Villin headpiece (HP35) — the last 36 amino acids of a much bigger protein, solved by X-ray crystallography as PDB 1VII. It folds in about 4 microseconds, which is why it is the one everybody uses to study folding.
The order of the amino acids, and nothing else — that is the only part your DNA specifies. Everything you are about to watch follows from it.
Fourteen backbone hydrogen bonds form. Twelve of them link a residue to the one four places along the chain — that spacing is what makes a spiral instead of a loop, and it is why three stretches turn into three α-helices. The other level-2 shape, the β-sheet, does not appear in this protein.
Now the side chains do the work instead. Three phenylalanines — oily, and water will not mix with them — end up buried in the middle with the helices packed around them. That is the whole chain's final shape.
Not here: this protein is one chain, so it stops at level 3. Level 4 is what happens when several finished chains assemble into one working unit — see the haemoglobin lesson, where one chain folds and the other three dock onto it.
Real folding is thermal chaos, not a smooth collapse: the order of events here is right, the path is not. And act 2 is drawn as the phenylalanines attracting each other, when the real cause is water pushing them together.