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. Scherk Logo56 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.
Scherk with handle
parallel planar ends doubly periodic 1 parallel planar ends doubly periodic 2
Similarly, the right hand surface can be seen as a fence of copies of the singly periodic Riemann minimal surface: parallel planar ends doubly periodic 3