Whitney immersion theorem

The Whitney immersion theorem is like playing with play-doh. Imagine you have a piece of play-doh (a surface), and you want to make it look like a more complicated shape. For example, you may want to make the play-doh look like a lion or a flower.

To do this, you can pack more and more play-doh on top of the original surface until you have the desired shape. Similarly, the Whitney immersion theorem tells us that, given any smooth surface, we can always find a way to pack extra dimensions on top of it, so that it looks like a more complicated shape.

For example, if we have a flat, two-dimensional surface, we can "pack" an extra dimension on top of it by bending and curving the surface in a particular way. If we imagine tracing out the path of a point on this bent surface, we get a curve in three-dimensional space. This extra dimension might not be visible, but it's there mathematically.

Overall, the Whitney immersion theorem tells us that we can always make a surface look more complicated by adding extra, hidden dimensions. It's like making a play-doh lion or flower, but with math!