# Why is a cylinder not perfectly symmetric as a sphere?

I have read that a cylinder not being perfectly symmetric is the reason behind Rayleigh instability: the process that makes bubbles out of a stream of water.

But a cylinder seems also perfectly symmetric to me

• The symmetry group of a sphere is a Lie group of dimension $3$. The symmetry group of a cylinder is a Lie group of dimension $1$. Mar 31, 2019 at 6:47
• (Or dimension $2$, if one means a [doubly] infinite cylinder.) Mar 31, 2019 at 7:06
• Also, that seems as good of an explanation as one could hope for. Perhaps write it up as an answer proper? Mar 31, 2019 at 7:07
• @Travis Can you explain this? Mar 31, 2019 at 8:09
• Sure: By a doubly infinite cylinder, I just mean a cylinder that extends infinitely far in both directions. Concretely, consider $\{(x, y, z) \in \Bbb R^3 : x^2 + y^2 = 1\}$, which is a cylinder of radius $1$ with symmetry axis the $z$-axis. Its symmetry group has dimension two, or, roughly speaking, there are two independent ways to move the cylinder without changing its appearance: One can slide it parallel to its axis of symmetry, or one can rotate it along that axis. If the cylinder is finite (or infinite in only one direction), only rotations are symmetries. Apr 1, 2019 at 2:34

This is because you can switch the axis $$x,y,z$$ and the sphere will appear always the same, while with the cylinder you will always have an axis which is different from the other two.