mathematical ideas in every day life: squaring the circle

I suspect, given where my thoughts currently are, that the following is going to be a bit rambling and somewhat labyrinthine. And even this, for a reason I’ll get to at the end, is not entirely correct, but I like the word, labyrinthine, and I’m feeling a bit indulgent today.

I was in a meeting a couple of weeks ago, and one of the other people in the meeting used the phrase, it’s like squaring a circle, to describe something difficult. I think the look on my face gave away my internal view, since everyone turned to look at me and started laughing in a friendly way at my evident discomfort. I decided that this was a situation in which discretion was truly the better part of valour, and I just let it slide.

I think that ‘squaring the circle’ and ‘we need to square the circle on this one’ have become commonly used phrases for something difficult, but this is not what these phrases should mean. And my apologies, gentle readers, but now I need to go into a bit of mathematics.

One of the classical geometry problems from antiquity, so from the ancient Greeks of roughly 2500 years ago, was the problem of squaring the circle. A reasonably constructed sentence, that of course tells you the reader no more than you knew about a question you didn’t know was in fact a question, except that it’s a old question. The Wikedia article is good, but I’ll summarise the essential bits. And this touches on what I started to do, and never finished, in the numerology of unrelated constants 1 and the numerology of unrelated constants 2, the completion of which has now gone back on the LIST OF THINGS TO DO.

So.  Take a circle in the plane, which has a radius R and which encloses an area of size A. The size of this area is given by the formula A = π R² and this is a formula that we have known since early days in school.

A question that you may wish to ponder is, why is this formula for area true?  It may be that you’ve never pondered this.  Of course, your teachers and the textbooks used state that this is the correct formula, but it isn’t until much later in our mathematical education that we develop the techniques that allow us to construct an argument to explain where such formulas come from.

Back to the question at hand.  The question of squaring the circle is this.  Given a circle C, of any radius R, construct a square S which encloses an area of the same size as the area enclosed by C, using only a compass (which allows for the construction of a circle of any center and any radius) and a straight edge (which allows for the drawing of lines of any length).  We can reframe this a bit.  We know that if S is a square whose sides of length s, then the area is enclosed by S is s².

And so, given the circle C of radius R enclosing an area of size A = π R², find a square S which also encloses an area of size s² = π R², again using only compass and straight edge.  And so things boil down to is, how to construct the square root of πusing only compass and straight edge.

And this is impossible to do, as we learned in the late 1800s.  The reasons are complicated, but the point that I would like to make is that this is actually impossible.  It’s not that it’s difficult.  It’s not that it might be possible but we don’t yet have an argument.  Rather, it cannot be done, and this impossibility is not something that sits well with most people.

Mathematicians I think are used to this notion of impossibility, and in fact I suspect that each of us, when we’re working on a question we’re finding difficult, always have the faint worry in the back of our minds that perhaps we’ve hit upon one of these impossible questions.  Almost always, it’s just that the question is hard or we haven’t yet seen the path to a solution.

As it turns out, there is a way of squaring the circle, if we allow remove the restriction of compass and straight edge.  But this other way involves doing violence to the circle in a way that is exceptionally hard to imagine.  Perhaps another day.

~ by Jim Anderson on 14 August 2017.

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