Roads, Vehicles, etc in 4D

[PatrickPowers]

Roads are easy. They would be 3D. A road then looks like a rod, possibly curved. Roads connect together like a jungle gym (dating myself: they went out of style long ago) or a tangle of wires.

One could build a computer game with 3D roads, as follows. The up-down dimension is of less interest, so use that steering in the extra dimension. To the driver a road then usually looks like a circle, with the 3D perspective view going off into the distance aligned with the vehicle. The game is to stay in the circle, not exceeding friction due to excess speed in the turns.

Scenery -- trees, buildings, streams, etc. -- would completely surround the road with the sky not visible. The 4th up-down dimension could be encoded via topographic lines and/or colors, if desired.

All this can be extended to arbitrary travel, without a road. Maps would also work like this.

The equivalent of an automobile would have 8 wheels for stability (I think.), two sets perpendicular to one another. A motorcycle or bicycle would have three, I think, two steering perpendicular in front and one in the rear. But I'm not sure about that.

What would the steering wheel look like, if any? Two wheels perpendicular to one another? That could work.

The equivalent of a child's tricycle would have how many wheels? My best guess is 5, three for stability and two for steering.

Trees would look roughly the same, since one would expect circular symmetry in the extra dimension. More room for branches, though.

[quickfur]

Roads (and other linear features like rivers) in 4D have the interesting feature that they do not divide the surface of the ground into two disconnected parts. This leads to the interesting conclusion that you never need to build bridges across rivers -- you just build your roads "around" them. The only time you need to build a bridge is when you need to cross a lake that's wide in at least 2 dimensions. Similarly, there is never a need for pedestrian cross-walks: you just build your pavement around the road, and nobody ever needs to step onto the road at all! Roads also do not divide the city into blocks, and you can build an entire building around a road without needing an overpass. It is also possible to have two (or more!) disjoint road systems that mesh together and cover the same 3D region, but never connect to each other.

Also, due to the 3D nature of the ground, traffic lights are never necessary: you just build your road system with ramps that split off the main roads and merge with the roads to be connected to, highway-style.

Then there's Keiji's planar-rail road system, which sounds strange at first but makes a lot of sense if you approach it from a native 4D dweller's POV. See the wiki for details.

All sorts of fun things are possible.

[Mrrl]

For the car it's enough to have 6 wheels (at the vertices of 3-prizm). May be, 8 will be more stable, but I'm not sure. But most of chairs in our world have 4 legs, so probably 4 front and 4 back wheels will give better result.
Child's tricycle will have 4 wheels: one for steering (it will be wide cubinder? Cartesian product of circle and square. Or duocylinder) and 4 back for stability.
Steering wheel will look like a sphere (like trackball) or joystick (like in aircraft). It will enable turns in two dimensions (right/left/in/out), but I'm not sure how they can implement turns of the car around front/up 2D plane. It may be done in the car (turn front wheels in spiral fashion), but not in vehicle with one front wheel.
I guess that there may be both kinds of professional bikes: with three (1 front+2 back) and two wheels. It may depend on the number of legs of tetronians: if they have 3 legs, they may prefer 3-wheel bike, and if they can walk on two legs, two wheels may be enough (they will have excellent the sense of balance).

[PatrickPowers]

For the car it's enough to have 6 wheels (at the vertices of 3-prizm). May be, 8 will be more stable, but I'm not sure. But most of chairs in our world have 4 legs, so probably 4 front and 4 back wheels will give better result.
Child's tricycle will have 4 wheels: one for steering (it will be wide cubinder? Cartesian product of circle and square. Or duocylinder) and 4 back for stability.
Steering wheel will look like a sphere (like trackball) or joystick (like in aircraft). It will enable turns in two dimensions (right/left/in/out), but I'm not sure how they can implement turns of the car around front/up 2D plane. It may be done in the car (turn front wheels in spiral fashion), but not in vehicle with one front wheel.
I guess that there may be both kinds of professional bikes: with three (1 front+2 back) and two wheels. It may depend on the number of legs of tetronians: if they have 3 legs, they may prefer 3-wheel bike, and if they can walk on two legs, two wheels may be enough (they will have excellent the sense of balance).

I believe that wheels would be a cubiner with two perpendicular axles. Actually, the "axles" would be a single piece, so it would make more sense to call it a singular axle. It's an axle, just shaped like an X, like an old fashioned tire iron. The point of this is to remove two degrees of freedom, leaving two degrees of freedom for the rotation of the cubinder.

