User:John Reid/Pi/Unrolled

The mathematical constant '''$$\pi \approx {3.14159...}$$ can be defined as the circumference of a circle with diameter 1. There are many mathematical formulas that can be used to compute π; for more, see the main article Pi. Below is a visual, animated illustration of the definition as a circumference.



A straight line may be marked off in equal measures, each the diameter of a given circle. This may be done legitimately with the classical construction methods of Euclidean geometry, using compass and straightedge.

Departing entirely from classical construction, we may mount the given circle on a wheel and roll it along the straight line, unrolling the circumference as we go. The wheel rim is covered with red paint, which is transferred to the road as the wheel travels. As shown by the animation. it travels an unusual, counterintuitive distance before making a full revolution: almost one-seventh again beyond three diameters. This is the number π.

The Greeks of Euclid's time sought to find a way, using only compass and straightedge, to construct π geometrically. As π can also be defined as the area of a circle with radius 1, and they tried to find a construction for a square of equal area, this problem is known as "squaring the circle". A solution to this problem requires not to construct a line of length π but of the square root of π, which is about 1.77245... . However, both π and its square root are transcendental numbers, whereas the classically constructible lengths are all algebraic. Thus, they were looking in vain: a classical geometric construction is not possible.