How to solve the system of inequalities when using topological proof to show that there...
How to solve the system of inequalities when using topological proof to show that there are exactly 5 platonic solids?
I have seen other answers explaining the topological proof up until the point of
and , are greater than or equal to three
Then they proceed to say that the 5 platonic solids have the only values that satisfy these conditions. My question is how do I solve for these values?
Answer & Explanation
We're looking for integer values of and that satisfy the three given equations , and .
Suppose both and are not equal to . Since they are integers, they must both be greater or equal to . But if that's the case, then and are both less or equal to , meaning that , contradicting one of the equations.
So to solve the equation either or must be equal to .
Let's consider only the case where is equal to . Then we have . Subtracting from both sides yields , which implies .
So if is equal to , must be an integer with , so must be either or or .
By a symmetrical argument, if is equal to , must be equal to or or .
So we have five cases remaining:
It is now easy (and necessary) to check that all these five cases satisfy the three given equations.
There isn't really a general method to do this kind of proof, it just requires finding a bunch of solutions, then a sufficient bunch of good arguments for why there aren't any more solutions.
Multiply both sides of the inequality by to get , which is equivalent to . Now and are integers , so it is easy to list the five possibilities:
which implies that