Today I'll show you how to make a hyperboloid cup for my first instructable (Paper Egg)
The great thing is you don't need to glue and cut (except of two for the right format). But maybe it's a bit hard for you to make it, unless you don't have folding experiences.
This model is based on the fujimoto cube method. Kasahara discovered the mathematical solution, how to use the method for several kinds of prisms and polygons. I would recommend to read the book "Amazing Origami" by Kasahara
Step 1: Crease Pattern Description
-red: valley folds
-blue: mountain folds
-black: cutting edges
Step 2: Folding Instructions
At first you prefold everything of the creasepattern. In the screenshot you can also see the description of the parts.
Step 3: The Bottom
At first you begin with the bottom. It's a kind of screwing and twisting from the both sides at the same time
Step 4: Cup Modelling
When you twist the bottom you get something that looks like a cup.
From the top you see the notations A, B, C (D is not important)
Bring A to the outside and B to the inside like shown in the pic.
Step 5: Close Gaps, Twist and Screw the Curved Surface
Bring C+A , D+B (inside of the cup) together and close the gaps. Twist the cup into a hyperboloid shape.
Step 6: Fold the Frame and Tuck
The last step is to fold the frame. Fold it, tuck it inside and you are finished!