mr_e_man wrote:What do you mean by 3D gears?
We already have 3D gears in 3D space, e.g. bevel gears or helical gears, in addition to the simple 2D gears in 3D space.
The Cartesian product of two 2D gears could be a 4D gear in 4D space, allowing double rotation. But I wonder if any kind of axle could support it. A 2D gear cannot be supported in 2D space; it needs 3D for the axle.
Frisk-256 wrote:another idea is to start with 2 3d spheres, and cut out parts of 1, attaching them to the other part in the same positions
mr_e_man wrote:What do you mean by 3D gears?
We already have 3D gears in 3D space, e.g. bevel gears or helical gears, in addition to the simple 2D gears in 3D space.
The Cartesian product of two 2D gears could be a 4D gear in 4D space, allowing double rotation. But I wonder if any kind of axle could support it. A 2D gear cannot be supported in 2D space; it needs 3D for the axle.
Frisk-256 wrote:Frisk-256 wrote:another idea is to start with 2 3d spheres, and cut out parts of 1, attaching them to the other part in the same positions
https://www.desmos.com/3d/nrxq8agewf
I made this, it lags a lot but It sort of shows my idea
gonegahgah wrote:In 3D gears are used to change the speed or direction of rotation.
Is that the same in 4D?
Frisk-256 wrote:One solution could be to just give the 3d gears a 4d thickness and pretend everything is normal, Just leave them as circles extruded 2 times, one to 3d and one to 4d, you could also expand the idea of bevel gears(think that's the right gear) that meet at an angle to 4d. Another idea is to start with 2 3d spheres, and cut out parts of 1, attaching them to the other part in the same positions, then extrude these warped spheres into 4d gears, this could also be extended to 4d bevel gears by extruding them with a scale to make them like truncated 4d cones with bumps. Another idea is to make the 2 gears both Clifford Torus with a looping grid pattern, which can rotate around 2 perpendicualr planes
PatrickPowers wrote:That's what I had in mind. In 3D a rotating shaft has a fixed plane of rotation. In 4D a rotating shaft can have a degree of freedom for the rotational plane. The hard part is changing both the speed (by a fixed ratio) and the plane of rotation with the same two gears. Can it be done?
This collaborative work is truly exciting!PatrickPowers wrote:I'm looking for a pair of gears that passes along changes in the orientation of the plane of rotation. This can never happen in our 3D world.
It's easy if the two gears can be the same size. Just have spherical gears and attach the gears to appropriate shafts.
gonegahgah wrote:This collaborative work is truly exciting!PatrickPowers wrote:I'm looking for a pair of gears that passes along changes in the orientation of the plane of rotation. This can never happen in our 3D world.
It's easy if the two gears can be the same size. Just have spherical gears and attach the gears to appropriate shafts.
As always, I’m approaching this from my "three" perspective. In my vision, a 4D car would have a steering system consisting of three steering circles — one for each hand. These circles would be positioned 4D-sideways from each other, all parallel to the driver's front, and at the same height.
Originally, I considered that a 4D steering system might involve a rack-and-pinion mechanism similar to those in 3D cars.
I believe Frisk-256’s concept of spherical gears is particularly useful in this context:
At first, I thought two racks would be sufficient, but now I believe three would be better to provide more comprehensive sideways coverage. By maintaining a straight track while curving the crosswise bevels to match the spherical cog, and incorporating a single spherical cog, we could achieve a well-connected and structurally reinforced system.
To enhance durability and compensate for the extra spatial complexity of 4D, each rack should be fairly wide. Additionally, they could be arranged in a crisscross pattern within the 3D side space of a car.
These racks would also need to transfer motion to three front wheels, which introduces another interesting challenge.
Would this modified rack-and-spherical-cog system achieve what you're envisioning, Patrick?
PatrickPowers wrote:gonegahgah wrote:PatrickPowers wrote:A single 2D disc-shaped rack is sufficient, as is a single 3D-sphere-surface (what mathematicians call a 2-sphere) steering wheel. The "pinion" gear can be spherical. The irregularity of the teeth in the gear is not a problem with rack and pinion because the rack can be made with the same irregularity.
PatrickPowers wrote:Frisk-256 wrote:Frisk-256 wrote:another idea is to start with 2 3d spheres, and cut out parts of 1, attaching them to the other part in the same positions
https://www.desmos.com/3d/nrxq8agewf
I made this, it lags a lot but It sort of shows my idea
Yes that's the idea. But since the gears are the same size that makes it easy. The two spheres have the same irregularities. If the gears have different sizes and different number of teeth I don't know if it can be done.
PatrickPowers wrote:A single 2D disc-shaped rack is sufficient, as is a single 3D-sphere-surface (what mathematicians call a 2-sphere) steering wheel. The "pinion" gear can be spherical. The irregularity of the teeth in the gear is not a problem with rack and pinion because the rack can be made with the same irregularity.
Frisk-256 wrote:I don't know if that would work, wouldn't things like holonomy mean that rotating it enough would eventuality make the rack move off the gear? Also how would the motion be transferred between the rack and more gears? And isint it impossible to map a sphere onto a plane?
mr_e_man wrote:So, Patrick, with "3D gear" you're referring not to the shape of the object, but to its freedom to rotate?
mr_e_man wrote:I agree with Frisk; a planar rack and spherical pinion probably wouldn't work, for continuous rotations. But it could work for discrete rotations.
mr_e_man wrote:Each gear could be a Platonic solid or planar tiling, with pyramidal teeth added to one surface and subtracted from the other surface. E.g. consider a kis-cube as the pinion, and a square tiling with pyramid-shaped indentations as the rack. Or consider a kis-octahedron and a kis-icosahedron, as two compatible spherical gears of different sizes.
Frisk-256 wrote:PatrickPowers wrote:gonegahgah wrote:PatrickPowers wrote:A single 2D disc-shaped rack is sufficient, as is a single 3D-sphere-surface (what mathematicians call a 2-sphere) steering wheel. The "pinion" gear can be spherical. The irregularity of the teeth in the gear is not a problem with rack and pinion because the rack can be made with the same irregularity.
I don't know if that would work, wouldn't things like holonomy mean that rotating it enough would eventuality make the rack move off the gear? Also how would the motion be transferred between the rack and more gears? And isint it impossible to map a sphere onto a plane?
PatrickPowers wrote:Yes, like steering an automobile. I'm not sure it would work. Suppose we have a ball rolling on a plane. Is the point of contact the same regardless of the path traveled by the ball? That is, start with the ball and plane both at (0,0). The ball rolls around the plane. When the wandering ball returns to the plane's (0,0), is its point of contact still the ball's (0,0)? That is, if the ball is rolling from A to B, does its orientation at B depend on the path? I think it does. Quaternions represent the rotation group of a sphere and quaternions don't commute. So it appears the 2D rack won't work. Is that right?
A little girl stole my tennis ball so I can't check it out via experiment. Sob.
gonegahgah wrote:If you have a single steering wheel connected to one Frisk pinion gear, you could attach three additional Frisk gears branching off in different lateral directions.
PatrickPowers wrote:The 3D spherical steering wheel is pretty much the same as our 2D one. To 4D people it looks like a ring. One handed driving is the same as it is to us. Every part of the sphere is about the same distance from the driver, who is separated from the wheel in the 4th dimension.
Let's look at the familiar 3D case first. A circle is 1D. At every point on the circle add a little 2D disc centered on that point and perpendicular to the circle. As a refinement, it can't be little disc, it has to be a little round wedgelike thing. The result is a 2D ring.
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