Mechanical Computer

WORK IN PROGRESS

Mechanical computer (simple ones also being called mechanical calculators) is a computer that uses mechanical components (e.g. levers, marbles, gears, strings, even fluids ...) to perform computation (both digital and analog). Not all non-electronic computers are mechanical, there are still other types too -- e.g. computers working with light, biological, quantum, pen and paper computers etc. Sometimes it's unclear what counts as a mechanical computer vs a mere calculator, an automaton or mere instrument -- here we will consider the term in a very wide sense. Mechanical computers used to be used in the past, mainly before the development of vacuum tubes and transistors that opened the door for much more powerful computers. However some still exist today, though nowadays they are usually intended to be educational toys, they are of interest to many (including us) as they offer simplicity (independence of electricity and highly complex components such as transistors and microchips) and therefore freedom. They may also offer help after the collapse. While nowadays it is possible to build a simple electronic computer at home, it's only thanks to being able to buy highly complex parts at the store, i.e. still being dependent on corporations; in a desert one can much more easily build a mechanical computer than electronic one. Mechanical computers are very cool.

{ Britannica 11th edition has a truly amazing article on mechanical computers under the term Calculating Machines: https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Calculating_Machines. Also this leads to many resources: https://www.johnwolff.id.au/calculators/Resources.htm. ~drummyfish }

If mechanical computer also utilizes electronic parts, it is called an electro-mechanical computer; here we'll however be mainly discussing purely mechanical computers.

Disadvantages of digital mechanical computers against electronic ones are great, they basically lose at everything except simplicity of implementation (in the desert). Mechanical computer is MUCH slower (speed will be measured in Hz), has MUCH less memory (mostly just a couple of bits or bytes), will be difficult to program (machine code only), is MUCH bigger, limited by mechanical friction (so it will also be noisy), suffers from mechanical wear etc. Analog mechanical computers are maybe a bit better in comparison, but still lose to electronics big time. But remember, less is more.

Some notable mechanical computers include e.g. the 1882 Difference Engine by Charles Babbage (aka the first programmer), Antikythera mechanism (ancient Greek astronomical computer), the famous Curta calculators (quality, powerful pocket-sized mid-20th century calculators) { These are really cool, check them out. ~drummyfish }, Enigma ciphering device (used in WWII), abacus, slide rule, Odhner Arithmometer (extremely popular Russian table calculator), Digi-Comp I (educational programmable 3 bit toy computer) or Turing Tumble { Very KISS and elegant, also check out. ~drummyfish } (another educational computer, using marbles).

Let's also take a look at how we can classify mechanical computers. Firstly they can be:

special purpose: Made to solve only limited set of problems, example being a mechanical calculator that can only perform a few operations like addition and subtraction. A special purpose computer may be easier to make as it doesn't have to bother with the flexibility needed for solving general problems.
general purpose: Full programmable Turing complete computer capable of solving very wide range of tasks efficiently. This is of course harder to make, so general purpose mechanical computers are rarer.

Next we may divide mechanical computers to:

analog: Working with analog data, i.e. continuous, infinitely precise values, typical examples are various integration machines. The analog approach is probably more natural and efficient in the mechanic world, so we encounter many of analog computers here (compared to the electronic world).
digital: Working with discrete values, i.e. whole numbers, bits etc.
analog-digital: Combination of both digital and analog, again this is more common in mechanic world than in electronic world.

And to:

autonomous: Computers that only require to be started and then work completely on their own, without human intervention.
semi-autonomous: Computers largely working on their own but still requiring some human assistance during computation, for example turning some handle to keep the parts moving.
computation helpers: Tools that only aid the man who is doing most of the computation -- typical examples are abacus, slide rule or integraph.

Basics

Analog computers are usually special purpose. { At least I haven't ever heard about any general purpose analog computer, not even sure if that could work. ~drummyfish } Very often they just solve some specific equation needed e.g. for computing ballistic curves, they may perform Fourier transform, compute areas of arbitrary shapes that can be traced by a pencil (see planimeter) etc. Especially useful are computers performing integration and solving differential equations as computing many practically encountered equations is often very hard or impossible -- mechanical machines can integrate quite well, e.g. using the famous ball and disk integrator.

