I just encountered this theorem for the first time but find it very difficult to picture.
This is how it's presented in my course in the simplest case
But for an example in the case where the manifold is $S^1$ and we are using the antipodal charts how would this embedding look like in $\mathbb{R}^4$? Is this the wrong way to approach this topic?
Edit: if we identify the circle as the set of angles $\theta$ then the antipodal map becomes $2tan(\theta/2)$ this can be scaled to an unit interval using the arctangent function so it can be reduced to a map of the form $\theta\mapsto\theta$. There will be two such maps one which is centered around $0$ the other around $\pi$. The bump function will be $e^{\frac{1}{1+\frac{x^2}{\pi^2}}}=b$ for short hand. The induced Whitney embedding will have to be something of the form $\theta\mapsto(\theta b(\theta),b(\theta),\theta b(\theta),b(\theta))$ I know there are some issues with the other chart but I chose to ignore them for now. If we look at what goes on in a single plane $\theta\mapsto(\theta b(\theta),b(\theta))$ I graphed this and got:
The way I picture what is going on is we have that as we go around $S^1$ we are moving in a circle (something homeomorphic to one at least) in two planes of $\mathbb{R}^4$. Am I going about it in the right way? How do I use this for a more general intuition?