4-D wheels

[quickfur]

Thanks quickfur, I like the picture.
Just a few questions to help me. Where are we looking top down from?

From the 4th direction, of course. Just like looking down on a 3D car from the top, you see the roof, doors on the sides, 4 wheels on the corners (well, kindof... you can tell where the wheels would go but probably you won't see the wheels themselves, but you could imagine it's one of those Formula 1 race cars with large, exposed wheels), the sides of the road, etc..

So the 3 dimensions in the picture represent the 3 dimensions of the ground in 4D. Perhaps you can pretend that the checkerboard cylinder is actually the roof of the vehicle, and the 8 grey cylinders are the tops of the wheels (since the actual wheels in 4D aren't just 3D cylinders, but cubinders), and the blue square is the top of the side door (since the door itself would be a tesseract, albeit a rather flat one).

[Keiji]

Hi quickfur, I was also wondering if it could be possible to have the twin rails closer to each other rather than on opposite sides of the vehicle.

That was actually what I originally intended. I only said the rails would be on either side of the vehicle to simplify the argument for 8 wheels being sufficient - in "reality", the rails would be under the vehicle, intersecting its shadow, just like our 3D trains are, but that would have made the argument more complicated.

Rails tend to imply - I would imagine - a fixed rail wheel separation distance. Which, on our roads wouldn't suit us as we have cars with different size wheel axles and trucks/buses with bigger (and more) axles to support larger vehicles.

Yes, there is a fixed separation distance in the confined lateral - between the set of wheels on one rail and the set on the other. However, the wheels can still be moved further apart in the navigable lateral, since they'd still be on the rails regardless and it would just be like moving wheels further apart on a 3D road vehicle. The vehicle could also be wider in the confined lateral, just that there would be more overhang past the rails. Finally, there's no reason it can't be longer, or taller, just like 3D road and rail vehicles.

We might also want some mechanism for going off road?

I mentioned that earlier, in the form of bi-modal vehicles.

[Keiji]

Re Quickfur's image: Perhaps the 4D version of North American cities would be like that, but it's very different to how I had imagined it!

Firstly, Quickfur has all the lanes physically separated. On roads, lanes are not separated usually, so that you can change lanes whenever you like. There would be no reason for this not to be the case in 4D.

So, there would just be one wide carriageway going in one direction, and another wide carriageway in the opposite direction, carrying several lanes. Sliproads (or slips for short - a "ramp" to you Americans - our word has the great advantage of not implying a height difference!) then connect carriageways to other carriageways at junctions.

[quickfur]

Re Quickfur's image: Perhaps the 4D version of North American cities would be like that, but it's very different to how I had imagined it!

Firstly, Quickfur has all the lanes physically separated. On roads, lanes are not separated usually, so that you can change lanes whenever you like. There would be no reason for this not to be the case in 4D.

True, I was just exploring one of the many possibilities. Plus, it was easier to model physically-separated lanes in povray than combined lanes that sometimes would split apart to allow room for a ramp/slip/whatever you call those things.

So, there would just be one wide carriageway going in one direction, and another wide carriageway in the opposite direction, carrying several lanes. Sliproads (or slips for short - a "ramp" to you Americans - our word has the great advantage of not implying a height difference!) then connect carriageways to other carriageways at junctions.

I'm not American, btw. I live in Canuckia. And I'm not even sure "ramp" is the right word to use here, I think I was a bit misled by the 3D layout that suggests a (false) analogy with actual highway on/off ramps, probably they are more accurately called "turning lanes", e.g., right-turn lane, left-turn lane.

One interesting aspect of this that we haven't considered is the large-scale topology of these planar rail "carriageways" as you Brits call them. For simplicity, let's say we put these 2D-wide carriageways between buildings (or city blocks, as the case may be). Then eventually we'll need some way for a vehicle to transfer to another carriageway that's 90° to the current one. This is easily achieved by having a suitable turning lane / slipway, of course. But an interesting effect occurs if a vehicle makes multiple such turns via a series of slipways, and eventually gets back on the original carriageway: there's a possibility that it will end up in a flipped orientation, such that left/right are now opposite of what they were originally!

[Keiji]

That shouldn't be possible, so long as a convention for driving direction is established.

Take the shadow of a dual carriageway (divided highway) like I described - the rails are two parallel planes split down the middle, into four.

The cross-section of this as one moves in the direction of traffic is then four lines like so

==== ====

We can assign a coloring to this, say, the top rails are red, and the bottom rails are blue. Let us also say that vehicles drive "into" the screen on the left carriageway as it appears here, and "out of" the screen on the right carriageway. If we look at it from the other direction, the top rails are blue, so we know the vehicles driving "into" the screen must be on the right instead. Similarly, if we rotate 180 degrees in the plane of the screen, the same thing happens. So we can establish that if the red rails are on top, vehicles driving "into the screen" are on the left.

What we now do is we make sure that all roads have a red rail and a blue rail in a similar way, and that red rails only ever join to red rails and blue rails only ever join to blue rails. This establishes a chiral property for left- or right- handed driving, and ensures that vehicles cannot get flipped around.

[quickfur]

That shouldn't be possible, so long as a convention for driving direction is established.

Take the shadow of a dual carriageway (divided highway) like I described - the rails are two parallel planes split down the middle, into four.

The cross-section of this as one moves in the direction of traffic is then four lines like so

==== ====

We can assign a coloring to this, say, the top rails are red, and the bottom rails are blue. Let us also say that vehicles drive "into" the screen on the left carriageway as it appears here, and "out of" the screen on the right carriageway. If we look at it from the other direction, the top rails are blue, so we know the vehicles driving "into" the screen must be on the right instead. Similarly, if we rotate 180 degrees in the plane of the screen, the same thing happens. So we can establish that if the red rails are on top, vehicles driving "into the screen" are on the left.

What we now do is we make sure that all roads have a red rail and a blue rail in a similar way, and that red rails only ever join to red rails and blue rails only ever join to blue rails. This establishes a chiral property for left- or right- handed driving, and ensures that vehicles cannot get flipped around.