As mere programmers let us focus more on digital computers now.

When building a digital computer from scratch we usually start by designing basic logic gates such as AND, NOT and OR -- here we implement the gates using mechanical principles rather than transistors or relays. For simple special-purpose calculators combining these logic gates together may be enough (also note we don't HAVE TO use logic gates, some mechanisms can directly perform arithmetic etc.), however for a highly programmable general purpose computer logic gates alone practically won't suffice -- in theory when we have finite memory (in real world always), we can always just use only logic gates to perform any computation, but as the memory grows, the number of logic gates we would need would grow exponentially, so we don't do this. Instead we will need to additionally implement some sequential processing, i.e. something like a CPU that performs steps according to program instructions.

Now we have to choose our model of computation and general architecture, we have possibly a number of options. Mainly we may be deciding between having a separate storage for data and program (Harvard architecture) or having the program and data in the same memory (intending for the computer to "reshape" this initial program data into the program's output). Here there are paths to explore, the most natural one is probably trying to imitate a Turing machine (many physical finite-tape Turing machines exist, look them up), probably the simplest "intuitive" computer, but we can even speculate about e.g. some kind of rewriting system imitating formal grammars, cellular automata etc -- someone actually built a simple and elegant rule 110 marble computer (look up on YT), which is Turing complete, but not very practical. So Turing machine seems to be the closest to our current idea of a computer (try to program something useful in rule 110...), it's likely the most natural way, so that might be the best first choice we try.

Turing machine has a separate memory for program and data. To build it we need two main parts: memory tape (an array of bits) and control unit (table of states and their transitions). We can potentially design these parts separately and let them communicate via some simple interface, which simplifies things. The specific details of the construction will now depend on what components we use (gears, marbles, dominoes, levers, ...)...

Concepts

Here we will overview some common concepts and methods used in mechanical computers. Remember the concepts may, and often are, combined. Also note that making a mechanical computer will be a lot about mechanical engineering, so great many concepts from it will appear -- we can't recount all of them here, we'll just focus on the most important concepts connected to the computing part.

Gears/Wheels

Gears (wheels with teeth) are a super simple mechanism popular in mechanical computers. Note that gears may be both digital and analog -- whether they're one or the other depends on our interpretation (if we assign importance to every arbitrary orientation or just a finite number of orientations that click the tooth into some box).

The advantages of gears are for example instant transfer of motion -- even if we have many wheels in sequence, rotating the first one instantly (practically) rotates the last one as well. Among disadvantages on the other hand may be the burden of friction (having too many gears in a row will require a lot of power for rotation and strong fixation of the gears) and also manufacturing a non-small number of good, quality gears may be more difficult than alternatives (marbles, ...).

Besides others gears/wheels can be used to:

Transmit power, i.e. delivering motion to components that need motion to work (even in computers that don't use gears themselves as computing components).
Do arithmetic: for example a differential can be used to instantly add two numbers (or actually to compute any linear combination, e.g. average, ... using the slide rule concept we can probably even implement multiplication, division etc.). A simple stepped-cylinder (kind of a "gear") alongside with normal gears can also be used to implement addition, subtraction and even multiplication (as explained e.g. in 1911 Encyclopedia Britannica; this principle was used e.g. by the old Russian calculators).
Represent a general digital value by how they are currently rotated, i.e. a gear with N teeth -- each one labeled with a value -- can hold one of N values depending on which of the values is currently under some pointer -- this is often used in mechanical calculators e.g. to display computed values. This has the advantage of being able to represent a digital number with one relatively simple part (the wheel) without having to encode multiple bits (i.e. many smaller parts). This may also be used to make a look up table -- imagine e.g. a wheel which by rotating looks up some value that may be represented e.g. by displacement (imagine spinning spiral) or holes on the wheel. If the gear represents a natural number, it naturally implements modulo increment/decrement (highest value will overflow to lowest and vice versa).
Represent one bit by turning either clockwise or counterclockwise.
Possibly represent values also in other ways, for example by speed of rotation, rotation vs stillness, position (gear traveling on some toothed slider, ...) etcetc.
...

