For the mathematical buffs and geeks out there:
We know that √2 is an irrational number. But has anyone thought of the physical implication of this?
By Pythagoras' Theorem, it means that a right isosceles triangle with both sides of unit length x would have a hypotenuse of length √2x.
In a mathematical world, the three sides of the triangle would be continuum's. However, in the physical world, we know that the lines forming the three sides are really made up of discrete atoms.
Let us assume for simplicity that the lines are exactly one atom in thickness. This means that each of the two mutually-perpendicular sides would be made up of nx discrete number of atoms in a straight line. We shall also further assume that successive atoms are equidistant from one another.
It therefore follows that the hypotenuse would have to be formed by √2nx number of discrete atoms. But because √2 is an irrational number, √2nx discrete atoms is a physical impossibility.
Let us then assume that we attempt to achieve a discrete number of atoms on the hypotenuse, by splitting the atoms further into k equal parts, i.e. the number of atoms on the mutually-perpendicular side would be nx/k and that of the hypotenuse would be √2nx/k, where k is necessarily a positive integer. However, because √2 is already an irrational number, turning √2 into an integer would require an infinite number of moves of the decimal point to the right, i.e. k→∞ and the atoms are split so small, they become a continuum, which violates physical laws of atoms. Or, stated another way, only if the size of the triangle approaches infinity.
Therefore, a right isosceles triangle in the real world is a mathematical idealisation, but a physical impossibility. Neat, huh?
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