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
Here are the CPs for the cup and box.
Step 2: Prefolding
The following folding steps are for the 6 sided jar.
Prefold everything like shown in the CP.
Step 3: Form It Into a Cup
You begin with the bottom of the prefolded paper. Twist it from both sides at the same time and model a cup.
Step 4: Follow the Notations
You have 4 ends ABCD.
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)
Step 5: The Last Step
Close the gap and fold + tuck the frame.
Step 6: The Result
And you are finished!