4D Everyday Life : What Next?

[PatrickPowers]

Its my hobby to think about what everyday things would be like if our world were Euclidian 4D. I can't think of any more everyday things as yet unsolved. I've done

Animals
Transport from bicycles and skateboards to ships to automobiles helicopters and airplanes
Structures from corrals to buildings
Cooking and food
Many different sports
Climate and mapping and calendar of a 4D planet (assumed to be orbiting a sun)
Astronomy, Stonehenge and modern
I don't quite have primitive pre-chronometer navigation, working on that.

I don't have the math to mess with quantum mechanics or waveforms.

[I don't believe any of this is possible, it's an intellectual exercise.]

What next? Any suggestions? I'd rather puzzle about this stuff than waste my life surfing the Inet.

[quickfur]

• How to tie knots
• How to weave 4D fabric to make clothing
• How would buttons and zippers work? (This is a lot more tricky than it sounds.)
• How 4D writing would work (what form would letterforms take, and how practical are they to write, and how they would differ from printed characters where the printing machine can stamp arbitrary shapes onto the 3D surface of 4D paper).
• How sundials would work: what curve(s) would a 4D sun's shadow trace out throughout the year, and what layout of sundial markings would be needed to tell the time of year.
• Navigation: how to tell direction if you're placed in an arbitrary spot on the surface of a 4D planet with no recognizable landmarks nearby? Would you use a compass (would it be sufficient)? What form must a compass take in order to be actually useful?

[PatrickPowers]

Thanks for your quick and thoughtful response.

• How to tie knots
  
  First I observe that real world friction knots have nothing to do with theoretical mathematical knots. Friction knots depend on the compressibility of the material. I figure that while in ND knots are trickier to tie, they are still possible. Here in 3D we can tie the knot loosely then tighten it. In ND you have to get the knot in place first which is harder but seems doable. You have to get the knot so its freedom of motion is less than the radius of the string, or something like that, and then you can tighten it more.
  
  
• How to weave 4D fabric to make clothing
  
  I gave up on that one. I couldn't figure it out but also couldn't show that it was impossible. Using tape instead of thread doesn't seem to work. Maybe 4D people have to rely on chain mail :-).
  
  These are very good ones.
  
  
• How would buttons and zippers work? (This is a lot more tricky than it sounds.)
• How 4D writing would work (what form would letterforms take, and how practical are they to write, and how they would differ from printed characters where the printing machine can stamp arbitrary shapes onto the 3D surface of 4D paper).
  
  You can have a pen with a 3D nib so it seems to me you can draw anything that can be printed. I figured that books could have one page, that would be great for printing. Letterforms are so arbitrary I stopped short of inventing them. If you have 3D vision as opposed to our 2D vision then there's no problem telling them apart.
  
  
• How sundials would work: what curve(s) would a 4D sun's shadow trace out throughout the year, and what layout of sundial markings would be needed to tell the time of year.
• Navigation: how to tell direction if you're placed in an arbitrary spot on the surface of a 4D planet with no recognizable landmarks nearby? Would you use a compass (would it be sufficient)? What form must a compass take in order to be actually useful?

We assume that Hyperearth has a magnetic field, which isn't all that likely. The magnetic field comes from a geodynamo and the small radius of Hyperearth seems to disfavor that. But it's no worse than assuming Hyperearth orbits a sun so away we go. The magnetic field depends on the "horatio", the horological ratio of the periods of the two rotations of the planet. If they are the same then the magnetic field tends to be the same in every direction and hence little use for navigation. I assumed that Hyperearth has a horatio of four. Then the magnetic field could have two planes, one four times as strong than the other. Your compass then aligns with the stronger plane. But that just gives you that plane and its complement. Here on Earth that complement is a unique direction. On Hyperearth the complement is just another plane so it's not as useful. I haven't figured out what to do with the weaker plane. Maybe you magnetize your compass so that it has a weaker field that aligns with this weaker plane, or maybe it's more stable if the weaker compass field is the opposite of the weaker plane. I don't see how this does any good for navigation though.

On any ND planet each star is above a certain latitude that changes very slowly. That would help too but wouldn't solve all your problems. Once you get chronometers and ephemerides then all your problems are solved assuming you can see the sky, so the interesting part is before that. The ancient Egyptians focused on where on the horizon a star would rise, that might be the way to go. The Vikings seemed to have used polarization of light, that might help.

Looking back at the history of Earth ancient people mostly used landmarks -- ships stayed in sight of shore and used maps of the landmarks, that sort of thing. If they ventured into unknown lands they hired guides. There are records of armies being deliberately misled by guides, I think that happened to Marc Anthony but that may have been fiction. Up until recently the Tibetians forbade entry to all foreigners. They didn't want spies making maps of Tibet. It was part of their defense strategy.

