In basic physics mechanical constructs are described as composites of three elemental devices: the axial wheel, the inclined plane, and the lever. The method usually used to define the concept of 'simple machines' is to take many commonly seen mechanical devices (pulleys, knives, corkscrews) and show how they are composites of the three basics devices... rather then stating an explicit definition for the concept. This leads to a question...

In a two dimensional world, there would be no simple way of constructing an axial wheel; the axle could not be attached to anything that did not pass through the entire radius of the wheel, thereby preventing rotation.

Thus there seems to be only two simple machines in a two-dimensional Newtonian universe. This raises the question: if there are two in 2D, three in 3D, are there more than three in higher dimensions?

I could accept that that 2D is simply a degenerate case... after all, it appears that there would be no simple machines at all in both 1D and 0D... but it does not seem obvious to me one way or the other, especially in the absence of an explicit definition.

Therefore: Consider an n-dimensional setting which approximates Newtonian dynamics on a relevant scale, and treat mass as a continuous property of an object. How many simple dynamical machines can be defined in this space? Any thoughts would be appreciated.


Ariels and others commented that the axle was useless in 2D... barring some kind of additional axiomatic structure to the 2D universe (i.e. an attractive force such as magnetism, or more generally the supposition of sets of points necessarily at rest with respect to each other regardless of any continuous spatial connection between them.)

Given that, the best you can do in so far as transportation is concerned is a system consisting of a cart on two rollers, which can only move a finite distance before it breaks into discrete pieces... This was essentially my point; our conception of the wheel-as-conveyance cannot be extended to 2 dimensions; it degenerates into an apparatus of drastically limited usefulness (admittedly rollers aren't all bad; they were apparently good enough for pyramid building.)

But what is the wheel, really?

  /     \
 / o     \
|    X    |
 \     o /
Imagine this is a flat wooden disk rotating about a fixed "X" with two protruding wooden prongs facing the screen (symbolized as "o"). Provided that the prongs are symmetrically placed about X, a force V applied to the upper prong causes the lower prong to exert a force of -V on whatever it happens to be touching.

This is really just a lever. The only 'difference' is the unstated assumption that a lever's motion is restricted by the pivot. Because if, in 2 dimensions, the pivot must be attached to another stationary object (the 'ground', or in the case of transportation, the 'cart'), then the lever is capable of a freedom of motion approaching 360 degrees (restricted by the spatial extent of the pivot), but it does not possess continous freedom of rotation about the pivot. So a wheel is simply a lever with this restriction removed.

Looking at the wheel-as-transportation, this is in fact a 'complex' machine which involves the interaction of two necessary components: the wheel-as-lever as defined above (with the 'prongs' becoming all opposite points on the edge of the wheel), and a separately hypothesized 'frictional medium' (that is, an area of space in which there is a 'frictional force' Dependant on the velocity of an object relative to some fixed medium which acts is opposition to the direction of motion).

Therefore our definition of the simple machines could be restated explicitly as follows:

The simple machines: 1) The Inclined Plane: An object which when subjected to a force produces a perpendicular force while conserving work.

2) The Lever: An object which when subjected to a force produces a parallel and opposite force while conserving work.

Addendum: By affixing various levers of equal length to a common pivot, a wheel is created.

Basically there no longer seems like any point in listing the wheel separately. By combining the above two simple machines (regarded as operations on force vectors), every possible work vector can be transformed into every other work vector, thus enabling you to do anything mechanical you could possibly want to, at least in two dimensions (maybe it also trivially follows for higher dimensions, I'm not sure?)

But disregarding that, the issue of dimensionality seems to relate more to ensuring the range of motion of the constituents of a machine then to the inherent properties of those machines.

For example, In 2 dimensions a cart could be envisioned which would be capable of a finite linear displacement (i.e. rollers and cart or some similar mechanical apparatus). The 'rollers-cart' system is only capable of an arbitrary linear displacement if another mechanical element is introduced to displace the constituents of the system back to some default coordination (i.e. a guy on top of the cart to lift rollers from the back to the front).

Thus the question originally proposed may still stand, and can be restated as follows:
In higher dimensions, does the fact in which there is 'more room' with which to configure mechanical components produce any qualitatively new configurations?

...With the impetus for this question being the example of the continuous range of motion affordable by the wheel in 3D, due to the presence of another axis on which to displace the pivot.

The simplest possible suggestion is an immediate extension of the above example: the spherical motor. Although I believe substantial work has been done (references anyone?) on constructing a spherical motor using electromagnetic forces, it seems obvious enough that a you cannot have a simple mechanical 3-sphere which can freely apply force along any axis, whereas in 4 dimensions you could, because you now have somewhere to attach the axis (although I'm not 100% sure on this because of the fact that rotation in 3d is now inherently a vector rather then a scalar... and I can't quite visualize a 3-spherical motor in 4-space in any convincing way.)

Regardless of whether this example is meaningful or interesting, there seem to be other indications of how things could get interesting; consider for example the corkscrew. This is essentially a inclined plane wrapped into a helix, a construction which has no meaningful equivalent in 2 dimensions. Although I can't quite seem to phrase it in any clear way at the moment, this seems to me is some way suggestive of qualitatively new applications of the inclined plane, given another dimension to work with.

And even the definition of simple machines I gave above seems somewhat inadequate to me; the lever seems fairly straightforward; the essential components are simple three arbitrary points in space: one is defined to be the pivot, and the other two are then defined such that their distance from each other and from the pivot remains constant regardless of any other motion. Thus when holding the pivot in space and applying a force to either of the other points, the other is subject to an opposite force (whose magnitude depends on the ratio of the points' distances to the pivot). The inclined plane on the other hand seems confusing to me... In physics you almost always consider it using gravity as a force framework, and I have a hard time performing an elemental decomposition similar to the one above (little help?).

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