pituusneliöinti

Pituusneliöinti, also known as squaring the circle, is a geometric problem of constructing a square with the same area as a given circle using only a compass and straightedge in a finite number of steps. This problem has been a subject of mathematical inquiry for centuries, dating back to ancient Greece.

The area of a circle with radius $r$ is given by the formula $A_{circle} = \pi r^2$. The area of a square with side length $s$ is given by $A_{square} = s^2$. For the areas to be equal, we must have $s^2 = \pi r^2$, which implies $s = r\sqrt{\pi}$.

The challenge of pituusneliöinti lies in the tools allowed. A compass and straightedge can only construct lengths that are algebraic numbers, meaning they are roots of polynomial equations with integer coefficients. The number $\pi$, however, is a transcendental number, meaning it is not a root of any such polynomial equation.

Due to the nature of transcendental numbers, it has been proven that pituusneliöinti is impossible using only a compass and straightedge. This impossibility was definitively established in 1882 by Ferdinand von Lindemann, who proved that $\pi$ is transcendental. This means that no square with an area exactly equal to that of a given circle can be constructed using the classical Euclidean tools. While approximate solutions can be found, a perfect construction is mathematically impossible.