Step 3: Build and assemble the base plate.
Set your seat down on the plywood, and measure the smallest possible square for the chair to sit on. You'll want to leave a bit of room around the bolt holes of the chair, in order to maintain the plywood's strength.
Use a circular saw to cut out your plate, then center it on the base. Set the chair on the plate, and mock up your seat before you start drilling bolt holes. Make sure the plate places the seat on the base in a comfortable position, and that the chair is more or less level.
NOW you can drill. Mark the positions of the bolt holes for both the base and the seat and drill them out. (It might be easier to completely mount the plate to the base, then do the chair.)
Hardware: I used four 1/4" diameter bolts for the base, and four 5/8" bolts for the seat. All the bolts were 1.5" long. These are just for reference, as your chair and base will undoubtedly vary from mine.
Make sure to use washers between the bolts and the wood. The wood will experience quite a bit of force when you recline, so spread out the load wherever possible.