4D gears

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:01 am

In 4D it is possible to have 3D gears. There is a problem though : with 2D gears it is easy to space the teeth evenly, in 3D not so easy. Am I right in thinking that generally the teeth cannot be spaced evenly, so that the peak of every tooth is an equal distance from every one of its neighbors? If the spacing must be irregular, is it possible for the other gear to have the same irregular pattern so that the teeth mesh no matter what the gears do? It seems hard if the gears have different numbers of teeth, which is usually the case.
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Re: 4D gears

Postby Frisk-256 » Mon Feb 03, 2025 6:14 pm

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
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Re: 4D gears

Postby mr_e_man » Mon Feb 03, 2025 6:22 pm

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.
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Re: 4D gears

Postby Frisk-256 » Mon Feb 03, 2025 6:30 pm

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.

If we think beyond to 5D, the axle could stick out perpendicular to all of this.
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Re: 4D gears

Postby Frisk-256 » Mon Feb 03, 2025 6:46 pm

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
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Re: 4D gears

Postby gonegahgah » Mon Feb 03, 2025 9:44 pm

In 3D gears are used to change the speed or direction of rotation.
Is that the same in 4D?
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Re: 4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:28 pm

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.


I'd say that bevel gears are 2D gears embedded in a 3D space. What I had in mind was spherical gears with a shaft or axle, which make no sense in 3D. 3D for the sphere, 1D for the shaft.
Last edited by PatrickPowers on Tue Feb 04, 2025 8:59 am, edited 4 times in total.
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Re: 4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:32 pm

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.
Last edited by PatrickPowers on Mon Feb 03, 2025 11:54 pm, edited 1 time in total.
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Re: 4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:43 pm

Deleted.
Last edited by PatrickPowers on Tue Feb 04, 2025 1:36 am, edited 2 times in total.
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Re: 4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:47 pm

gonegahgah wrote:In 3D gears are used to change the speed or direction of rotation.
Is that the same in 4D?



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?
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Re: 4D gears

Postby PatrickPowers » Mon Feb 03, 2025 11:52 pm

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



Whew! This is over my head. The case I'm interested in is with a different shaft attached to each gear.
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Re: 4D gears

Postby gonegahgah » Tue Feb 04, 2025 10:50 am

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?

One potential solution is a "worm" gear, which is well-suited for altering both the speed and the direction of rotation—typically by 90 degrees. This principle extends beyond 3D; it can work in 4D or even higher dimensions.
Image
In 4D, both the cog and the worm gear would have teeth shaped like regular plane trapezoidal prisms, with the gaps between them following the same shape. These prisms would be slightly skewed along the bevel of the teeth to ensure smooth engagement as the worm gear progresses.

The key idea here is the use of a "regular polygon" as a basis for the gear’s structure. A spherinder (a 4D cylinder) could serve as the foundation, slightly deformed by the trapezoidal shape of the teeth. However, any fairly uniform regular shape—such as a square, pentagon, or hexagon—could also be used. The difference in structural strength between these shapes and a full circle would likely be minimal.

A useful way to visualize this is by analogy with lower dimensions. Consider a 2D cogwheel—it would be weak if used on its own in a 3D system. To strengthen it, we extend it along the third dimension, creating a 3D cogwheel. The same principle applies when extending a 3D cog into 4D: rather than obstructing the mechanism, the 4D extension reinforces it.

A common misconception is that this 4D "sideways" extension would interfere with rotation. However, just as a 2D cog extended into 3D does not disrupt its function, a 3D cog extended into 4D remains structurally sound. The extensions always stay the same distance into the extra dimension, even as the cog rotates, merely providing additional strength.

Ultimately, there’s little fundamental difference between 2D, 3D, and 4D cogs in terms of function. While 3D introduces additional cog variations, each still operates within a specific plane of rotation. It’s unclear whether 4D, 5D, or higher dimensions would allow for fundamentally new types of cogs if their primary purpose is simply to change direction and speed. They would all likely function in a similar manner.
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Re: 4D gears

Postby gonegahgah » Tue Feb 04, 2025 11:25 am

I did ask ChatGPT if it thought that 4D could add new concepts to cogs and it gave me the following.
I haven't studied it yet but I put it out there for consideration.
I'm a little doubtful without having looked at the following but maybe I could be wrong hopefully...
The following is totally ChatGPT; even the final paragraph!