The problem with this is that there is no fixed "top" and "bottom" rail, because in 4D, they lie on the lateral dimensions, which can rotate arbitrarily. For simplicity, let's take a rectangular section of a red/blue rail pair. So these are two rectangles lying on parallel 2D planes, which, in turn, lie on some 3D section of the ground. To make this more precise, let's say W is the vertical dimension, which means the ground spans the X, Y and Z directions. Say the rail pair stretches from -X to +X, and they are separated in the Y direction: say, the red rail lies in the +Y direction, and the blue rail lies in the -Y direction. So then the width of the rails lie in the Z direction. Now let's say we're standing at one end of the rail pair, say at the -X direction, and we're looking down the rails in the +X direction. Since up is +W, that still leaves a single degree of freedom in orientation (rotation in the YZ plane).

Consider the case where we're oriented such that our +Y and +Z laterals line up with the +Y and +Z dimensions of the ground. So then we see the red rail in the +Y direction, and the blue rail in the -Y direction. Let's say we decide on the rule that when red lies in the +Y direction and blue lies in the -Y direction, then vehicles will travel in the +X direction.

Now remember, since we have one degree of freedom in our lateral orientation, we can rotate about the YZ plane. Suppose we rotate 180° while standing on the same spot and facing the same direction. Then, from our POV, +Y rotates to -Y, and -Y rotates to +Y, but +X remains unchanged (since it lies in the stationary space of the YZ rotation). So now we see the red rail in the -Y direction (relative to ourselves) and the blue rail in the +Y direction, and we conclude, from our rule, that vehicles should travel in the -X direction.

Contradiction.

Now consider the following 3D diagram of some roads around a 3D city block:



I didn't have time to add everything I wanted to add in this diagram (modelling curved shapes in povray is very tedious), but let's imagine that there are ramps that allow cars to transfer between the lanes.

Let's say we start at the bottom of the left face on the red lane, moving up (in the 3D projection, that is, not up in the 4D sense). Let's say we turn onto the red lane on the right face. The easiest way to build the ramp would be to first allow a 90° right turn on the left face, then a 90° turn from the left face to the right face, which then lets you merge into the red lane on the right face.

Suppose now we want to merge onto the red lane on the top face. The simplest way to do this is to have a branch on the red lane on the right face, that turns 90° to the left, then 90° from the right face to the top face. This then aligns our orientation so that we can merge onto the red lane on the top face.

Lastly, suppose we merge back to the red lane on the left face (a little distance further up that where we started). Again, the simplest way to do this is if there was a branch from the red lane on the top face, that lets you turn 90° to the left, then turn 90° from the top face to an extension of the left face -- since otherwise you'll end up going against the direction of traffic in the red lane. Then you can merge back to the red lane on the left face and continue going up. But once you do this, though, you'll be driving on the red lane in a flipped orientation from when you first started.

//

Note, however, that your original idea of having red rails (not to be confused with the red lanes in my picture, now we're talking about the individual rails that pair up to form a single lane) join only to red rails, and blue rails join only to blue rails, is actually workable -- but just not in the way you described it. The idea here is that we can set things up so that the red rail, if we consider its entire extent throughout the entire road network, forms a 2D manifold that's disjoint from the blue rails. You can imagine this as painting one rail of a pair red, and the other rail blue, and only ever connecting segments of rails that match in color. This then guarantees that no matter how twisted the road network may be, the red and blue manifolds are disjoint, and since the vehicle's wheels cannot jump from a red rail to a blue rail, it will always have a fixed orientation when travelling past any given segment of the road network.

The path that I described earlier is actually a kind of "Mobius twist" where the entire extent of the red rails actually includes the blue rails, so like the Mobius strip that only has a single surface, this makes the road network non-orientable, and thus allows you to travel through the same segment of road in two different orientations. If we distinguish between the rails in a rail pair, and only connect red with red and blue with blue, then this twist can never happen. The catch is that this distinction cannot be determined geometrically, since which pair lies "on top" vs "on the bottom" (or "on the left" vs "on the right") is ambiguous in 4D. So you have to physically mark the rail segments in some way, maybe by actually painting one rail red and the other blue, or by having incompatible interconnects so that red and blue rails won't fit together when you try to join them, in order to be able to tell, when you're adding new roads, which rail should go where. (Or making the two types of rails different from each other, like having different sizes, with corresponding differently-sized wheels on all vehicles (say red rails are higher so red wheels are smaller, and blue wheels are bigger), so that it's impossible to join red and blue rails and impossible to put a vehicle on the road in the wrong orientation (the mismatch would be obvious when the vehicle can't stand upright when you put large wheels on the large rail and small wheels on the small rail).)

[wendy]

The design of the four-dimensional wheel and steering can be thought of like this.

A wheel is in the general shape of a circle. It is meant to rotate entirely in the space of up/forward, with no across rotation. A rotation outside the up/forward axis would be felt as a rotation of the cabin.

Looking from above the vehicle, we see then a number of wheels which appear as thickish rods, mounted on a frame. These rods point in the direction of travel. The change in direction is effected by making a number of these rods point to the new direction. This means that the rods can freely move over the half-sphere in the same way that 3d wheels can be turned over a half-circle.

Placing the steering-ring (yes it's an N-1 sphere), over the wheel, so that the top of the ring is at the front of the wheel, the modal action is that when we pull the ring in a direction of 'across-ness', then the wheel will turn towards that direction. As in 3d, the intensity of the turn depends on how hard and long one holds the steering wheel in a given direction, and the actual turn is affected by the nature of the ground underneath.

[gonegahgah]

I have to admit that this diagram seems to depict to me two conventional 3D roads simply rotated 180° sideways to each other?
Is that interpretation incorrect?

[quickfur]

I have to admit that this diagram seems to depict to me two conventional 3D roads simply rotated 180° sideways to each other?
Is that interpretation incorrect?

