Doubly Periodic Surfaces 



The most basic doubly periodic surface was discovered by Scherk. It comes in a family with varying angle between the top and bottom planar ends. This family gives all the embedded doubly periodic surfaces of genus 0.  56 K Quicktime Movie 
For higher genus, the zoo grows. To the right you can see a Scherk surface with a handle. Below are two different doubly periodic surfaces with parallel top and bottom ends. They belong to a family which shears the handles along the ends. The right hand one can be interpreted as a fence of singly periodic Scherk surfaces whith planar ends glued together. 

Similarly, the right hand surface can be seen as a fence of copies of the singly periodic Riemann minimal surface: 