1. Multi-Planar Gears (Unique to 4D and Higher)

Concept:
In 3D, gears transfer rotation within a single plane. In 4D, however, a gear could simultaneously transfer rotation between multiple planes at once.

Design:
Instead of just having one plane of gear teeth (as in 3D), a 4D cogwheel could have multiple sets of teeth along different rotational planes.
Imagine a 3D bevel gear, but instead of just one slanted surface, it has multiple slanted surfaces allowing for different rotational interactions.
A 4D multi-planar gear could be a single gear engaged with multiple other gears in separate planes simultaneously.

Operation:
The gear teeth could be aligned such that when the gear rotates, it drives motion in multiple perpendicular directions at once.
One possible realization: A single gear might be meshed with one gear in the XY plane and another in the ZW plane. Turning the gear would drive both motions at once, possibly at different ratios.
By adjusting how these gear teeth are arranged, it might be possible to continuously shift the power transfer between different planes during operation.

Implications:
Multi-axis power transmission: One gear could simultaneously drive multiple independent rotational axes.
Variable coupling: A system could switch between which planes receive more or less torque, possibly making adaptive machinery that shifts power between different directions as needed.

2. 4D Bevel Gears

Concept:
In 3D, bevel gears allow rotation to be transferred between axes that are not parallel, most commonly at 90°. In 4D, we could generalize this idea to enable motion transfer between any two planes of rotation in 4D space.

Design:
A 3D bevel gear has conical gear teeth that allow for smooth engagement at an angle.
A 4D bevel gear would need a more complex curved structure to allow engagement along multiple possible rotational planes instead of just one.
Instead of a simple cone, the surface might be a 3D spherical section or even a spherinder section to accommodate rotations in 4D space.

Operation:
Instead of just engaging along a single fixed plane (like 3D bevel gears do), a 4D bevel gear could engage in multiple planes at once or shift dynamically between them.
Depending on the alignment, it could allow for power transfer between three or more independent axes, rather than just two as in 3D.

Implications:
More efficient multi-axis gearing: Instead of needing multiple gears in sequence to change rotational direction through complex arrangements, a single 4D bevel gear could accomplish this in one step.
Adaptive gear shifting: A machine might be able to dynamically adjust which rotational plane receives the most power, leading to new forms of mechanical efficiency.
Higher-dimensional mechanical linkages: This could be useful for theoretical 4D machinery, robotics, or even future simulations where 4D physics is modeled.

Comparing the Two:

Feature Multi-Planar Gear 4D Bevel Gear
Primary Function Transfers rotation across multiple planes simultaneously Redirects rotation smoothly between different 4D planes
Number of Planes Involved 2+ (can interact with multiple gears at once) Typically 2 or 3, but more adaptable than 3D bevel gears
Adaptability Can be dynamically adjusted to shift power between axes Allows smooth transition between different rotational planes
Engineering Challenge Requires designing teeth that mesh across multiple planes Requires curved or flexible engagement to handle multiple rotational axes

Final Thoughts
Both of these concepts suggest new forms of mechanical linkages that wouldn't be possible in 3D. While the fundamental behavior of gears remains the same (transferring rotational motion), 4D introduces new ways to couple, redirect, and distribute motion, which could lead to entirely new mechanical systems.

I find the multi-planar gear particularly exciting because it allows a single gear to influence multiple independent rotations at once, something that is difficult or impossible in 3D without additional linkages. Meanwhile, the 4D bevel gear makes me wonder whether smooth, dynamically shifting gearing mechanisms could be developed in higher-dimensional physics models.
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Re: 4D gears

Postby PatrickPowers » Tue Feb 04, 2025 11:28 am

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. If the gears have different numbers of teeth then you have problems, as I don't know if it's possible to space the teeth in the same way on both gears.
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Re: 4D gears

Postby gonegahgah » Tue Feb 04, 2025 9:59 pm

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.
This collaborative work is truly exciting!