Strictly speaking that's not accurate. But that's partly my fault; the way I depicted the wheels doesn't quite convey how they might both rest on the rails and have a protruding rim that is confined by the inside of the rails (just like train wheels in our 3D world, in fact). Furthermore, the rails protrude outwards from the ground in the 4th direction, so that the inside surface of the rails are not what is supporting the vehicle; they are the analogue of the inside part of 3D railway tracks that serves to confine the wheels of the vehicle from rolling off the tracks. The supporting surface is actually the volume inside those two plates (which is analogous to the top of the railway tracks in 3D).

One way to think of it is to imagine slicing the above model in half with a plane that bisects the cylinder and both grey plates. The cross-section would look like two narrow rectangles (rails) with a large rectangle in between (the vehicle) -- representing the overhead view of a 3D train car resting on the tracks. So it's the volume inside the grey plates that corresponds with the top of the 3D railway tracks; the wide area of the grey plates are something that has no analogue in 3D -- they are what permit a 2D freedom in the lateral/forward dimensions in 4D. In 3D, there is no such extra lateral dimension.

[Keiji]

Note, however, that your original idea of having red rails (not to be confused with the red lanes in my picture, now we're talking about the individual rails that pair up to form a single lane) join only to red rails, and blue rails join only to blue rails, is actually workable -- but just not in the way you described it. The idea here is that we can set things up so that the red rail, if we consider its entire extent throughout the entire road network, forms a 2D manifold that's disjoint from the blue rails. You can imagine this as painting one rail of a pair red, and the other rail blue, and only ever connecting segments of rails that match in color. This then guarantees that no matter how twisted the road network may be, the red and blue manifolds are disjoint, and since the vehicle's wheels cannot jump from a red rail to a blue rail, it will always have a fixed orientation when travelling past any given segment of the road network.

This is basically exactly what I was trying to describe, and I don't see what you're trying to achieve with the rest of your post. You've essentially agreed with me that coloring the rails in that way fixes the orientation, avoiding the problem you brought up.

[quickfur]

Note, however, that your original idea of having red rails (not to be confused with the red lanes in my picture, now we're talking about the individual rails that pair up to form a single lane) join only to red rails, and blue rails join only to blue rails, is actually workable -- but just not in the way you described it. The idea here is that we can set things up so that the red rail, if we consider its entire extent throughout the entire road network, forms a 2D manifold that's disjoint from the blue rails. You can imagine this as painting one rail of a pair red, and the other rail blue, and only ever connecting segments of rails that match in color. This then guarantees that no matter how twisted the road network may be, the red and blue manifolds are disjoint, and since the vehicle's wheels cannot jump from a red rail to a blue rail, it will always have a fixed orientation when travelling past any given segment of the road network.

This is basically exactly what I was trying to describe, and I don't see what you're trying to achieve with the rest of your post. You've essentially agreed with me that coloring the rails in that way fixes the orientation, avoiding the problem you brought up.

I think I got thrown off by your describing how to determine travel direction based on the orientation of the cross-sections of the rails (red on top, blue on bottom, vs. red on bottom, blue on top). That doesn't work in 4D because you can spin your orientation around laterally to flip that configuration. But the separation of red rails to blue rails is a topologically-sound fact.

[Keiji]

Even so, I don't see how it doesn't work. The vertical dimension is fixed, leaving 3 dimensions through which you can rotate. Now, one of those dimensions is again fixed by the fact that you set red to always have a greater coordinate than blue in that dimension. That leaves two dimensions through which to rotate, akin to rotating a ⥮ symbol in the plane. No matter how you rotate it, you cannot transform it into a ⥯. The ⥮ represents vehicles driving on the left, and ⥯ represents vehicles driving on the right. As long as you color your rails, you have an established driving side.

[quickfur]

Even so, I don't see how it doesn't work. The vertical dimension is fixed, leaving 3 dimensions through which you can rotate. Now, one of those dimensions is again fixed by the fact that you set red to always have a greater coordinate than blue in that dimension.

Except that rotating around your lateral dimensions (keeping the vertical and forward vectors fixed) flips the sense of that coordinate. The forward vector cannot be parallel to that coordinate, because you won't be able to drive forward otherwise. So it has to be one of your lateral vectors, which can be flipped by a simple orientation change.

[gonegahgah]

Strictly speaking that's not accurate. But that's partly my fault; the way I depicted the wheels doesn't quite convey how they might both rest on the rails and have a protruding rim that is confined by the inside of the rails (just like train wheels in our 3D world, in fact). Furthermore, the rails protrude outwards from the ground in the 4th direction, so that the inside surface of the rails are not what is supporting the vehicle; they are the analogue of the inside part of 3D railway tracks that serves to confine the wheels of the vehicle from rolling off the tracks. The supporting surface is actually the volume inside those two plates (which is analogous to the top of the railway tracks in 3D).

One way to think of it is to imagine slicing the above model in half with a plane that bisects the cylinder and both grey plates. The cross-section would look like two narrow rectangles (rails) with a large rectangle in between (the vehicle) -- representing the overhead view of a 3D train car resting on the tracks. So it's the volume inside the grey plates that corresponds with the top of the 3D railway tracks; the wide area of the grey plates are something that has no analogue in 3D -- they are what permit a 2D freedom in the lateral/forward dimensions in 4D. In 3D, there is no such extra lateral dimension.

Hi quickfur, yes the wheel depiction was confusing me... I've just woken up (bleary eyed) so I'll come back later today and study your helpful words. Cheers.

[Keiji]

Even so, I don't see how it doesn't work. The vertical dimension is fixed, leaving 3 dimensions through which you can rotate. Now, one of those dimensions is again fixed by the fact that you set red to always have a greater coordinate than blue in that dimension.

Except that rotating around your lateral dimensions (keeping the vertical and forward vectors fixed) flips the sense of that coordinate. The forward vector cannot be parallel to that coordinate, because you won't be able to drive forward otherwise. So it has to be one of your lateral vectors, which can be flipped by a simple orientation change.

That's what I'm saying - you're keeping the wrong vectors fixed!

