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Monday, August 29, 2016

Ford Circles

(via WikimediaCommons)

Awhile back I mentioned Alfred Posamentier’s latest volume “The Circle,” and around now it should be showing up in bookstores -- another great little geometry offering from Dr. Posamentier (and Robert Geretschlager). One of so many interesting tidbits in it is about “Ford circles”:

Imagine you have two tangent circles sitting atop a number line, one tangent to that line at “0” and the other tangent at “1.” Now in the space between these circles draw another circle tangent to both the “parent” circles and to the number line as well — it will touch the number line at the 1/2 position. You can keep iteratively drawing such circles (to infinity) in the space created with each new (smaller) circle. Now, quoting from the book:
“Of course, the circles get very small very quickly. As it turns out points of tangency of all these infinitely many circles with the number line have a quite unexpected property. The points of tangency are precisely the rational numbers in the interval between 0 and 1. No circle created by this process touches the number line at an irrational point, and every rational number is the point of tangency for some circle created in this manner.”
Pretty amazing, and a nice demonstration of one area of mathematics, plane geometry, connecting to other areas of infinity and number theory. Further, these circles relate back to Farey sequences.
Here’s one of several treatments of Ford circles on the Web:

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