Laser-cut Hinges (a.k.a. kerf hinge, living hinge)
Designing a pattern to allow flexibility in a rigid material can take a lot of trial and error. However, with a bit of persistence, as well as an understanding of a few rules, you can turn a solid sheet of plywood into a flexible work of art.
There are a number of constraints that will affect whether or not your design will work. After a lot of attempts, I have tried to codify the rules that seem to govern how to make a strong, functional hinge in plywood. I used 3 mm thick birch plywood - thicker material will need more material removed, and thinner stock should be fine with less taken out.
When you pick a 'repeat unit', you need to orient it correctly. The long axis of that unit (green) needs to be perpendicular to the way that you want the hinge to bend (red).