This past week I was shown something that peaked my interest! That something was Aristotle’s wheel. To begin I will show you what Aristotle’s wheel looks like!
Can you notice the issue in this picture? Think Mathematically…
Maybe a video will help? Aristotle’s wheel
Or if you prefer a Geogebra animation
I will give you some time to let that sit in your head, maybe you have already guessed what seems to violate what you know about circles, but maybe you have not. I often times enjoy checking into history, so perhaps that will lead us to the issue, but also the solution!
To begin there is much argument over who thought of this problem first, but we are more concerned with the mathematics behind it rather than who found the problem. The story must begin with Aristotle( unless you have found better historical journals about this topic, please comment and post the link to the comment). The problem first appeared in Aristotle’s the Mechanica. Aristotle’s wheel stumped him, and Galileo along with many mathematicians throughout history as well. James Cardan was one of the greats to take a shot at Aristotle’s wheel, but all he did was identify a point to point correspondence, but did little to develop that idea (Drabkin, 1950). It took over one thousand years from the time the problem was conceived to the time that people could explain the paradox clearly.
Aristotle’s wheel is a paradox because there are two wheels one within another that are spinning. The issue comes in that the smaller wheel covers the same distance as the larger wheel when they are both rotated together by one full revolution, but the smaller wheel has a different circumference than the larger wheel right? So how can this be? Here’s the wheel again.
Check the wheel again and think about cars, friction, and any other concepts of rotation that you know of.
The Solution or what I understand of it!
The solution has come about in two major approaches. The physics approach contends that the inner wheel has the same line length because there is much greater friction on a sliding object than a rolling circle. Thus, the outer wheel appears to have the same length as the inner wheel because the inner rim has more friction acting upon it, so the inner wheel turns in smaller turns than the outer wheel, but overall turns the same length.
The point to point correspondence approach contends that regardless of the rotation rate, the inner part of the wheel’s points on the line can connect to the outer line with a one to one correspondence. Thus, I believe an appropriate connection would be to say that the points on the inner wheel are an injection with the points on the outer wheel of Aristotle’s wheel as they rotate forward. So though, the outer wheel appears to have a much longer circumference the inner wheel can still form a one to one correspondence with the points of the outer wheel. Thus, the outer wheel appears to match the length of the circumference of the inner wheel when they are rolling from one space to another.
So either by a point to point correspondence and an understanding of the slipping of the inner wheel we see that Aristotle’s wheel is a paradox which is true and does not violate mathematical laws though it appears to do so. Aristotle’s wheel has vexed mathematicians through the ages, but it has great implications for students and scholars today. The problem shows us that what we see is not always what is. By unraveling the deeper context and possible ways to view problems, students and scholars alike can solve problems that seem to contradict the laws of mathematics or the elements set in place in other subjects. Aristotle’s wheel is a wonderful problem to help students think abstractly and to help them persevere in problem solving via working at a problem that initially makes no sense. I gave this problem to eight grade students this week and with support the students were able to grasp the paradox, and begin working toward a solution. I am looking forward to making a lesson plan on the paradox in the future because this problem challenges students to develop flexibility and apply many mathematical practices!
“The curious paradox is that when I accept myself just as I am, then I can change.”
Drabkin, I. E.. (1950). Aristotle’s Wheel: Notes on the History of a Paradox. Osiris, 9, 162–198. Retrieved from http://www.jstor.org/stable/301848