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.