The duplication of the cube is a similar problem known as the Delian problem.
The story goes that the natives of Delos in Greece consulted the oracle in the face
of a plague they were suffering. They were told to construct a new altar, twice
the volume of the existing one.
Imagine the Delian altar began as a three-dimensional cube with all sides
equal in length, say a. So they needed to construct another cube of length b with
twice its volume. The volume of each is a^3 and b^3 and they are related by b3 =
2 a^3 or b =^3 √2 × a where^3 √2 is the number multiplied by itself three times that
makes 2 (the cube root). If the side of the original cube is a = 1 the natives of
Delos had to mark off the length^3 √2 on a line. Unfortunately for them, this is
impossible with a straight edge and compasses no matter how much ingenuity is
brought to bear on the would-be construction.
Squaring the circleSquaring the circle
This problem is a little different and is the most famous of the construction
problems:
To construct a square whose area is equal to the area of a given circle.
The phrase ‘squaring the circle’ is commonly used to express the impossible.
The algebraic equation x^2 – 2 = 0 has specific solutions and.
These are irrational numbers (they cannot be written as fractions), but showing
the circle cannot be squared amounts to showing that π cannot be a solution of
any algebraic equation. Irrational numbers with this property are called
transcendental numbers because they have a ‘higher’ irrationality than their
irrational cousins like √2.