Let's take a closer look at the Platonic solids, one at a time. First, we'll look at the tetrahedron. If you haven't already explored the interactive polyhedra, you might want to go look at them before finishing this article.
When you're examining the tetrahedron, count the number of vertices V it has. Then, count the number of edges E it has. Next, count the number of faces F it has, and calculate the formula V-E+F. What number did you get? You have calculated the Euler Number for the tetrahedron!
Now, how about the cube? Can you calculate V-E+F for the cube too? What about the octahedron? It can be a little trickier to calculate the Euler Number for the icosahedron and the dodecahedron, but it can be done!
Did you get the same Euler Number for every Platonic solid? You can check your answers if you want to.
Something the Platonic solids have in common is that they are all uniform, regular, and convex (if you want to read the definitions of those words, see the vocabulary page). Also, you can imagine that they basically look like balls, which have been flattened in different ways. Mathematicians might say that the Platonic Solids are "topologically equivalent" to the sphere (ball). What does that mean? Have you ever played with a ball of modelling clay, and shaped it with your fingers so that it looked like a six-sided die? "Topologically equivalent" is kind of the same idea--that you could reshape and deform one object until it looks like another one, without ripping or tearing it.
Anyway, when shapes are topologically equivalent, it turns out that they have the same Euler Number. That's why all the Platonic Solids have the same Euler Number!