How to solve the system of inequalities when using topological proof to show that there...

Chant6j

Chant6j

Answered

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?

Answer & Explanation

Kiana Cantu

Kiana Cantu

Expert

2022-07-06Added 22 answers

We're looking for integer values of p and q that satisfy the three given equations p 3, q 3 and 1 p + 1 q > 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 1 p and 1 q are both less or equal to 1 4 , meaning that 1 p + 1 q 1 4 + 1 4 = 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 1 3 + 1 q > 1 2 . Subtracting 1 3 from both sides yields 1 q > 1 6 , which implies q < 6.
So if p is equal to 3, q must be an integer with 3 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:
( p = 3 , q = 3 ) , ( p = 3 , q = 4 ) , ( p = 3 , q = 5 ) , ( p = 4 , q = 3 ) , ( p = 5 , q = 3 ) .
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

prirodnogbk

Expert

2022-07-07Added 6 answers

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

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