[DonSoreno]

I'll just add my 2cents regarding bicycles:

Assuming the cyclist has n legs (perhaps n=3, or n=4),
you simply offset the phase (angle) of each pedal by 2*pi/n.
This arrangement would lead to a more consistent torque, compared to 2 legs.

I think, the front wheel should have 3 degrees of freedom, which can be achieved by using a spherindrical bearing for the head tube (front tube).

Furthermore, the way bicycles are stabilized in 3d, is by rotating the front wheel towards the direction (left or right), that the bike is tilting.
In 4d, the bike could tip over towards any vector in the 2d left-right-ana-kata plane. It could then be stabilized by rotating the front wheel towards that direction (rotation in forward-sideways plane, where sideways is the direction, in which the bike is tilting.)

The rough pattern wouldn't resemble a sine curve as in 3d, but instead 2 orthogonal sine curves --> roughly trace out an ellipse in the left-right-ana-kata plane
(assuming equal frequencies, the frequencies being mainly dependent on reaction speed & the strength of gravity.
Somebody could probably figure out a wave equation for this.)

Such a bike could also roll (rotate in the left-right-ana-kata plane), due to the spherindrical bearing.

All the other components: pedals, chain, wheels, could just be prisms of their 3d counterparts.<

Obviously the tricycle approach, that was talked about elsewhere could also work.

[DonSoreno]

What about trains?

Should they have 3 rails (very minimal) or 4 rails (much nicer symmetries, more stability.)

In any case, all 3 or 4 wheels would probably be connected by a single axis, in order to stabilize the train using hunting oscillations.
https://en.wikipedia.org/wiki/Hunting_oscillation

--> hunting oscillation of the wheelsets would oscillate in the entire left-right-ana-kata plane, instead of only left right.
Thus under ideal conditions, the wheelsets would oscillate with an elliptical wave on each straight piece of track.
Each turn would perturb this oscillation, and on the next straight track, it would oscillate allong a different ellipse.

However, the hunting oscillations cannot stabilize a rolling wheelset.

Rolling cannot be stabilized, by having the wheels rotate at different speeds.
One solution would be to shape the wheels, so that cannot roll.

Like all other means of transport, trains also benefit greatly from the fact that each track (a line) does not partition the space.
Thus train stations can be built at ground level, which may be cheaper in some sense, than the elaborate structures we have to use in 3d.

[PatrickPowers]

Trains ... never gave that any thought.

"One solution would be to shape the wheels, so that cannot roll." That seems pretty easy.

[PatrickPowers]

It seems that here on Earth celestial navigation was pretty crude until the chronometer was invented. You headed across the Atlantic. The Americas were unmissable so you made landfall then used a map to figure out where you were.

The compass came into use in the 16th century and the chronometer in the 18th so there was a relatively short interval of that. Even after the chronometer was invented there were some pretty basic things that they didn't figure out until the 19th century. https://en.wikipedia.org/wiki/Circle_of_equal_altitude. Navigation is harder on 4D Earth so they would have done even worse. Also, things are closer together so there is less need for long-distance navigation. All in all this makes me feel better at not being able to make much progress in 4D.

[quickfur]

Keiji came up with the 2D rails system, which is a hybrid of free-driving cars and trains. Basically, you have a T-shaped rail spanning 2 dimensions, confining the vehicle to a 2D span of movement (instead of the entire 3D area a free-driving car would have), but within the confines of this rail, the vehicle can freely turn and move within the confined 2D area, covering the area a 3D free-driving car can cover. This partial confinement eliminates a lot of the complication of constructing a steering mechanism that would be required to navigate roads with 3 degrees of freedom, reducing it to a 2D area that a simple steering wheel (the same as we have in 3D) can be used for steering. The confinement can be used for stability so that vehicles will never overturn; the 2D area can be exploited for allowing drivers to choose which way to go at forks.

Junctions are never needed in 4D, since there the 3D surface of the ground has enough degrees of freedom that one can just use a system of on-ramps and off-ramps for main roads, the way freeways and highways are built in 3D, except that thanks to the extra degree of freedom no elevation is ever needed. This allows a road system without traffic lights and intersections, which should reduce the risk of accidents.

This hybrid system can be used for both small private vehicles and large transports that could serve as the analogue of trains. Thanks to the extra dimension of space in 4D, it's possible to build multiple disjoint road systems that never intersect each other yet cover the same geographical area. (Think of two interlocking cubic honeycombs that span an entire 4D city. A vehicle travelling on either grid can get close enough to any point in the city for you to walk the rest of the distance, yet the two grids do not intersect and vehicles from one system cannot reach the other. No bridges or overpasses are needed either.) So you could have one 2D-rail system for private vehicles, with lots of freedom for drivers to decide where to go, and another 2D-rail system for train analogues with restricted branching and fixed routes. The two systems can service the same area and never interfere with each other.

