Pi Day (or if you prefer, Tau Day, on June 28) is coming around again, and we will soon find ourselves measuring the circumference of circles, comparing circumferences to diameters, discovering a seemingly magical ratio, and then celebrating our efforts with plenty of pie.
But here’s the thing that bothers me about π day: we celebrate a special ratio (when in fact there are many special ratios in geometry) and marvel at the fact that we’ve found a number that “goes on forever” (when in fact almost all numbers have an infinite number of decimal places), but no one seems to take objection to the fact that the measurement we’re trying to do is IMPOSSIBLE!
Let me repeat. The task of accurately measuring the circumference of a (curved) circle using a (straight) diameter is impossible. You can’t measure π.
Of course it’s impossible, you say. π is irrational. If you measure a circle’s circumference in units of diameter and do a perfect job, the number you’ll get goes out to an infinite number of decimal places. Rulers can’t mark precision out to an infinite number of decimal places. So after we make the measurement, there’s no way to recite the exact number that we measured.
That’s true, of course. Any quantity can only be measured to a certain precision in reality. What I mean is that π is impossible to measure abstractly, on the 2D Cartesian plane. I contend that if you try to measure the circumference of a unit-diameter circle, you can’t even point to the exact value of the circumference — that is π — on the x-axis. And the reason for this is not explained by the irrationality of π.
There are plenty of irrational numbers whose exact position you can find, even if you can’t recite those numbers’ exact values using our decimal system. √2 is an example.
We know from the Pythagorean theorem that √2 is just the length of the diagonal of a unit square.
Now to find the exact location of √2 on the x-axis, rotate the square about the lower left vertex at (0,0) so that the diagonal lies along the x-axis.
Done! We can now see the exact location of √2 on the number line.
But it’s not so easy when we try to find the exact circumference of a circle.
We can imagine “unrolling” a circle with unit diameter so that the circumference lies along the x-axis. Then our exact location for π is the point on the x-axis that the fully unrolled circle lands on.
Let’s try unrolling the circle by dividing the circumference into 6 arcs and lining up each of the arcs with the x-axis.
When we unroll the circle with six arcs, each arc spans a distance of .5 on the x-axis. ((The chord length of an arc with angle measure 60º is equal to the circles radius.)) We tried to measure π, but we only got to 3! We’re 0.14159…, etc. short of the exact value.
You see immediately the reason we come up short: we’re using straight line segments to approximate the distance along curved arcs. Since the straight line distance is shorter than the curved distance, the sum of the straight line distances is less than π.
We can try again, this time breaking the circle into a larger number of arcs. That will get us closer to π, but it won’t get us all the way there. No matter how small the arcs, there will always be a bit of unmeasured circumference.
π is fundamentally a different kind of irrational number from √2, and that’s why we have so much trouble finding its exact location on the number line. √2 is an irrational number but it is also algebraic, meaning that it is a root of a polynomial with integer (or rational) coefficients. This means we can measure the distance √2 using analytic geometry.
But π isn’t algebraic. It’s transcendental. And with transcendental numbers there are no such guarantees. You can measure all you want, but on π day you’d better take care to eat your pie before it disappears!