Chant6j

2022-07-05

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
$1/p+1/q>1/2$ and $p$, $q$ 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?

Kiana Cantu

Expert

We're looking for integer values of $p$ and $q$ that satisfy the three given equations $p\ge 3$, $q\ge 3$ and $\frac{1}{p}+\frac{1}{q}>\frac{1}{2}$.
Suppose both $p$ and $q$ are not equal to $3$. Since they are integers, they must both be greater or equal to $4$. But if that's the case, then $\frac{1}{p}$ and $\frac{1}{q}$ are both less or equal to $\frac{1}{4}$, meaning that $\frac{1}{p}+\frac{1}{q}\le \frac{1}{4}+\frac{1}{4}=\frac{1}{2}$, contradicting one of the equations.
So to solve the equation either $p$ or $q$ must be equal to $3$.
Let's consider only the case where $p$ is equal to $3$. Then we have $\frac{1}{3}+\frac{1}{q}>\frac{1}{2}$. Subtracting $\frac{1}{3}$ from both sides yields $\frac{1}{q}>\frac{1}{6}$, which implies $q<6$.
So if $p$ is equal to $3$, $q$ must be an integer with $3\le q<6$, so $q$ must be either $3$ or $4$ or $5$.
By a symmetrical argument, if $q$ is equal to $3$, $p$ must be equal to $3$ or $4$ or $5$.
So we have five cases remaining:
$\left(p=3,q=3\right),\left(p=3,q=4\right),\left(p=3,q=5\right),\left(p=4,q=3\right),\left(p=5,q=3\right).$
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.

prirodnogbk

Expert

Multiply both sides of the inequality by $2pq$ to get $2q+2p>pq$, which is equivalent to $\left(p-2\right)\left(q-2\right)<4$. Now $p-2$ and $q-2$ are integers $\ge 1$, so it is easy to list the five possibilities:
$\left(p-2,q-2\right)\in \left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(2,1\right),\left(3,1\right)\right\},$
which implies that
$\left(p,q\right)\in \left\{\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(4,3\right),\left(5,3\right)\right\}.$

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