A question about topology from an ignorant logician, so please be kind if this is obvious!
We all know that the Klein bottle, unlike the torus, cannot be embedded in 3-space. And we all know (because we were told) that it can be embedded in 4-space. I can see how to embed the torus in 3-space, using only quadratic stuff (I can do that with stuff I learnt at school) but how does one embed the Klein bottle in 4-space? I am asking because I think if I had an equation to pour over I might start to get a feel for how the injection works and what the bottle looks like. In particular, I would like to know if the isometry group of the embedded bottle looks like the isometry group of the embedded torus in 3-space. I'm hoping it does, because that would put flesh on the assertion that there isn't really any self-intersection.
I asked this of a visiting topologist but she didn't know off the top of her head so I'm trusting that this isn't a stupid question!