This is a design for an "origami style" plywood chair. After doing some initial prototypes with various sheet goods and continuous hinges I settled on this design as a good balance of affordability and quality.
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Step 1: The Parts
There are many kinds of plywood out there but if you're actually going to make a nice piece of furniture without applying edge banding you really need to make the investment in a good sheet. One sheet of 1/2" latvian birch plywood ran me about $80 and two chairs will fit easily into one sheet. The continuous hinges, which I use to make the folds, are a more cost driven decision. After experimenting with more costly hinges I settled on your garden variety 1 1/2" wide continuous hinges from the local hardware store. A 6' length runs about $15 to $20 dollars and five lengths should easily make two chairs. I used a 3/8" button head screws for the continuous hinge. These are NOT the screws that come with the hinge. It would be nice if I could countersink the screws but it turns out continuous hinges typically countersink the opposite side from the face we use in this project. I've spoken with the manufacturers about reversing the countersink but at this point, big surprise here, it would kill the economy of the project. Thankfully the button heads screws on the backrest are so deep in the "v" of the backrest you don't feel them. I think you should be able to buy all the screws for $20. All of this to say that if you're careful you should be able to make two chairs for less than $100 each. I'm planning on finishing mine in Tung Oil but you can certainly pick any number of finishes.
Step 2: Some Background on the Geometry...
If you decide that you want to use different hinges and plywood be aware that there is a very specific performance criteria to your coupling. Different couplings of hinges and plywood thickness yield different possible shapes. Above are some images showing the dihedral angle limitations of the coupling and the design itself. Notice the hinge/plywood folding capacity is paired tightly with that of the design. In doing this folded plate exercise I needed to reconcile what I could dream up with what the materials would let me get away with. It was fun coming up with a shape that worked with my kit of parts.
Step 3: How It Can Be Made
There are two ways basic ways this geometry can be fabricated.
1) Print out the attached image and make the pieces with a jigsaw and a router. Please let me know if you need something with greater resolution.
2) Use a CNC machine. I've attached the VCarve file here as well. Good Luck!