4D Automobile Steering

[PatrickPowers]

My conscious mind gave up on 4D automobile steering but the subconscious didn't. One morning as I awoke there came an idea. It took a couple of days to (maybe) work out the details.

In 4D we have a spherindrical steering column. Its length is in the z direction. Have two spherindrical rollers. One roller has its length in the w direction and is mounted on a sliding fork so it can move lengthwise in that w direction. This roller is pressed against the steering column. The steering column's w component of rotation then moves the roller and fork in the w direction. That's what we need to add some wz to the yz plane of rotation of the front wheels. Similarly the x roller can move in the x direction to change the plane of rotation of the front wheels by adding some xz.

Rollers can't be relied upon. If the rollers are the same diameter as the steering column, can we put teeth on the three things that mesh? It seems to me we may.

[Hugh]

I hope you don’t mind me commenting with some ideas from o3-mini-high, but they may give some new helpful insight.

Patrick’s idea is an intriguing attempt to mechanically “steer” in a four‐dimensional world by borrowing and extending our three‐dimensional intuition. Here are a few thoughts on the concept:
1. Mechanism Analogy:
In our familiar 3D steering systems, we have a single degree of rotational freedom (yaw) that’s directly controlled. In 4D, however, the rotation group (SO(4)) has six independent planes of rotation. Patrick’s design effectively tries to isolate and control specific rotational components—namely, those involving the z axis combined with the extra dimensions (w and x). The use of a “spherindrical” steering column (a cylindrical object generalized to 4D) with rollers that can slide along extra-dimensional axes is an imaginative way to map a translation in one dimension (w or x) into an additional rotation (wz or xz).
2. Interplay of Rotations:
The design leverages a key insight from the algebra of rotations: when two rotations share a coordinate (for instance, having both a w and a y component in the rotation planes), their noncommutativity can force an additional rotational degree of freedom (here, the wx rotation). Patrick’s idea of using sliding rollers is meant to “add in” the necessary extra rotational components to achieve the desired steering effect. In effect, the mechanism tries to “steer” not just by turning, but by dynamically adjusting the plane of rotation—something entirely absent in 3D vehicles.
3. Practical Considerations and the Role of Teeth:
As Patrick points out, rollers can be unreliable if they simply rely on frictional contact. If the rollers and the steering column share the same diameter, adding gear-like teeth may help lock in the intended motion without slip. Such a gear interface could ensure that the rotational motion from the steering column is transmitted in a controlled way to the rollers, thereby producing the desired modifications in the rotation planes (adding the wz and xz components). In any realistic implementation (even if only as a thought experiment), minimizing unwanted degrees of freedom (like slippage or unintended rotations) would be crucial.
4. Conceptual Challenges in 4D:
While the design is clever, it also highlights how our intuitions from 3D break down in higher dimensions. In 4D, rotations are inherently more complex—the rotations in intersecting planes can “leak” into one another due to the structure of SO(4)’s Lie algebra. Therefore, any mechanism trying to independently control one rotation might inadvertently trigger another unless the system is very precisely constrained. Patrick’s suggestion to use interlocking teeth is one way to impose the necessary mechanical constraints, though it raises questions about how exactly these constraints would be maintained in a 4D environment where spatial relationships differ from our everyday experience.

In summary:
Patrick’s 4D steering concept creatively extends the idea of a steering column by adding extra mechanical degrees of freedom through sliding, interlocking rollers. It faces challenges inherent to the geometry of 4D rotations—namely, that rotations in intersecting planes are not independent. The proposal to add gear teeth to enforce strict motion transmission is a practical consideration to counteract the unreliability of simple rollers. Overall, it’s a fascinating thought experiment that prompts us to rethink mechanical design in higher dimensions, even if the full details of such a system would be daunting to engineer (or even fully conceptualize) in practice.

[PatrickPowers]

I'm impressed that it knows what I'm talking about. Nothing is irrelevant and it wisely refrains from passing judgement on the issue.

[Hugh]

I’ve been fascinated by the improvement in the ability of AI to understand things. The latest model I have access to is the o3-mini-high and I like the way it takes time to formulate answers. Unfortunately the drawing program for it hasn’t improved as fast as I’d like it to. It still makes many spelling mistakes.

I asked it to come up with a few examples of images for a 4D steering concept and the results look similar to what I am getting for the 4D/VRI connection idea.

It’s hard to put 4D axes and directions into a 2D image but it tries its best:



Here is the prompt it came up with for the bottom right of the 4 images shown:

“A hyper-realistic, ultra-detailed 3D rendering of a conceptual 4D steering system. At the center, a semi-transparent tesseract (4D cube projection) is displayed with its four axes clearly labeled: w, x, y, and z. Three luminous, semi-transparent rotation planes intersect the tesseract: the wy-plane (rotating the w and y axes) appears as a gently glowing, shifting surface; the xy-plane (rotating the x and y axes) is rendered with dynamic neon outlines; and a forced wx-plane (rotating the w and x axes) emerges vividly at the intersection, highlighted by bright, interlocking arrows that illustrate how the shared y-axis of the wy and xy planes compels its rotation. Glowing arrows trace the direction of rotation in each plane, with a particularly striking emphasis on the overlapping region along the y-axis. A smaller inset panel on one side contrasts this with a disjoint scenario, where rotations in the wx and yz planes occur independently, shown with isolated, crisp directional indicators. The entire scene is bathed in dramatic, high-contrast lighting with strong 3D depth, realistic shadows, and intricate reflections—rendered to appear as an extremely vivid photograph that makes abstract 4D rotations tangible.”