This scales up to higher dimensions. An axle always removes N-2 degrees of freedom in order to get something controllable. The wheel is round in one plane and square in every other plane. The difficulty isn't in getting something to round to roll, the difficulty is getting it to roll in the direction one desires and no other. So an axle in 4D has 2 subaxles. In 11D it has nine, and so forth.

So, is it possible to ride a unicycle in N dimensions with such a wheel? I'm not sure. I don't really understand how a unicycle works in 3D.

[gonegahgah]

A unicycle is possible (with a stronger 4D wheel) but you would need to be even more skillful as there are more directions to fall...
Two wheel motorcycles are also possible but probably not practical as you aren't just leaning left and right but a whole 360° of sideways directions against falling and against the turn.
At least turning is linear but staying upright while riding is fighting all 360° at once which are all universally assisted by the gyroscopic affect.
The gyroscopic effect is shared over the whole 360° of sideways so it may not be as effective at stopping us from falling sideways.

Or, please correct me, would it still fight any sideways movement equally; no matter which of the 360° of sideways we fall?

My vote is for - where free standing - 6 wheel cars as we tend to trend towards minimalist constructions when making things.
It should be all you need to prevent a car tipping sideways or back and forth...

Wheels can be any design on their (to us 3D) footprint so the manufacturing processes would tend to dictate this.
It's a toss up between a square or circle cross-section of the footprint (our 2D cross-section of a tyre's footprint being a line) I would imagine depending upon which is easier to make.
Either can provide an equivalent braking footprint but the circle is the most compact in extremity size... So I favour the circle...

You might need a dual 'steering arm' for each wheel with an incorporated means of these arms rotating on each other via a ring. I'll try to draw a diagram.
Here's a link to show a regular steering arm:


The increased complexity (and weakness) may possibly be overcome by the greater strength 4D materials have against inertia. Maybe?
The steering wheel itself might need to incorporate a ring connection to allow two steering shafts to extend from it with each controlling a separate track/tie rods.
The steering wheel would be turned like ours but the turn direction could be any of 360° rather than just left or right...
We would think of their steering wheel as a sphere handle whereas ours is a ring. The 4th dimension direction extends with arms to the steering wheel centre.

[gonegahgah]

Here's a similar diagram I have done up:


I've added what looks like an additional tie rod but in the diagram it appears to go up rather than forwards/backwards.
Actually it's going ana/kata to the other one.
With the dual tie rods the tyre can be pushed left/right and ana/kata. There is a ring mount added so that the pushes can occur without locking each other out of their respective movements.
The wheel would be fitted to the axle via two pivots 90° to each other. As I said a single 4D steering wheel could be attached to two steering shafts via a similar reverse process...

Does that sound okay?

[PatrickPowers]

It sounds OK, but I'm no expert. Perhaps I will get a freshman physics mechanics text and work through the problems as if they were 4D. I'm not sure that even that would help. It can't hurt.

[gonegahgah]

Hi Partick. I'm just rethinking how to depict it. Remembering that you need to have at least 3 tyres and not 2 as in our world.
Also these will need to be organised in a equal triangle formation so it appears that we may need three track/tie rod combinations; and not two as originally stated.
This makes for a bit more of a complicated coupling between steering wheel and the (now) three steering shafts.
The coupling needs to allow each rod to rotate around their linear axis without impeding the others.
I think that is physically possible even in a 3D world but in this case with limited scope.
In a 4D world the scope is not so limited so that the steering wheel has a much freedom of movement in the 360° of their sideways as do our steering wheels in our spinning the wheel only left or right.

Here is a depiction of part of the tyres in 4D showing only forwards (towards us) and 360° of sideways (up is borrowed to depict the extra sideways degrees). No up direction is depicted.
There is a dotted line showing parallel to where the 3 wheel axles lie. Because the axles are off the ground and we are not showing any up direction then in this picture they are hidden in the unshown up direction.


It's clearer to see with this, though I'll will try to clarify later, that the track/tie rods can also lie (hidden also) in the up direction at the same height of the the axles.
To do so they would lie directly behind the wheel axles as do ours for our single front axle cars.
Does that start to make it a little clearer Patrick?

I just added the picture to the right as well.
This shows the 3 front axles at mid-axle height. It also shows the track and tie rods behind the axles (though the front is now towards the screen).
Each track/tie rod pair pushes its pair of wheels/tyres along their (the rod's) linear arc when the steering wheel moves in the same linear steering circular direction.
Working in conjunction they all move the three steering tyres into a synchronised direction causing the car to turn towards that direction...