Like I mentioned before, the Fujimoto method is one of the most elegant solution without glueing and cutting. Now I'll demonstrate how to fold a hexagonal 6-sided cup. It's easier to fold than the 12-sided hyperboloid cup.
You can use it for your pencils. Besides this construction is rock solid, it can hold a dictionary.
The same method is used for the other geometry shapes. If you study the pattern of the antiprism, the bottom of the prism is shifted by 1/2 length and the sides of the object are converted into equilateral triangles so that it does result as an antiprism. The folding structure of the closure is equivalent to the lower half part of the bottom.
The cube-octagon hybrid is combined with octagon + square frame (base).
Step 1: Crease Pattern
Step 2: Prefolding
Prefold everything like shown in the CP.
Step 3: Form it into a cup
Step 4: Follow the notations
The pic shows the top of the cup. Unwrap and bring A to C and D to B (goes automatically)
Then it should look like in the 2nd Screenshot. (C+A, B+D)