- there are some cool results on embeddings, like: every smooth n-dimensional manifold embeds in ℝ²ⁿ:
https://en.wikipedia.org/wiki/Whitney_embedding_theorem
A dimension-counting argument suggests that if you try to embedded it in ℝ²ⁿ you may be stuck with places where it crashes into itself at isolated points called 'double points'. But then a more sneaky trick, called the Whitney trick, can be used to eliminate these double points. A low-dimensional example is illustrated on the Wikipedia article - but it's so easy to embed a circle in ℝ² that this example doesn't fully illustrate the genius of the Whitney trick.
