I'm interested how the rotational motion from the big wheel is transferred to the bolts that slide into the door frame. Is it some sort of screw arrangement, worm gear, or something else entirely? Thanks in advance.
As others have said, I think most of them are connecting rods. I think I've seen a very old one that had a rack and pinion. I think Roger Shoaf and Stormn Morman are locksmiths and maybe if you ask nice they will give you a professional opinion.
Many years ago I worked for Diebold and had the opportunity to see lots of vault door mechanisms. Some of the bolts were connected to the front by simple levers and links, some were connected by rack and pinion system and one famous one I remember had no external handles, dials, levers or anything else. The outer skin of the door was one unblemished sheet of brushed stainless steel. It was a very large door and it was awe-inspiring to be there to watch it open as if by magic when the internal time clocks ran down in the morning.
There is a restaurant in town that is in an old bank building. They have taken the vault door out and put it in backwards so that the insides are exposed. The mechanism is quite beautiful. There seems to be three time clocks that enable the door to be opened at the set time. I assume that if any two of the clocks fail the door will still open.
Three timers is the norm and if any one of the three functions correctly the door can be opened. The door with no outer dials or handles had five clocks and they were set and checked each day before closing by two people. The bolts were operated by a spring mechanism that was cocked by a removable handle from the inside. When the door was closed the bolts were locked by the spring operating half way then when the clocks ran down the bolts would release by the spring mechanism operating the rest of the way. The door was hung very slightly off vertical so it would "run" very slowly to the open position.
I have a picture of a large round door. There are about 36 bolts around the inside that engage the frame. It looks as though there is a large cam that has a lobe for each bolt. As the handle is rotated, the cam moves and the bolts recede. I assume they must be spring-loaded. It would only take a few degrees of rotation to move the cam enough to retract the bolts. Also, there is a roller on the end of each bolt that contacts the cam. It must take a lot of effort to move such a large mechanism. Older square doors have a system of levers to move the bolts on all four sides of the door. It is fascinating to watch it work. The handle used to lock or unlock the door has a heavy and solid feel, and the bolts slam in or out with a solid thump. I got to see one up close and personal recently at an old high school that was being cleaned out. Jim Don't raise the bridge, lower the water.