For the record we have:

* vertical
* confined lateral (in which red > blue)
* navigable lateral
* frontal

I am fixing the vertical and confined lateral, and rotation in the plane formed by navigable lateral and frontal cannot change the driving side.

You are fixing the vertical and frontal for some reason and rotating the others, which of course is going to change the driving side.

[quickfur]

Even so, I don't see how it doesn't work. The vertical dimension is fixed, leaving 3 dimensions through which you can rotate. Now, one of those dimensions is again fixed by the fact that you set red to always have a greater coordinate than blue in that dimension.

Except that rotating around your lateral dimensions (keeping the vertical and forward vectors fixed) flips the sense of that coordinate. The forward vector cannot be parallel to that coordinate, because you won't be able to drive forward otherwise. So it has to be one of your lateral vectors, which can be flipped by a simple orientation change.

That's what I'm saying - you're keeping the wrong vectors fixed!

For the record we have:

* vertical
* confined lateral (in which red > blue)
* navigable lateral
* frontal

I am fixing the vertical and confined lateral, and rotation in the plane formed by navigable lateral and frontal cannot change the driving side.

You are fixing the vertical and frontal for some reason and rotating the others, which of course is going to change the driving side.

OK, I think I see where we aren't getting each other.

I was approaching it from the POV of a road architect, who's standing on a patch of empty land where new rails are to be laid, and trying to decide which rail goes where. So he would be facing some arbitrary direction, let's say looking in the planned direction of travel on the new road, which we can call the forward vector. Since he's standing upright, the vertical vector is also fixed. But this isn't enough to fix his orientation, as there's still an additional degree of freedom in the lateral space. Let's call the forward vector +X, and the lateral directions Y and Z (which are relative to the architect's current orientation). He could arbitrarily decide that the red rail should go in -Y and the blue rail should go in +Y. But his chief engineer, standing next to him and looking in the same forward direction, could have another orientation, wherein the builder's -Y = the engineer's +Y, and vice versa. Or worse yet, the engineer's Y axis is the architect's Z axis, so they both have a completely incompatible idea of where the rails should be laid. They are both looking in the same direction (i.e., the prospective direction of travel), yet they can't agree on the orientation of the rails -- this indicates that you can't uniquely determine the lateral orientation just by the direction of travel alone. You have to make a judgment call, or use some kind of arbitrary rule based on some external fixed frame of reference, in order to resolve this ambiguity. (Note that 3D right-handed/left-handed rules don't help here, because the position of the rails in the pair relative to each other are colinear, so together with the forward vector they only span 2 dimensions, which is not enough to uniquely determine a unique lateral orientation.)

Now let's say the architect's decision takes precedence, so the engineer has to reorient himself to line up with the architect's orientation so that he would lay the rails according to the architect's idea of where it should be. Some time in the future, after this road has been laid, a decision is made to extend it to reach a nearby town. Now, the nearby town already has some rails laid, part way out in this direction, so now the task is to extend both rails so that they join up. Since whoever laid the rails on the nearby town didn't consult the architect in which orientation the red/blue rails should be, they simply made their own arbitrary decision. Now suppose that it just so happens that their choice of +Y happens to be the opposite of the current road's +Y, and similarly for -Y. When the engineers now extend the current road towards the nearby town's road, they notice that the rails of both roads more-or-less lines up, so they could simply connect them. But unless each rail of the rail pair are specifically marked as red/blue (i.e., +Y or -Y), they wouldn't know which one was which, and if they simply connected them with the minimum twist, then the current road's +Y would be connected to the nearby town's -Y, and vice versa, thereby introducing an orientation flip into the road system.

The only reliable way to prevent this is to physically mark the rails in some way -- e.g., actually painting them red/blue, or using different joints for each rail so that rails of unlike color won't fit together. Then, in the above scenario, the engineers will realize that a twist is necessary in order for the rail pairs to match up correctly, so they would be forced to ensure that the +Y segment connects with the +Y segment of the nearby town, and ditto for -Y, thus avoiding the orientation flip.

[Keiji]

Yes, you've got it now

Incidentally, if one has to introduce an 180 degree twist into the highway network to correct an archaic flip from before tetronians started painting their rails ( ) what forces would this exert on the driver and passengers?

For us, if we were driving along and suddenly had our car turn 180 degrees upside down (around the frontal), we'd feel it due to gravity, if it turned end to end (around the vertical), we'd feel it due to moving in that direction, and if we were unlucky enough for it to turn head over heels (around the lateral), we'd feel both. But a 180 degree twist for a 4D vehicle, which occurs only in the two laterals, therefore "around" the vertical and frontal, would not have either of those reasons to be felt. Is there some other reason that would limit the speed of such a twist, or could it be done very quickly without any negative side effects?

[quickfur]

Yes, you've got it now

Incidentally, if one has to introduce an 180 degree twist into the highway network to correct an archaic flip from before tetronians started painting their rails ( ) what forces would this exert on the driver and passengers?

Well, this requires the big assumption that Newtonian physics (or some approximation thereof) works the same way in 4D.

This kind of flip has no equivalent in 3D, but perhaps the closest analogue would be if you're riding an accelerating rocket, lying on your back and looking up the axis of the rocket, and the rocket was spinning on its long axis. You'd feel the angular momentum around the axis of rotation, or, if the rocket were to suddenly rotate 180°, you'd feel the torque.

For us, if we were driving along and suddenly had our car turn 180 degrees upside down (around the frontal), we'd feel it due to gravity, if it turned end to end (around the vertical), we'd feel it due to moving in that direction, and if we were unlucky enough for it to turn head over heels (around the lateral), we'd feel both. But a 180 degree twist for a 4D vehicle, which occurs only in the two laterals, therefore "around" the vertical and frontal, would not have either of those reasons to be felt. Is there some other reason that would limit the speed of such a twist, or could it be done very quickly without any negative side effects?