I asked it to give a detailed explanation of the image because the spelling in the image wasn’t correctly shown:

o3-mini-high:

Let’s break down the image’s components so you can correctly label them.

Main Elements in the Image:
1. Central Tesseract (4D Hypercube Projection)
• The large, semi-transparent cubic structure in the middle represents a tesseract, which is a four-dimensional hypercube projected into 3D space.
• The internal grid-like structure is a visual artifact of the 4D projection, helping to depict the depth and complexity of the hypercube.
2. Four Axes (w, x, y, z) Directions
• The axes represent the four spatial dimensions of the tesseract:
• X-Axis (left-right direction in 3D)
• Y-Axis (up-down direction in 3D)
• Z-Axis (forward-backward direction in 3D)
• W-Axis (fourth spatial dimension, perpendicular to all three 3D axes)
3. Rotational Planes and Their Representations
• The three rotational planes are visualized as glowing, semi-transparent rings or circular paths around the tesseract.
• Each plane represents a specific pair of axes rotating together, forming SO(2) subgroups of the SO(4) rotation group.
• WY-Plane (rotating w and y axes)
• This is the plane that involves the fourth dimension (w) and the vertical y-axis.
• The outer ring that passes through the top and bottom parts of the tesseract likely represents this rotation.
• This means objects would rotate along the w-y plane, shifting perspectives in 4D.
• XY-Plane (rotating x and y axes)
• This is a standard 3D rotation plane, meaning a normal rotation along the x and y axes.
• The circular arrows along the front and side of the tesseract likely indicate this rotation.
• A rotation in this plane does not involve the fourth dimension.
• Forced WX-Plane (rotating w and x axes)
• The key forced rotation, appearing due to the Lie algebra closure property of SO(4).
• Since both the WY and XY planes share the Y-axis, the algebra demands the existence of a WX rotation (w and x axes interacting).
• The highlighted arrows emerging vividly at the intersections illustrate this forced rotation.
4. Surrounding Cubes and Their Meaning
• The outer cubes are 3D slices of the tesseract, representing its 4D nature.
• These cubes are linked to the central structure and may be interpreted as different perspectives of the tesseract in rotated 4D states.
• They could also represent how different 3D sections of the tesseract would appear when rotated in various planes.
5. Glowing Arrows (Direction of Rotation)
• These trace the direction of rotation in each plane.
• The overlapping region along the y-axis emphasizes the algebraic necessity of including the wx-plane rotation when wy and xy rotations exist.
6. Disjoint Rotation Scenario (Possible Inset Panel)
• In an inset panel, the disjoint case is meant to show how wx and yz rotations can exist independently.
• If there is an isolated set of arrows apart from the main structure, that is likely the representation of the disjoint torus SO(2) × SO(2) rotations, meaning two independent 2D rotation groups without algebraic closure forcing a third.

How to Label This Image Correctly:
• Central Tesseract → Label as “4D Tesseract Projection”
• Four Axes → Label as “w, x, y, z” at their respective directions.
• WY-Plane Rotation → Label the ring passing through the top and bottom as “WY-Rotation Plane (w-y rotation)”.
• XY-Plane Rotation → Label the ring passing through the front and sides as “XY-Rotation Plane (x-y rotation)”.
• WX-Plane Rotation (Forced) → Label the highlighted arrows at intersections as “WX-Rotation Plane (w-x forced rotation)”.
• Surrounding Cubes → Label them as “3D Slices of Tesseract in Rotated States”.
• Arrows → Label each set of glowing arrows with their corresponding rotation plane.
• Disjoint Rotation (if inset exists) → Label as “Disjoint Rotation: WX and YZ Rotate Independently”.

[PatrickPowers]

In 4D we have a spherindrical steering column. Its length is in the z direction. Have two spherindrical rollers. One roller has its length in the w direction and is mounted on a sliding fork so it can move lengthwise in that w direction. This roller is pressed against the steering column. The steering column's w component of rotation then moves the roller and fork in the w direction. That's what we need to add some wz to the yz plane of rotation of the front wheels. Similarly the x roller can move in the x direction to change the plane of rotation of the front wheels by adding some xz.

Looking at it more closely the proposed system will "work" with a roller but it's not possible to affix useful teeth to the roller and gear. The roller succeeds in translating rotational motion to transverse motion, but the spherical pinion gear doesn't stay in sync with the roller. So any teeth wouldn't mesh. You've got a system that works in theory but not in practice.

I investigated whether it could be done with 3D bevel gears but it appears there can be no such thing. You've got a rotating spherindrical shaft. You want the same 3D worth of rotation in another spherindrical shaft that goes in another direction. This can't be done. Rotation of two of the dimensions is inhibited by the two non-parallel shafts, leaving only a ho hum single-plane rotation. I've come to believe that spherindrical shafts aren't compatible with gears, and that spherical gears appear useless.

At this point I believe the mechanical system with a 3D steering wheel isn't practical. How about a joystick mounted in the wx plane. That seems to be very easy and direct. Get the unnecessary y dimension out of the control system and all these problems disappear. The joystick can be somewhat long and you may use both hands so there is a fair amount of leverage and power available in a purely mechanical system. That's what was done in pre-electronic aircraft.

What set me on the wrong path is that 3D steering wheels seem to work just fine on simple gearless systems like quadcycles (4D version of the tricycle) or bicycles. I thought it would scale up. Nope.