below, two horizontal lines trace the bottom of the circle and appear to be the circles' circumferences.

The paradox comes from the assertion that both horizontal lines are the same length despite the fact that each circle has a different circumference. To solve this paradox, we have to first accept the fact that the inner circle has a smaller circumference than the outer circle. So, the top line must be shorter than the bottom line if they do in fact measure their respective circle's circumference. Referencing the image above, notice that the red lines give the impression that the circumference of each circle is rolled out onto the horizontal lines. It is this illusion that is fallacious: while the outer circle is actually unrolling its circumference onto the lower line, the inner circle is both unrolling and sliding. This fact is masked by the visual effect of the red line being transferred from the inner circle to the upper line.

Further, the line within the circles representing their radii suggests even more that the illusion is correct because it helps to reinforce the appearance that each circle's circumference is being unrolled at the same rate. This too is fallacious since it suggests that, simply because an analogous point is traced on similar circles, the circumferences must be equal.

In reality, at least one of the circles would have to be dragged to some degree if the other is assumed to be rolling along a surface. For example, imagine a car wheel. While the car is in motion, it is clear that both the tire and hubcap have the same angular velocity, otherwise either the tire or hubcap would be spinning faster than the other and, unless you have spinners, this isn't going to happen. Rather, the hubcap, if its rotation and linear movement were recorded, would be seen to be dragged along the path of the tire. That is, the linear movement the hubcap traced would be longer than the rotational distance it would have traced had the hubcap been independent of the tire. If the red lines were removed from the above animation, this would be more apparent.

Achilles and the Tortoise

Zeno proposed a paradox that states Achilles cannot win a race against a tortoise. Suppose Achilles is racing a tortoise and the tortoise, being slower than Achilles, is given a head start. When the race starts, Achilles will eventually reach the point where the tortoise starts. By this time, the tortoise will have moved some distance ahead. Achilles will then have to run further to reach the tortoise but by then, once again, the tortoise has made it some distance more. The paradox states that the fast-running Achilles will never reach the slow tortoise since every time Achilles progresses and reaches a point where the tortoise used to be, the tortoise has made it some distance more. 

Clearly, a fast runner is able to win against a slow runner. So a solution to the problem must exist. Unfortunately Zeno didn't expect us to have the aid of an idea from calculus, convergence. Even though the tortoise is continually moving, it's at a slower pace and Achilles will eventually catch up. This is because, although we can mathematically express the distance Achilles has travelled as a faction of the distance the tortoise has travelled, this fraction will converge to 1.

To see this, suppose after the first measurement, the tortoise has moved far enough so that Achilles has travelled 0.9 the total distance of the tortoise (such a point must exist since Achilles is moving faster). After the second measurement, Achilles has covered 0.99 the total distance of the tortoise. Then 0.999. Then 0.9999, and so on. This will eventually (or, more realistically, very quickly) converge to 1, implying that Achilles has caught up to the tortoise. Simply stated, the infinite number of small distances Achilles has to travel is offset by the infinitely decreasing amounts of time it takes Achilles to reach the previous point of the tortoise. At this point Achilles can take the lead. Many of Zeno's paradoxes are solved in a similar manner.


  1. These + last reflection + math of today = 15/15 dailies.

    I loved having you do them this way - do you think that would have worked well for most or are you just more comfortable with this mode?

    1. I'd imagine doing something similar would work well for some people, but it somewhat limited what I was able to do for my daily work. For instance, I stuck to mostly writing about topics rather than creating actual projects (drawings, etc.) because the blog format made it easier that way. I liked it because I had all my work recorded in one place and I didn't have to keep track of (or forget) what I did or didn't do. It just seemed like it'd be easier for both you and me doing it this way.