[DonSoreno]

I dislike this 2d "plane" rail network, simply because it is waste of material.

Furthermore such a 2d plane, would then again partition the ground (-> 3d hyperplane), at least locally.

That being said, there may be cases where arranging the roads/lanes in 2d subspaces is useful.

I think 3d steering can be made to work, using spheroidal gear and spheroidal bearings (or ball bearings with glomes).

I've made a previous post, where I at least described the rough steering mechanism.


Spheroidal gears would be an interesting topic. We could probably use the platonic solids, or other polyhedra as gears.

Most spherical gears that actually exist, have to be driven by ordinary 2d gears, and hence have very weird shapes:

[PatrickPowers]

By golly, 3D gears really exist.

https://newatlas.com/technology/abenics-versatile-active-ball-joint-gear/

[quickfur]

Whoa that's amazing! Now all we need to do is to extrude that and we'd have 4D gears.

Actually, we'd have 3D gears extruded to 4D. But now that I think of it, 4D gears can do even better: we can take advantage of the Hopf fibration of the 3-sphere to construct a native 4D gear that will take advantage of all 4 degrees of freedom! This is basically the same toroidal division of a planet's surface into cuboids and triangular prisms. Basically take a discrete subset of the fiber bundle and divide each great circle into a set of serrated teeth, then construct the mating gears in a similar way to what's shown in that video, and now you have a way to construct a drive train out of these 4D gears.

[mr_e_man]

The pictured 3D gears don't allow the "sliding motion" to be transmitted. They essentially have only 2 degrees of freedom.

all 4 degrees of freedom!

Which?

(Which 4 of the 6 degrees of freedom of rotation in 4D?)

[mr_e_man]

I dislike this 2d "plane" rail network, simply because it is waste of material.

Furthermore such a 2d plane, would then again partition the ground (-> 3d hyperplane), at least locally.

That being said, there may be cases where arranging the roads/lanes in 2d subspaces is useful.

The rails wouldn't be that wide. They would be mostly 1D, with just enough 2D width to allow several lanes.

[DonSoreno]

However wide they may be, we can always remove material, so that only a set of linear rails remain, e.g. a 2d grid.

[quickfur]

The pictured 3D gears don't allow the "sliding motion" to be transmitted. They essentially have only 2 degrees of freedom.

But that's all you need, isn't it? 2D meshed gears only transmit torque in a single direction; these gears allow bidirectional torque to be transmitted.

all 4 degrees of freedom!

Which?

(Which 4 of the 6 degrees of freedom of rotation in 4D?)

Well actually, it's 3 degrees of freedom, corresponding to motion along the surface of a 3-sphere (or rather, the duocylindrical decomposition of 3-sphere).

[DonSoreno]

We could associate each gear to a "spherical" prism (Is there a term for this? cylinder,sphere,duocylinder, extruded sphere, glome, N-sphere,... ),
and investigate them based on the tangent space at the surface, where they are stuck together.

N-d gears, which can ideally be represented by N-d spheres (corresponding to (N-1)-Spheres),
can in some sense transmit the rotation group SO(N) (or SO(N-1), I gotta look it up.)

big sphere: center C (coordinates), radius R.
small sphere: center c (coordinates), radius r.

Two spheres that are in contact, then have gear ratio R/r

These spheres meet at a tangent plane. Said tangent plane is orthogonal to the vector (C - c).

These gear ratios are only for rotations, for which the center-to-center vector (C-c) lies incident to their rotation bivector.

Roll rotation, that is rotations with rotation bivectors orthogonal to (C - c), are transmitted with a 1:1 ratio, irrespective of the gear ratio.

I'd say, (extruded) (N-1)-dimensional gears, are the best choices to consider, for N-dimensional analogues of steering mechanisms (cars,bikes, bicycles,...),
because we generally need ways to steer left, right, ana, kata and roll, which corresponds.

For propulsion purposes, we can stick to prisms of 2d gears, for all higher dimensions.


Regarding 4d gears, we could consider the cartesian product:

2d-gear X 2d-gear. (>similar to a Duocylinder)

Which can transmit two torques independently. Furthermore a duocylinder can have two radii:
r1,r2
and given another duocylinder, with radii:
R1, R2

We would have two gear ratios: R1/r1, R2/r2.

The duocylinders can then transmit both rotations independently, with two different gear ratios.

(unlike a spherical gear, which transmits both, using a single gear ratio.)

Maybe somebody can come up with a use case for this.

[mr_e_man]

However wide they may be, we can always remove material, so that only a set of linear rails remain, e.g. a 2d grid.

If there are gaps in the rails, you don't have the freedom to change lanes at any time.