4D gears

[PatrickPowers]

Assuming that what you're looking for is 3D gears in the sense that they have evenly spaced teeth and rotate in 2 dimensions, analogous to 2D gears rotating in 1D, then the only viable configurations would be those with Platonic solid symmetry. These are the only symmetries your gear teeth can be in, and be equally spaced from each other in a way that the meshing gear won't get stuck with mismatching teeth.

Unfortunately, due to the Platonic solid symmetries being finite and discrete, you will not get full 2D freedom; the perfect matching meshing of gear teeth will only happen along the circles of symmetry of the gears. E.g. along the great circles of icosahedral symmetry. If your meshing gear is spinning in a direction other than this, you will still get the mismatching gear teeth problem.

With a 3D gear like a durian I found nothing but problems. The second Big Problem is this. Suppose that the shafts of the gears aren't colinear. Then each gear will have a plane in which the other gear cannot rotate, leading to much wailing and gnashing of teeth. I gave up on 3D gears, it is one of those things that just doesn't work in 4D. I think that's interesting. Note that a very simple gearless system like in our tricycles works just fine in 4D. It's gears that are the problem.

So how to steer an automobile? Use a joystick restricted to the sideways plane and copy what World War One airplanes did here in this world with our mundane 2D gears. I've forgotten the details but rightly or wrongly concluded it avoids all this trouble.

That brings up the question, can a 4D airplane have a 3D joystick? I haven't got time to deal with it right now.

[quickfur]

Yeah, Keiji proposed the concept of the planar rail system years ago, and I think it was a total stroke of genius. Combine our 3D world's monorail system with the extra dimension, and now you have a 2D surface area you can drive over using a 3D-style steering wheel without any problems with gimble lock and all of that jazz. You get the benefit of both freedom (in 2 dimensions) and stability (in the other dimension). Just designed your roads like the highway / freeway system here, except in 3D, and you can have a road system that does not have intersections or traffic lights, only merging lanes. Thanks to the extra dimension of space, none of this requires overpasses or elevated roadways, the entire thing can be built on ground level. You don't even need a bridge to cross rivers since in 4D you can just build your road around it.

On a road with only 2 degrees of freedom, you only need wheels that rotate in a single plane, so no problematic 3D gears are needed; extrusions of existing 2D gears work just fine. This simplifies motor / transmission / wheel design, makes everything much more practical.

[DonSoreno]

I want to make this stuff more mathematically rigorous.

I'll extend it to N-dimensional space (N = 1,2,3,4,..., countable infinite,...)



Regarding the shape / Structure of the Gears

All rotatopes of dimension N can be used as gears.

Each rotatope is basically represented by a partition of the set of basis vectors, each partition corresponding to a spheroid.
(in the wiki: http://hi.gher.space/wiki/Rotatope.

E.g. the 4d rotatopes
tesseract: {{e1},{e2},{e3},{e4}}
cubinder: {{e1,e2},{e3},{e4}}
duocylinder: {{e1,e2},{e3,e4}}
spherinder: {{e1,e2,e3},{e4}}

Furthermore each rotatope can have a Radius for each of the spheroids into which it factorizes.
Thus we can represent each rotatope by a "weighted" partition of the basis vector (not taking into account, that the rotatopes may be tilted w.r.t. the basis.)

In N-dimensional space a general axle/shaft still has (N-1) dimensions, orthogonal to its length.

We can transmit an arbitrary partitioning of (N-1) dimensions, through such a shaft.

E.g. in 4d we have the following options, represented by their respective rotation groups:

-> transmitting SO(3) (basically 3d rotations, quaternions.)
-> transmitting SO(2) x SO(1) (only allowing rotation in one 2d plane.)

In 5d, such a rod could transmit:
SO(2) x SO(2) (two rotations along orthogonal bivectors; the meet of said bivectors is empty: b1 ∧ b2 = 0)=
SO(3) x SO(1) (--> spherical rotations.)
SO(4) (--> transmitting glome rotations.)



(The rotation group SO(1) is useless, since there are no continuous rotations in 1 dimension.)



collinear prisms that are some (N-1)-dimenjsional rotatopes extruded along an axis, can then transmit torque.
They must be the same rotatope, except the radius of their respective hyperspheres may be different, leading to gear ratios for each rotation group SO(n).

E.g. in 5d, we could have two duocylindrical gears, that transmit two rotations:
SO(2) x SO(2)
with two different gear ratios.




Gear surface / teeth
All rotatopes could act as gears based on friction.

Equipping 3d and higher dimensional hyperspheres with teeth is difficult.

I think, teeth may be represented by spherical harmonics (--> fourier transform on spherical manifolds),
but this is just speculation on my part.

Or we could use some tiling of the sphere, that can be mapped to a plane with minimal defects. E.g.:
https://www.researchgate.net/figure/Applying-the-query-symmetry-class-Stellate-and-normal-true-and-maximal-true_fig3_343124534