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.
Image

I believe Frisk-256’s concept of spherical gears is particularly useful in this context:
Spherical Cog.jpg
Spherical Cog
Spherical Cog.jpg (14.69 KiB) Viewed 956 times

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?
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Re: 4D gears

Postby PatrickPowers » Tue Feb 04, 2025 11:24 pm

gonegahgah wrote:
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.
This collaborative work is truly exciting!

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.
Image

I believe Frisk-256’s concept of spherical gears is particularly useful in this context:
Spherical Cog.jpg

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?


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.
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Re: 4D gears

Postby Frisk-256 » Wed Feb 05, 2025 1:38 pm

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?
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Re: 4D gears

Postby Frisk-256 » Wed Feb 05, 2025 1:41 pm

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.


different sized gears would be hard
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Re: 4D gears

Postby Frisk-256 » Wed Feb 05, 2025 1:45 pm

An interesting thing with the 2d gear extruded 2 times is that it could have and axle that is an entire plane, if the gear was in the xy plane, and extruded into z and w, the axle could be a plane in z and w extruded into x and y to give it thickness
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Re: 4D gears

Postby mr_e_man » Wed Feb 05, 2025 8:09 pm

So, Patrick, with "3D gear" you're referring not to the shape of the object, but to its freedom to rotate?


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?

I agree with Frisk; a planar rack and spherical pinion probably wouldn't work, for continuous rotations. But it could work for discrete rotations.

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.
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Re: 4D gears

Postby PatrickPowers » Wed Feb 05, 2025 11:09 pm

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?


The gear in question has a 3D spherical shape, is flat in the 4th dimension, and is attached to a shaft with a 3D spherical cross section. Then the gear and shaft have a degree of freedom viz. their plane of rotation that we can't have here in 3D.


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.


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.

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.


That looks good. The set of allowable gear ratios would be limited but certainly better than nothing.
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Re: 4D gears

Postby gonegahgah » Thu Feb 06, 2025 12:12 pm

I should mention I get ChatGPT to rewrite my ideas. It conveys things much better than me!

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?

First, thanks to Patrick for prompting me to rethink this concept. Initially, I shifted to using a single steering wheel with its one pinion, with three racks, but one issue that remains is cross-rotation and its inherent limitations.

To address this, I've returned to the idea of using three separate steering wheels. Each steering wheel is constrained to rotate only within a specific 60° sector (plus its opposite) in the 4D sideways space. For example:
Steering Wheel 1: Rotates from a forward vertical tilt to either 0° or 180° sideways.
Steering Wheel 2: Rotates from a forward vertical tilt to either 60° or 240° sideways.
Steering Wheel 3: Rotates from a forward vertical tilt to either 120° or 300° sideways.

By similarly constraining the corresponding rack to these sideways angles, each rack-and-pinion pair would function analogously to traditional systems, while still accommodating any ratio differences between the rack and the pinion.
Instead of using dual rods, we would implement reinforced triangular rods that connect to the pivot arms — each controlling the designated side angle for every wheel.

With three such triangular rods, each linked to a specific pivot arm of each steered wheel, the system would collectively enable the wheels to steer in all sideways directions.
This arrangement ensures that the tires, pivot arms, steering rods, and rack work in unison to prevent any impossible or conflicting wheel rotations.
For instance, it would be mechanically impossible for all wheels to simultaneously turn to 0°, 60°, and 120° if those angles aren’t mutually achievable by the vehicle’s design.

The central idea is to constrain each steering subsystem to operate solely in one of three opposite directions. With proper reinforcement, this approach should be both mechanically feasible and effective.
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Re: 4D gears

Postby PatrickPowers » Thu Feb 06, 2025 1:09 pm

I'm going to give up on steering 4D automobiles. I couldn't have invented the system they have here in 3D so doing it in 4D seems out of reach. I have plenty of other ideas, it won't be missed.
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Re: 4D gears

Postby gonegahgah » Thu Feb 06, 2025 1:25 pm

I had another idea…

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.
Each of these Frisk gears would have its own rotation pin, and you could place a track against each pin to guide movement in only one specific direction.

