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→The classical unsolved problems
==The classical unsolved problems==
==The classical unsolved problems==
There were three very famous geometrical problems that they couldn't solve. Listed by their popular names, they are:
There were three very famous geometrical problems that they couldn't solve. Listed by their popular names, they are:
*"Squaring the circle"—Given a circle, construct a straight line whose length is equal to the circle's circumference. Equivalently, construct a square with the same area. Equivalently, given a square (or rectangle), construct a circle with the same area or the same perimeter. This can be seen to be the problem of constructing the ratio <math>\pi</math>.
*"[[Squaring the circle]]"—Given a circle, construct a straight line whose length is equal to the circle's circumference. Equivalently, construct a square with the same area. Equivalently, given a square (or rectangle), construct a circle with the same area or the same perimeter. This can be seen to be the problem of constructing the ratio <math>\pi</math>.
*"Duplicating the cube"—Given the length of the side of a cube, construct the length of the side of a cube with twice the volume. This can be seen to be the problem of constructing the cube root of 2.
*"Duplicating the cube"—Given the length of the side of a cube, construct the length of the side of a cube with twice the volume. This can be seen to be the problem of constructing the cube root of 2.
*"Trisecting an angle"—Given an arbitrary angle, divide it into 3 equal angles. Some angles (e.g. 90 degrees) are easy to trisect, but the problem is to do it for any angle. The ancient Greeks could bisect arbitrary angles (hence divide them by 4, 8, 16, ...) but could not trisect them. [[Pythagoras]] himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. This problem is equivalent to solving cubic equations.
*"Trisecting an angle"—Given an arbitrary angle, divide it into 3 equal angles. Some angles (e.g. 90 degrees) are easy to trisect, but the problem is to do it for any angle. The ancient Greeks could bisect arbitrary angles (hence divide them by 4, 8, 16, ...) but could not trisect them. [[Pythagoras]] himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. This problem is equivalent to solving cubic equations.