       1         1
  __  ,-,  ___  ,-,  _______
 |   { o }     { o }   ,-,  |
 |    '-;,    ,-;-'   { o } |
 |    _|||___{ o }_____;-;  |
 |            '-'.-.  { o } |
 |_____________ { o } _'-'__|   
                 '-' 
                  1
                   
                   
       0         1
  __  ,-,  ___  ,-,  ______
 |   { o }     { o }-,     |
 |   ,;-' ,-,   '-{ o }    |
 |  _|||_{ o }_____;-;     |
 |        '-'  .-.{ o }    |
 |___________ { o }'-'_____|   
               '-' 
                0

NXOR (equality) gate implemented with gears (counterclockwise/clockwise rotation mean 1/0); the bottom gear rotates counterclockwise only if the both input gears rotate in the same direction.

Marbles/Balls

Using marbles (and possibly also similar rolling shapes, e.g. cylinders, disks, ...) for computation is one of the simplest and most KISS methods for mechanical computers and may therefore be considered very worthy of our attention -- the above mentioned marble rule 110 computer is a possible candidate for the most KISS Turing complete computer. But even with a more complicated marble computer it's still much easier to build a "marble maze" than to build a geared machine (even gears themselves aren't that easy to make).

Basic principle is that of a marble performing computation by going through a maze -- while a single marble can be used to evaluate some simple logic circuit, usually (see e.g. Turing Tumble) the design uses many marbles and performs sequential computation, i.e. there is typically a bucket of marbles placed on a high place from which we release one marble which (normally by relying on gravity) goes through the maze and performs one computation cycle (switches state, potentially flips a memory bit etc.) and then, at the bottom (end of its path), presses a switch to release the next marble from the top bucket. So the computation is autonomous, it consumes marbles from the top bucket and fills the bottom bucket (with few marbles available an operator may sometimes need to refill the top bucket from the bottom one). The maze is usually an angled board onto which we just place obstacles; multiple layers of boards with holes/tunnels connecting them may be employed to allow more complexity. { You can build it from lego probably. ~drummyfish }

NOTE: Balls may be used to perform computation also in other ways than described here, very notable is e.g. the billiard ball computer (which also has a great advantage of performing reversible computation). However here we will focus on the traditional "marble maze" approach.

The disadvantage here is that the computation is slow as to perform one cycle a marble has to travel some path (which may take many seconds, i.e. in the simple form you get a "CPU" with some fractional frequency, e.g. 1/5 Hz). This can potentially be improved a little, e.g. by pipelining (releaseing the next marble as soon as possible, even before the current one finishes the whole path) and parallelism (releasing multiple marbles, each one doing some part of work in parallel with others). Advantages on the other hand include mentioned simplicity of construction, visual clarity (we can often make the computer as a flat 2D board that can easily be observed, debugged etc.) and potentially good extensibility -- making the pipeline (maze) longer, e.g. to add more bits or functionality, is always possible without being limited e.g. by friction like with gears (though usually for the cost of speed).

Some things that can be done with marbles include:

Flipping/setting bits: A marble running through some specific part of the maze can flip something over, which we may interpret as flipping a bit. A simple rotating "T" shape can be used to make a one bit flip-flop (see below).
Branching: If/else/switch branching can be implemented simply as a marble taking one road or another on a crossroad, which can be decided by some moving part connected to some bit elsewhere.
Rotating gears: A marble may rotate a gear by precise number of teeth, this can be used e.g. to implement a shift on memory tape (see pictures below).
Weight/count representing a value: instead of encoding a number with flippable bits we may instead use a small bucket into which marbles fall -- the number of marbles in the bucket then encode the number stored. We may e.g. introduce a limit number (if x > N), i.e. weight at which the bucket becomes heavier than a counterweight and opens some new path for the marbles.
...

