# Dimension by Exponentiation

1. Jan 16, 2004

### Antonio Lao

What is done in the following might be considered as “not mathematical.” Nevertheless we must be allowed to use crazy ideas if only for the sake of exploring the uncharted domains of human mathematical logic.

This is not to prove again the famous math problem called Fermat’s Last Theorem. The proof was done by Andrew Wiles in the 1990s. But use it in relation to dimensional analysis.

The algebraic equation is $$a^n +b^n = c^n$$ The exponential number n, in our discussion, is the dimension. For the case where n=2, the equation is called the Pythagorean theorem and its solutions are the Pythagorean triples. For our purpose, we only need one of these triples: a=3, b=4, c=5. The equations becomes $$3^2+4^2=5^2$$. The left-hand (LHS) side is always equal to the right-hand side (RHS).

When n=3, the RHS is bigger than the LHS. When n=4, the RHS is even bigger than the LHS. It can be noted that, as n increases from 2 to infinity, the RHS becomes progressively bigger and bigger. We can make the conclusion that at n=infinity, the principle of holism is vindicated. The whole is greater than the sum of its parts. And no matter how many parts are added together, the sum is always less than the whole.

Likewise, it can be shown that if n is less than 2, the reverse happens, the RHS becomes smaller than the LHS. And if we allow n to take on fractional value, the RHS becomes progressively smaller and smaller than the LHS. So that when n=0, the quantum nature of number becomes apparent (1 + 1 > 1).

2. Jan 16, 2004

### Sariaht

Well spoken Antonio.

3. Jan 16, 2004

### krab

No. The only conclusion that can be drawn is that $3^n +4^n = 5^n$ is not true for $n\neq 2$.

4. Jan 16, 2004

### Antonio Lao

Thanks for all your replies.

For the next discussion is that of an n-sphere and an n-cube.
What spatial dimension makes their volumes equal?