Gravity isn't the reason we feel it when a vehicle turns over or turns around. What is felt is the torque that causes the orientation-flipping rotation (and also the rotation itself). The kind of lateral torque you'd experience in 4D has no real equivalent in 3D, but perhaps the closest is if you're lying on your back in a rocket (and pretend that that's your "upright" position). You'll certainly feel it when the rocket suddenly rotates. It won't change your facing direction or the direction of gravity, but it's a lateral orientation change that I'm pretty sure can be felt. Not to mention the visual rotation of your surroundings, which will probably induce vertigo if done too quickly.

Such lateral orientation changes (in 4D) would be perfectly safe, of course, since you aren't at risk of falling out of the vehicle due to gravity or flying off the road like will happen if you turn a corner too quickly, but I don't expect it will feel very comfortable either, if it's too sudden. But a gentle reorienting twist is probably OK, and may even feel interesting.

[Keiji]

Incidentally, if one has to introduce an 180 degree twist into the highway network to correct an archaic flip from before tetronians started painting their rails ( ) what forces would this exert on the driver and passengers?

Well, this requires the big assumption that Newtonian physics (or some approximation thereof) works the same way in 4D.

I think if we're building planar rail transport on a tetraspace planet, we're already making that assumption.

[quickfur]

Incidentally, if one has to introduce an 180 degree twist into the highway network to correct an archaic flip from before tetronians started painting their rails ( ) what forces would this exert on the driver and passengers?

Well, this requires the big assumption that Newtonian physics (or some approximation thereof) works the same way in 4D.

I think if we're building planar rail transport on a tetraspace planet, we're already making that assumption.

Sure, though strictly speaking the workings of a planar rail is mostly geometrical -- the vehicle is mechanically restrained in its range of motion by the rails due to their respective geometries (and the assumption that objects don't overlap). Here, however, we're talking about dynamics: the forces felt and so on, which requires a few more assumptions that merely things fitting together geometrically. E.g., planar rails will continue to operate more-or-less the same way under Aristotelian physics, but the forces felt would be completely different than in Newtonian physics.

[wendy]

I think things have slipped somewhat here. Building a road of two rails like recently illistrated is pretty much the same cost or more as painting the edges of a laneway (rather than the C/L).

A wheel is meant to rotate in the 2-space formed by 'height + forward'. When you divide space by height, you get a line segment. If there were rotation coming from the wheel around this line-segment, its net effect is to rotate the cabin of the vehicle, which is not necessarily a good thing.

The wheels that lead direction, are supposed to move with some freedom like a joy-stick. In two dimensions, this is more like a wiper. But in 3d, you can think of a wheel in steering as a joystick. The 360 degrees of angle is perpendicular to the space the wheel is in.

The steering ring (it is the surface of an N-1 sphere), can be imagined as a circle around the wiper, or a sphere around the joy stick. The normal position is set by springs in both cases, has the top of the wheel pointing up when the wiper or joy-stick projection of the wheel is straight forward. Turning the steering ring in a given direction acts against the springs, the direction is to the L/R or to the 360 angle the turn is desired to, and the hardness of the pull on the steering ring gives the hardness of the turn. Some of you folk have been in motor-cars. Just imagine the driver sitting in front of a steering sphere, formed by rotating the steering wheel from selecting LR to a full circle. The wheels stay pretty much the same, but the steering movement is more like a joy-stick in place of a wiper.

[Klitzing]

The steering ring (it is the surface of an N-1 sphere), can be imagined as a circle around the wiper, or a sphere around the joy stick. The normal position is set by springs in both cases, has the top of the wheel pointing up when the wiper or joy-stick projection of the wheel is straight forward. Turning the steering ring in a given direction acts against the springs, the direction is to the L/R or to the 360 angle the turn is desired to, and the hardness of the pull on the steering ring gives the hardness of the turn. Some of you folk have been in motor-cars. Just imagine the driver sitting in front of a steering sphere, formed by rotating the steering wheel from selecting LR to a full circle. The wheels stay pretty much the same, but the steering movement is more like a joy-stick in place of a wiper.

The thing is that most tend to be led from their expereance of the steering ring in cars. But there is some conversion ratio implemented which allows for multiple rotations for still some finite angle of wheel displacement. You'd better consider a bycicle steering rod. That one needs only a 180 degrees interval, where +/-90 degrees rotation would mean an apprupt sharp effect to the side, which is impossible to be driven directly because of inertia.

Similarily in 4D you would just need for the upper half of a sphere where the joy stick could be pushed/pulled.

--- rk

[quickfur]

I think things have slipped somewhat here. Building a road of two rails like recently illistrated is pretty much the same cost or more as painting the edges of a laneway (rather than the C/L).

A wheel is meant to rotate in the 2-space formed by 'height + forward'. When you divide space by height, you get a line segment. If there were rotation coming from the wheel around this line-segment, its net effect is to rotate the cabin of the vehicle, which is not necessarily a good thing.

The wheels that lead direction, are supposed to move with some freedom like a joy-stick. In two dimensions, this is more like a wiper. But in 3d, you can think of a wheel in steering as a joystick. The 360 degrees of angle is perpendicular to the space the wheel is in.

The steering ring (it is the surface of an N-1 sphere), can be imagined as a circle around the wiper, or a sphere around the joy stick. The normal position is set by springs in both cases, has the top of the wheel pointing up when the wiper or joy-stick projection of the wheel is straight forward. Turning the steering ring in a given direction acts against the springs, the direction is to the L/R or to the 360 angle the turn is desired to, and the hardness of the pull on the steering ring gives the hardness of the turn. Some of you folk have been in motor-cars. Just imagine the driver sitting in front of a steering sphere, formed by rotating the steering wheel from selecting LR to a full circle. The wheels stay pretty much the same, but the steering movement is more like a joy-stick in place of a wiper.

Yes, this is the so-called "spaceship steering" model (probably not a very good name, as it is 3D-centric nomenclature; a better name perhaps is "free-form steering" as opposed to the constrained steering proposed here with the planar rails).