     :
 #   o   #     #       #     #       #     #       #
 ##     ##     ##     ##     ##     ##     ##     ##
 ###   ###     ### : ###     ###   ###     ###   ###
 #  \ /  #     #  \o/  #     #  \ /  #     #  \ /  #
 #   O   #     #   O   #     #  oO   #     #   O   #
 #   #\  #     #   #\  #     #  /#   #     #  /#   #
 #   #   #     #   #   #     #   #   #     # : #   #
 #   #   #     #   #   #     #   #   #     # o #   #
   0   1         0   1         0   1         0   1

Marble falling into a flip-flop will test its value (fall out either from the 0 or 1 hole) and also flip the bit -- next marble will fall out from the other hole.

\:  marble slide
 \o
  \           hole      sliding plane
=============----===============VVVVVVVVVVV====  <----->
  |    |    |    |    |    |       -''-
  | b0 | b1 | b2 | b3 | b4 |      { () } 
  |    |    |    |    |    |       '--' gear
             bits

Above a gear is used to select which hole an incoming marble will fall into (each hole may contain e.g. a flip-flop bit shown above). This may potentially be used to e.g. implement random access memory.

Fluids

Whether the use of fluids/gases (water, air, steam, maybe even sand, ...) is still considered mechanical computing may be debatable, but let's take a look at it anyway.

Power: flowing fluid (steam, water stream, falling sand, ...) can be the source of movement in the mechanism.
Hydraulics can easily transmit movement: fluid in a tube under pressure can transfer movement on a long distance and can be curved in any way, this may be much simpler way of transferring movement than e.g. a sequence of many gears.
Logic: fluids can also implement logic gates, see fluidics.
Weight/volume can have significance: similarly to marbles, the amount of water in a bucket may record a value, we may employ weights, overflows etc. to incorporate this into computations.
Electronics emulation: it's known many electronic concepts can be imagined with water pipes instead that deal with similar concepts (pressure ~= voltage, flow ~= current, resistor ~= narrower pipe, ...). By this we may possibly emulate very simple electronics without actual electricity.
...

Other

Don't forget there exist many other possible components and concepts a mechanical computer can internally use -- many things we leave out above for the questionability of their practical usability can be used to in fact carry out computation, for example dominoes or slinkies. Furthermore many actually useful things exist, e.g. teethed cylinders/disks may be used to record plots of data over time or to store and deliver read/only data (e.g. the program instructions) easily, see music boxes and gramophones; punch card have widely been used for storing read-only data too. Sometimes deformed cylinders were used as an analog 2D look up table for some mathematical function -- imagine e.g. a device that has input x (rotating cylinder along its axis) and y (shifting it left/right); the cylinder can then at each surface point record function f(x,y) by its width which will in turn displace some stick that will mark the function value on a scale. To transfer movement strings, chains and belts may also be used. Random number generation may be implemented e.g. with Galton board. If timing is needed, pendulums can be used just like in clock. Some mechanical computers even use pretty complex parts such as mechanical arms, but these are firstly hard to make and secondly prone to breaking, so try to avoid complexity as much as possible.

BONUS THOUGHT: We have gotten so much used to using our current electronic digital computers for everything that sometimes we forget that at simulating actual physical reality they may still fail (or just be very overcomplicated) compared to a mechanical simulation which USES the physical reality itself; for example to make a simulation of a tsunami wave it may be more accurate to build an actual small model of a city and flood it with water than to make a computer simulation. That's why aerodynamic tunnels are still a thing. Ancient NASA flight simulators of space ships did use some electronics, but they did not use computer graphics to render the view from the ship, instead they used a screen projecting view from a tiny camera controlled by the simulator, moving inside a tiny environment, which basically achieved photorealistic graphics. Ideas like these may come in handy when designing mechanical computers as simulating reality is often what we want to do with the computer; for example if we want to model a sine function, we don't have to go through the pain of implementing binary logic and performing iterative calculation of sine approximation, we may simply use a pendulum whose swinging draws the function simply and precisely.

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