Would this allow the rotation pins of the Frisk gears to transfer lateral movement at different ratios?
Multiple (triangular) rods would still be necessary, but that wouldn't be a problem.

The steering wheel itself has one main rotation pin, while each of the three additional Frisk gears would have their own rotation pins to engage.
Is this possible?
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Re: 4D gears

Postby mr_e_man » Thu Feb 06, 2025 7:03 pm

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.

Right. Orientation depends on the path. I think that's the idea of "holonomy".

Check out Simon Tatham's "Cube" game.
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Re: 4D gears

Postby gonegahgah » Thu Feb 06, 2025 10:42 pm

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.

One potential issue with using a single steering wheel is holonomy.
If one hand tries to steer at an angle that isn’t parallel to the other, it would induce unwanted rotation within the grip of the other hand(s).
Unless the steering wheel itself can freely rotate in all possible grip placements, this could feel unnatural.

A similar cross-rotation issue might arise with each of the three additional Frisk rotation pins due to their fixed-direction movement racks.
This suggests that independent — but coordinated — steering wheels might be necessary, one for each hand.

You could still drive with two hands when necessary or, with enough determination, even control it with just one hand on one of the wheels (Edit: Hmmm, probably not one hand?).
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Re: 4D gears

Postby PatrickPowers » Fri Feb 07, 2025 1:17 am

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.

In 4D surface of a sphere is 2D. A little round wedge is 2D. Take the surface of a sphere and at every point add a wedgelike thing centered at that point. You get a 3D ring.
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Re: 4D gears

Postby gonegahgah » Sat Feb 08, 2025 6:17 am

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.

That wasn't my mistake Patrick. From my previous rethink inspired by your pushback I had understood that already.
My mistake was to think that the 4D being's other hand that was not in a direct circle of turn being done by the first hand would stay in place.
That was purely wrong. When we steer our wheels with two hands both hands move with the wheel and twist. Same in 4D on a spherical "ring" wheel.
It doesn't stay rooted in place and it follows along and twists with the steering wheel as needed.
And if it goes too far for one hand what do we do? We lift that hand and move it to another accessible spot on the wheel.
Simple really. That definitely was a Dorothy moment sadly.

3D Slices of 4D Steering Wheel
Image
Leftmost: Shows the sideways angles that the steering wheel could be in plus the alignment from above for the 3 of 4 up-tilting spokes that support the outer steering wheel rim to the central "disc".
Right 3: Here you can see the front of 3D slices of the 4D steering wheel when we rotate our sideways from x to w by, in this case:
- 0° starting at 0 rotation from x-w we can see the up-tilting spoke towards the right... then mostly nothing in-between except the down support until
- 60° we can now see the spoke that was 60° towards ana has now come around to our 3D left... then mostly more nothing except the down support until
- 120° we can now see the spoke that was almost kata at the start has now come around to our 3D right... then mostly more nothing except the down support until
- etc. (Note: at different twists of the steering wheel the down support will also not be visible in our 3D slice. However here I have kept the steering wheel in one position throughout while rotating 4D-sideways to illustrated).

Anyhow, that still leaves us with the problem of the "racks" and "rods" pushing different corresponding points relative each tyre wheel to affect them to turn in coordinated fashion towards each sideways direction.
I might have an idea now and I'll try to draw it...
Attachments
Steering Wheel .jpg
3D Slices 4D Wheel
(50.95 KiB) Not downloaded yet
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Re: 4D gears

Postby gonegahgah » Sun Feb 09, 2025 7:26 pm

Another interesting diagram from Wikipedia showing how the wheels turn slightly different angles to improve the travel of the wheels.
The inside wheel in the turn follows a shorter tighter path than the outside wheel during the turn.
https://en.wikipedia.org/wiki/Ackermann_steering_geometry
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