I’m not sure if everyone is aware of the concept of “squaring a circle.” The traditional definition of “squaring the circle” is to go back the ancient Greek mathematician “Oenopides” and his challenge to the Pythagoreans of the time to convert a circle into a square of identical area using nothing but geometric tools (protractors and straight-edges, basically). At the time Pythagoreans believed that you could solve pretty much any mathematical problem geometrically.

This is one of the things that absolutely keeps me enthralled with mathematics. Such a simple request, but the pursuit of this request spawned entire branches of mathematics.

As it turns out the effort to create a square of equal area to a circle drags one kicking and screaming into concepts as profoundly strange as “irrational” and “transcendental” numbers. To even begin to understand such concepts leads inevitably to calculus and the mathematics of infinite numbers. (For example, did you know that there is a distinction between “countable” and “uncountable” infinite sets?)

The reality of the challenge is that it is physically impossible to solve the problem using geometric tools. This is actually a quite profound result, which gave physicists and mathematicians their first glimpse into the concept that our physical reality cannot be described in entirely “rational” ways. It is very hard to describe how bizarre this result was when it was finally accepted. Among other things it forces us to recognize that the value of pi is not representable in any finite value, nor is it representable in any ratio of existing numbers NOR is it representable in any numeric sum or product of rational numbers, nor is it representable by any exponential equation involving rational numbers.

Or to put it plainly, the value of pi, which is central to geometry, CANNOT be exactly expressed algebraically. And, to make matters worse, it turns out there are OTHER such numbers, in fact an infinite number of them.

I think what has made me contemplate the irrationality of pi and the impossibility of squaring circles is the Obama administration’s attempts to “solve” our economic problems. Hopefully it won’t take us fifteen hundred years to understand enough of the problem that we can make some headway on it.

## 2 users commented in " Squaring circles…. "

Follow-up comment rss or Leave a TrackbackThis is interesting Cosmic. Having barely passed Algebra I, it’s a little over my head. ðŸ˜‰

But I know for sure that I can square a circle easily with Algebra (or at least get close enough for my purposes). I didn’t take enough mathematics to know about this particular problem, but I’m pretty sure that the great minds needed an Advil after trying to solve it with geometry.

I’m in total agreement with you. BarackO’ knows less about solving economic problems than I do about physics. We’re in trouble big!

You can reach any arbitrary approximation of pi but choosing a regular polygon with n sides where n is large enough to provide the approximation you need. But you can never do it exactly even if you draw a circle as large as the galaxy and put a polynomial with a googol of sides in it. It still won’t be “exact.”

but in general you can get “close enough” with a couple of octagons, one inside the circle, and one outside the circle, then averaging the result. That’s more or less how the ancients first approximated the value of pi.

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