Paper-folding is a somewhat obsessive pastime of mine. The following are some patterns I've come up over the years. I'm sure some of them have been discovered before, but I include them because they are geometrically interesting. Each design uses only a single sheet of rectangular paper and nimble fingers.

I have indicated the root system associated to each model, i.e. the smallest angle between fold lines. Types A2, B2, and D2 refer to angles of 60, 45, and 90 degrees, respectively.


Leaf shape (B2).




Negative curvature (B2).




A cusp (B2).




A tiling of a cylinder with saddles (D2).




Two views of a dome (D2). This is an example of weaving pleats to make the ends of a piece of paper hold together. The spiral at the top also contributes to the object's cohesion. It is possible to modify this construction in order to make a sphere.





A collapsible tower with cyclic rotational symmetry of order 4 (B2). This piece involves introducing angled corners perpendicular to a simple row of mountain and valley folds. It can be made to have cyclic rotational symmetry of any order.




A seven-pointed star (B2).




A torus (A2).




Two views of a hyperboloid of one sheet (D2).





Assorted Rolls (B2).