The issue here isn't that we merely lift the 3D way of driving into 4D without regard to the extra lateral dimension; that possibility has already been entertained. The main thrust here is that the 4D equivalent of the 3D railway, in which the vehicle is confined in one of its lateral dimensions, still has a leftover lateral dimension in which free steering may be permitted; and based on this we can construct a hybrid road system in which vehicles are both confined and have free steering -- each within its own lateral degree of freedom. Thus, we can combine the advantages of both the railway system and the free-form driving in 3D into something that, hopefully, proves superior to either extreme. In densely-packed areas such as cities, such a system is arguably superior to "free-form steering" with 2D lateral dimensions, because of safety concerns and navigability.

In the wilderness, however, the full 2D lateral needs to be covered somehow, so here the direct analogue of the 3D steering wheel driving applies, i.e., via a joystick.

[gonegahgah]

This is very interesting Keiji the more I study it.
We wouldn't even need spherical^cylinder wheels for the rail wheels. We would need extra spherical^cylinder off-road wheels for the off road driving.
The rail wheels need only be prism^cylinders and you would only need a single steering rod for the rail wheels.
You would still need dual steering rods for the off-road wheels.

I imagine the one steering wheel would work for both sets of wheels so although we only steer towards one dimension on the rails the steering wheel would also have to steer towards all lateral directions.
All sets of wheels I imagine would tend to be engaged at once - a bit of a energy waste - to ensure smoother transactions between rail and rail missing roads.

I don't tend to think it would be a joystick. Even analogue joysticks would be impractical for our roads. Jet pilots yes; but cars?
I tend to think they will want something that they can wrap their three hands around; just as our steering wheels allow us to wrap two hands around them.
Just as we tend to drive with both our hands (well many of us) I tend to think that they would utilise all of their three hands at once.

Alighting onto and off the rail system would be interesting because you would have to correctly align your vehicle to the rails.
We would also need some design in the system to ensure that all drivers align with the same orientation instead of an opposite 'sideways cum upside-down' to each other.
Maybe one track with 'commencer' grooves or something.

Steering would certainly be much easier on the rails. Being metal rails and metal rail wheels we would have to accept the sound of 'clickety-clack' as part of our driving experience.
I also wonder what effect rail buckling would have on the system?

Also I wonder if we would add guard rails at the at the open-sides or alternatively raised rail 'edges' to help prevent derailment.
Turning directly towards the later at speed would allow deliberate rail detachment.

You would need at least three rubber wheels at front and at back for the off-road driving plus four or more rail wheels front and back placed in line with the tracks.
So a minimum of seven wheels front and seven wheels back between the two varieties.
That would make for interesting car designs...

[quickfur]

This is very interesting Keiji the more I study it.
We wouldn't even need spherical^cylinder wheels for the rail wheels. We would need extra spherical^cylinder off-road wheels for the off road driving.
The rail wheels need only be prism^cylinders and you would only need a single steering rod for the rail wheels.
You would still need dual steering rods for the off-road wheels.

Prism^cylinder = cylinder prism = circle X square = cubinder. Keiji has already mentioned this in the planar rails wiki page.

I imagine the one steering wheel would work for both sets of wheels so although we only steer towards one dimension on the rails the steering wheel would also have to steer towards all lateral directions.

That seems to be a waste. If most vehicles are rails-only, then why bother with the additional complexity of a wheel / joystick that can point in 2D, when the rails always confine you to only 1D?

For dual-mode cars, I can see the rationale, but the working assumption is that those are rare.

All sets of wheels I imagine would tend to be engaged at once - a bit of a energy waste - to ensure smoother transactions between rail and rail missing roads.

What's wrong with a control (lever or something) that lifts the rail-wheels off the ground when it's not being used, and vice versa? 3D amphibious vehicles already have something like this.

I don't tend to think it would be a joystick. Even analogue joysticks would be impractical for our roads. Jet pilots
yes; but cars?

But that's precisely the point, on our roads we only need to steer in a single lateral dimension, but on a 4D road, you need to steer in two lateral dimensions. And jet pilots have to steer in two dimensions, essentially -- and the joystick has proven to be an effective solution for that.

I tend to think they will want something that they can wrap their three hands around; just as our steering wheels allow us to wrap two hands around them.
Just as we tend to drive with both our hands (well many of us) I tend to think that they would utilise all of their three hands at once.

Take a look at an aircraft's cockpit. You can design a joystick to have handles for all 3 (or however many) hands to hold on to. It doesn't have to be literally just a stick!

Alighting onto and off the rail system would be interesting because you would have to correctly align your vehicle to the rails.
We would also need some design in the system to ensure that all drivers align with the same orientation instead of an opposite 'sideways cum upside-down' to each other.
Maybe one track with 'commencer' grooves or something.

Well, if the two rails are designed to have different wheel sizes fit on them, that would solve the problem, as you won't be able to get the rail wheels to "lock" onto the rails if you try to drive on it in the wrong orientation.

Steering would certainly be much easier on the rails. Being metal rails and metal rail wheels we would have to accept the sound of 'clickety-clack' as part of our driving experience.

Nothing says we can't design the rails to be noiseless.

I also wonder what effect rail buckling would have on the system?

That's an interesting thought. Never thought of that before. Maybe the rails would be installed in sections, to give enough room for expanding/contracting sections so that they won't buckle? If they do, that could be disastrous, since adjacent sections of rails may become detached from each other, and cause passing vehicles to get derailed.

Also I wonder if we would add guard rails at the at the open-sides or alternatively raised rail 'edges' to help prevent derailment.
Turning directly towards the later at speed would allow deliberate rail detachment.

Another issue to be addressed is the momentum of vehicles when traversing a curve. Probably some built-in banking would be needed to prevent derailment (just like our 3D roads, esp. highway ramps, are banked at sharp turns in order to prevent speedy cars from flying off the road).

You would need at least three rubber wheels at front and at back for the off-road driving plus four or more rail wheels front and back placed in line with the tracks.
So a minimum of seven wheels front and seven wheels back between the two varieties.
That would make for interesting car designs...

