Platonian solids and homeomorphism

Are all Platonic solids homeormorphic to one another? Since they all have the same Euler characteristic $\displaystyle \chi=2$ It is commonly known that if topological spaces have the same Eulers characteristic then they are homeomorphic but on the other hand this sometimes does not work....for example a point is not homeomorphic to a line even if their Euler characteristics are equal...

$\displaystyle \chi(point)=\chi(line)=1$

Please can someone give me a good and easy explanation of it?

Thank you