Blog 3 - Aristotle's Wheel

I think so far this year, the most mindblowing thing that I have experienced in this class was Aristotle's wheel. There aren't too many things that I take home from my math classes and ask my roommates opinions, but this was one of those things.

I became intrigued by this paradox because it simply had never crossed my mind. Walking to my car, I looked at every wheel/tire on a car that I passed. I just couldn't figure out how the heck that thing worked. So I took it to my trusty friend Wolfram Alpha and got some answers. The popular graphic of Aristotle's wheel seems as though it is illustrating that two circles with drastically different circumferences unroll themselves and end up having the same length. However, this is not the case. Aristotle's wheel is really illustrating one-to-one correspondence.
The site that helped me grasp this concept stated the following "The cardinalities of points in a line segment of any length (even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same." To me, this means that the red lines that seem to be representing the length of the circumferences of both circles are really illustrating the cardinality of both points. 
Something else that I found while researching this strange paradox, was a graphic that someone created after they isolated both the outer and inner circles onto their own lines. In this depiction of Aristotle's wheel, the circle on the top line is the smaller circle and the bottom circle is the larger, outer circle. It appears that the smaller circle is actually half the length of the outer circle. The outer circle has already rolled the length of the smaller circle when the line starts to change to green. The lines are traveling the same distance but at much different velocities. Overall, I think that this illustration is showing that the circles are both traveling along the same line, however they are taking different paths/speeds to get there. 

I would definitely like to spend a little bit more time looking at this in class and trying to find out as a class WHY the first illustration makes it seem like they are the same length. Each time I look at this picture, it still looks as though they are the same length, but this is just an illusion.

Comments

  1. John GoldenNovember 18, 2017 at 8:30 PM

    I'll say as a teacher, you leaving class wanting to know more and pursuing it, is about the best possible outcome. How I think of is just like W|A said - as cardinality. Even though the length of the circumferences is different, the number of points is the same. It's our misconception about how points work that creates the paradox. The wheel turning is really a function mapping the points of the inner circumference to the outer circumference.

    Nice writing on a problem of interest.

    C's 5/5

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