Such would be rare, though, since we're expecting the majority of cars to be railed, and only a minority of specialized vehicles would be dual-mode. It may very well be that only service vehicles need to be dual-mode, since wilderness explorers generally don't need to be driving on city roads, and vice versa, so it's hard to justify the cost of building a more complicated wheel/steering mechanism. It may be more expensive to build a dual-mode vehicle than to build two single-mode vehicles, for example.

[gonegahgah]

Prism^cylinder = cylinder prism = circle X square = cubinder. Keiji has already mentioned this in the planar rails wiki page.

True, sorry I meant it more as a realisation than something new. I knew there would be a proper name for it.

That seems to be a waste. If most vehicles are rails-only, then why bother with the additional complexity of a wheel / joystick that can point in 2D, when the rails always confine you to only 1D?

I think in the transition periods its going to be a challenge. We went from bush bashing to tracks to dirt roads to asphalt roads. They would then likely add planar rails.
We still have the occasional dirt roads and some people still like to go bush bashing. Also we like to park off road as well.

What's wrong with a control (lever or something) that lifts the rail-wheels off the ground when it's not being used, and vice versa? 3D amphibious vehicles already have something like this.

I would like to agree with you on this but I just wonder at the ease of this. Certainly something automatic would be nicer but how does it gauge and smoothly handle the transition?

But that's precisely the point, on our roads we only need to steer in a single lateral dimension, but on a 4D road, you need to steer in two lateral dimensions. And jet pilots have to steer in two dimensions, essentially -- and the joystick has proven to be an effective solution for that.

Flying a plane is different to driving a car not just because you can pitch and roll but because you are driving through unfettered space.
I would shudder at the thought of sharing a road with someone driving with a joystick. It just doesn't have the sensitivity of a steering wheel. I'm more with Wendy on this one.

Take a look at an aircraft's cockpit. You can design a joystick to have handles for all 3 (or however many) hands to hold on to. It doesn't have to be literally just a stick!

I guess that's true.

Well, if the two rails are designed to have different wheel sizes fit on them, that would solve the problem, as you won't be able to get the rail wheels to "lock" onto the rails if you try to drive on it in the wrong orientation.

I did think of this but then I thought they would have to turn at different speeds. That might make the engineering more complex?

Nothing says we can't design the rails to be noiseless.

Our modern trains are getting better at hiding this.

That's an interesting thought. Never thought of that before. Maybe the rails would be installed in sections, to give enough room for expanding/contracting sections so that they won't buckle? If they do, that could be disastrous, since adjacent sections of rails may become detached from each other, and cause passing vehicles to get derailed.

Scary.

Another issue to be addressed is the momentum of vehicles when traversing a curve. Probably some built-in banking would be needed to prevent derailment (just like our 3D roads, esp. highway ramps, are banked at sharp turns in order to prevent speedy cars from flying off the road).

Fortunately we still have the height dimension. The interesting thing I guess in 4D is that you can have two different torques occurring simultaneously (and probably rotational torque I guess).
If you have a road/rail bank would you tend to slide in the non-torque direction which also shares the same bank? Would they need to engineer for both at the same time?

Such would be rare, though, since we're expecting the majority of cars to be railed, and only a minority of specialized vehicles would be dual-mode. It may very well be that only service vehicles need to be dual-mode, since wilderness explorers generally don't need to be driving on city roads, and vice versa, so it's hard to justify the cost of building a more complicated wheel/steering mechanism. It may be more expensive to build a dual-mode vehicle than to build two single-mode vehicles, for example.

I wonder at the cost too and how much it would add to fuel usage?

[quickfur]

Prism^cylinder = cylinder prism = circle X square = cubinder. Keiji has already mentioned this in the planar rails wiki page.

True, sorry I meant it more as a realisation than something new. I knew there would be a proper name for it.

No problem, one thing about higher dimensional geometry is that quite often you rediscover things that are already known, but in your own particular way, which often causes you to name it in a different way from the "accepted" name. For example, I (re)discovered something I dubbed the "double torus", that ultimately turned out to be nothing other than the elusive (to me, at the time!) duocylinder. Since I made the (re)discovery while exploring a completely different direction from the way I first learned of it, I didn't know until later that they were actually the same object.

That seems to be a waste. If most vehicles are rails-only, then why bother with the additional complexity of a wheel / joystick that can point in 2D, when the rails always confine you to only 1D?

I think in the transition periods its going to be a challenge. We went from bush bashing to tracks to dirt roads to asphalt roads. They would then likely add planar rails.
We still have the occasional dirt roads and some people still like to go bush bashing. Also we like to park off road as well.

True, so dual-mode vehicles might be very common during the transition period when planar rails are first introduced to when they become ubiquitous. And some hobbyists might insist on keeping dual-mode vehicles around, even if everyone else have moved on to rail-only vehicles.

So it seems that the first planar rail vehicles might have needed hybrid controls, or at least controls that resemble free-range vehicles (har har) in order to garner widespread adoption. Once the majority of the population become comfortable with railed vehicles, the controls would get simplified from their hybrid form, so they would still resemble the original "spaceship-steering" controls but perhaps with one dimension locked / absent, rather than being something engineered to be a rail-only control from the start. "Historical accidents" like this happen all the time in our 3D world.

What's wrong with a control (lever or something) that lifts the rail-wheels off the ground when it's not being used, and vice versa? 3D amphibious vehicles already have something like this.

I would like to agree with you on this but I just wonder at the ease of this. Certainly something automatic would be nicer but how does it gauge and smoothly handle the transition?

Well, to answer that, we'd have to actually design a planar rail implementation first. As in, the exact physical specifications of it -- the materials used, details of how the wheels should lock onto the rails, the relative heights of various parts of the system, etc..

But that's precisely the point, on our roads we only need to steer in a single lateral dimension, but on a 4D road, you need to steer in two lateral dimensions. And jet pilots have to steer in two dimensions, essentially -- and the joystick has proven to be an effective solution for that.

Flying a plane is different to driving a car not just because you can pitch and roll but because you are driving through unfettered space.
I would shudder at the thought of sharing a road with someone driving with a joystick. It just doesn't have the sensitivity of a steering wheel. I'm more with Wendy on this one.

