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So, I can only create perfect solids with some numbers of sides (3 and 5). Why is that? Some symmetry caused by having 3 dimensions?

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Don't forget the cube with quadratic sides. In general these are called platonic solids [1] and there are only 5 of them. Why only 5 ? It has to do with angles and how you can unfold such a solid. (so it's somewhat caused by getting something from 3d-space into the 2d-plane)

[1] http://en.wikipedia.org/wiki/Platonic_solid

Also, it's not just about the angles. It's about the number of connections between vertices too. There just aren't any other solutions.

To convince yourself, here's a fun game you can try with pencil and paper: http://ferkeltongs.livejournal.com/28364.html (disclosure: that's my blog)

I feel bad for the poor heptagon, which has neither a flat tiling nor a solid. At least on the hyperbolic plane everyone gets to take part in a tiling.

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