It occurred to me that there is a simple way to build lookup-table (LUT)-based mechanical finite state machines with planar fabrication. I think this is considerably less dense than than the heightfield- based LUTs I’ve written about previously. It represents the transition matrix as a maze of channels cut into a surface, with a single follower traversing the maze under the influence of springs and friction.
Press the button on the back of a pushbutton ballpoint pen, and the tip extends; press it again and it retracts. It’s a mechanical state machine with two states that takes no inputs. It works by having a rotating cylindrical thingy that slots into guides in the outer barrel of the pen; they force it to rotate when it moves longitudinally.
We can extend this mechanism to more general state machines, that take N discrete inputs and transition arbitrarily between M discrete states, each of which has one of P discrete outputs. The input is encoded as a distance to which you push the thing corresponding to the ballpoint pen button against the spring; the state is encoded in the rotation of the barrel; and the output is encoded as a distance to which the spring is able to push back once the input is withdrawn. And the mechanism is changed somewhat to be able to handle general state machines.
For simplicity of diagramming I will unroll the cylindrical thingy to be planar. Here’s a way to realize our ballpoint pen with my new scheme:
State State
1 2
+————————————————+
| |
| | | Output 1
| | |
| | |
| | | | Output 2
| | | |
| |\ /| |
| | \ / | | Follower (larger scale):
| | \ / | |
| | | | | | _______________
| | \/ | | (_______o_______)
| | /\ | |
| \ / \ / |
| \| |/ |
| # # |
| |
+————————————————+
The paths here are smoothly curved channels in the plane. An oblong “follower” with rounded ends begins in either the State 1, Output 1 or State 2, Output 2 position at the top of the diagram, and is then pushed down to the bottom. It’s free to move horizontally, but has to overcome some friction to do so. Where the channels branch, the branch points are wide enough that it is kinematically capable of following either branch; but the horizontal friction ensures that it simply continues straight down; it follows the curves down to one of the transition points marked #.
When the input pressure is removed, it begins to return back upwards, but because it’s at a branch point with an available vertical path it follows that branch rather than returning by the same path by which it came — and thus it ends up going to the other state. At the point that the two return paths cross, the intersection is not wide enough to allow the follower to turn to follow the wrong return path, like the shuttles in a trammel of Archimedes.
(If the plane were a cylinder, the two paths would not have to cross over each other, and that is in fact the approach taken by actual ballpoint pens. Also, they typically have a larger number of states, such as 8 or 12, but with only 2 outputs. And commonly they have a separate “nudge” in the thing propelling the follower vertically to push it into the return path, but that isn’t strictly necessary.)
When there are multiple possible inputs, there will be multiple return branch points on each path down, branching off at different heights:
| \| | | |
| # | | |
| \| | |
| # | |
| \| |
| # |
These different branch points are independently routable to arbitrary different next states. This lets you use any arbitrary transition function for your state machine. However, in the general case, all these separate M*N return paths need to be able to pass in parallel and cross over before getting back to the states, which puts a limit on how complex our state machine can be in a given space.
This mechanism provides fan-out, but not fan-in: unlike with the heightfield LUT, combining two separate inputs, such as two summands or multiplicands, must be done with some other mechanism. But such mechanisms can be fairly simple.
In addition to wrapping the transition maze around a cylinder, you can also wrap it around on a disc, as shown by the spiral cams in “Basic Mechanisms in a Fire Control Computer (1953)”, using rotation rather than translation as one of your dimensions of motion; and indeed, if you distort the other dimension into a series of arcs, you can use rotation rather than translation for both of them, while retaining the compactness of planarity.
This video also shows that my heightfield LUT was already invented, in analog form for continuous functions in 1953; apparently it’s called a “barrel cam”. The Equation of Time Cam, from the first prototype of the Clock of the Long Now, is an example that I had seen. It compensates for the changing time of solar noon throughout the year.
Investigating further, it turns out that, before 1953, the Jaquet-Droz automaton known as The Writer used a heightfield LUT, in the form of a stack of cams, to encode a vector font; this automaton is a pen plotter with a non-ASCII character set, in the shape of a small boy with curly hair. Three cam followers control the X, Y, and Z motions of the pen, and the axial displacement of the camshaft determines which character is produced. Wikipedia explains that the automaton was constructed between 1768 and 1774. It still works, more or less, and is demonstrated once a month at the Musée d’Art et d’Histoire de Neuchâtel.
In between the Writer and the Cold War barrel cams, the rotational dimension of the cam was generalized to be any continuous input variable, rather than just time; and the axial displacement dimension was generalized to be possibly continuous rather than discrete. I don’t know when between 1774 and 1953 these two generalizations occurred, nor whether they might have been present in other machines even before 1774.
This is interesting to me because, as I wrote in 2010, LUTs have major advantages for mechanical digital computation: they’re much easier to design than combinations of universal gates like NANDs or NORs, and they also involve many fewer moving parts. So it seems like you could have built a general-purpose digital computer in the 1700s, at least in Switzerland, if you had the idea that it was an interesting thing to want to build.