Can't the sensitivity be adjusted appropriately? There are fighter pilots who can fly in formation with their wing tips only inches away from the next plane (or so I've heard), so this isn't entirely impossible. Whether casually-trained civilians can handle this, though, is another question, so you may have a point there.

[...]

Well, if the two rails are designed to have different wheel sizes fit on them, that would solve the problem, as you won't be able to get the rail wheels to "lock" onto the rails if you try to drive on it in the wrong orientation.

I did think of this but then I thought they would have to turn at different speeds. That might make the engineering more complex?

You could make the wheels displaced vertically, but with the same rotational radius (sorta like lopsided wheels). That could work.

Or make the wheels with grooves, and one set of wheels has grooves of one size and the other has a different size, so you won't be able to fit them both into the rails if you're in the wrong orientation.

[...]

Another issue to be addressed is the momentum of vehicles when traversing a curve. Probably some built-in banking would be needed to prevent derailment (just like our 3D roads, esp. highway ramps, are banked at sharp turns in order to prevent speedy cars from flying off the road).

Fortunately we still have the height dimension. The interesting thing I guess in 4D is that you can have two different torques occurring simultaneously (and probably rotational torque I guess).
If you have a road/rail bank would you tend to slide in the non-torque direction which also shares the same bank? Would they need to engineer for both at the same time?

Whoa, you're right. There's a certain possibility that banking will not completely solve the derailment problem, because in addition to the centrifugal force in the forward/sideways plane, there's also a twisting force in the two lateral dimensions, and since the two overlap, one would be able to transfer to the other, so the vehicle might experience a twisting force as a compensation to the curvature of the rails. If this twisting force grows too large, the vehicle's wheels will fly off the rails, even though the vehicle itself is still travelling in the direction of the rails! (Of course, thereafter it will fly off the rails if the rails curve away from the forward direction.)

So the banked parts of the rails would have to be designed to minimize this twisting motion. This also makes it trickier to design joining lanes that must align the vehicle with the orientation of the target rails: some amount of twisting may be necessary so ample room will be needed to allow the vehicle to be "eased" into the new orientation gently. If you're driving too fast, this "gentle" twist may become not-so-gentle and your wheels will get dislodged from the rails. Yikes.

Such would be rare, though, since we're expecting the majority of cars to be railed, and only a minority of specialized vehicles would be dual-mode. It may very well be that only service vehicles need to be dual-mode, since wilderness explorers generally don't need to be driving on city roads, and vice versa, so it's hard to justify the cost of building a more complicated wheel/steering mechanism. It may be more expensive to build a dual-mode vehicle than to build two single-mode vehicles, for example.

I wonder at the cost too and how much it would add to fuel usage?

Extra machinery means extra weight, and for support structures like the wheel system, they usually would be made of very hard structural materials that can bear the weight of the vehicle, and that generally also means they are dense, i.e., very heavy. Carrying all this dead weight around (if you hardly ever use it) would waste a lot of fuel, indeed.

[wendy]

In general one would suppose that the tyres of a weekel (vehicle) would be in the shape of a bicurcular prism, the contact with the ground would be a circle. Even largish fridges run on small wheels, so i can't imagine much different in four dimensions.

I would not suggest using a joystick for steering. A steering ring is in 4d a sphere, but this would be flat for the driver, like a steering wheel is for us. Steering is effected in the same way: pulling the top of the ring towards the direction where the steering is to go. It's just in 4d, in place of a line, you have a circle. You are not using to wheel to ease backwards to climb upwards: there's a one-to-one mapping of the points on the equator of the steering-ring to the directions on the ground, perpendicular to the direction of travel, and in the direction of it. Moreover, it is mapped directly (ie c50 corresponds to c50 on the ground). The purpose of the lattitude control in 4d is the same as the 'hardness of turn' in 3d, turning the steering-ring hard will make the vehicle turn harder in the desired direction.

For trains, one has the same issues as ordinary trains in 3d, except that the design of the rail might need some work. If you suppose, just ordinary bullhead, for example, then the design of the wheel (which for the most part would be a bi-circular prism), would need to have a flange that stops the wheels wandering off the top of the rail. One would also need to consider what the frogs in the points might look like, because i can hardly imagine polish points (ie lifting the wagon from track to track) working.

[Keiji]

Yes, a 4D vehicle would have a spherical steering wheel, which would be fully utilised when off-rail, but one dimension would be locked when on-rail.

I would imagine the rail-wheels would have rims on the inside just like our 3D train wheels do (to keep them on the tracks). With that in mind, the tracks could come gradually inward from a wider gauge to the standard gauge, allowing the driver leeway as they mounted the rails, and the error would then correct itself as the gauge narrowed.

[quickfur]

Yes, a 4D vehicle would have a spherical steering wheel, which would be fully utilised when off-rail, but one dimension would be locked when on-rail.

While thinking over this, I suddenly realized that I was unconsciously lapsing into thinking in 3D again, and overlooking the fact that in 4D, you can wrap your hands round the outer and inner surfaces of a spherical shell. That is to say, what is to us a hollow sphere where the inside is inaccessible from the outside is to a 4Der a 3D analogue of a torus. Just as in 3D we can grip a torus (i.e. steering wheel) at any point around its perimeter by wrapping our fingers around it, so in 4D we can grip a hollow sphere at any point around its surface area by wrapping our fingers "around" it (not in the sense of around the entire sphere, but around the outer and inner surfaces of the sphere, since in 4D you can rotate around a 2D plane). So a 4D steering wheel can be easily constructed to be a hollow sphere mounted on spokes that connect it to a central shaft, and the driver would be able to grip it anywhere there isn't a spoke and rotate it in any direction at will. So there's no need for joysticks at all!

I would imagine the rail-wheels would have rims on the inside just like our 3D train wheels do (to keep them on the tracks). With that in mind, the tracks could come gradually inward from a wider gauge to the standard gauge, allowing the driver leeway as they mounted the rails, and the error would then correct itself as the gauge narrowed.

Sounds like a sound idea.