At first I came to an idea to divide a vase into two parts (P1 and P2, see 3rd screenshot).
You have 2 objects that look like a Hyperboloid and an Ellipsoid. One object has a 'negative' (slender) and the other one a 'positive' (bulbous) curve.
The approximation and the unions of the segments turns into a blocky vase.
Step 1: CP-Samples of the vase
red: valley folds
blue: mountain folds
black: cutting lines
Step 3: Variations