Clock For 4D Planet

[PatrickPowers]

I made a a clock for a 4D planet. There are two planes of rotation with differing periods. You get a clock like this. It is a little analog computer that outputs the height of the sun. Let's hope that the uploading of the image works.

IMG_20190821_082333126.jpg (84.38 KiB) Viewed 5291 times

The yellow jewel gives the height of the sun. The horizontal lines measure the height. If the jewel is over the white, its daylight. Over the black, its night. All sorts of weird things happen. The sun can set, "bounce" off the horizon, and rise back up

The red and blue hands are hour hands for the two periods of rotation. In this case the clock is for a location 8 degrees away from the equator, in the direction of the blue plane of rotation. That is why that hand is longer. The blue plane has more influence.

It is also possible to read time. It's whatever numbers the hands are pointing at. The left side of the clock is am, the right pm. The red hand comes first. That clock reads 6am/4:30am. The jewel is at 5:30, halfway up in the sky, so it is bright daylight.

There are also anticlockwise planets, but clocks look exactly the same there. They just rotate in the opposite direction.

[quickfur]

According to Wendy, a rotating 4D planet, regardless of whether it started off with two different rates of rotation or not, would eventually settle into a Clifford rotation (two equal rates of rotation), because that is the state where there is no tidal tension between the two rates of rotation.

A Clifford rotation is peculiar in that it is, to borrow a term from chemistry, racemic: the spiralling motion of any given point on the planet becomes symmetric with the motion of any other point via a symmetry operation, so that you can no longer tell where the two "rotational poles" are. Every circle traced out by a fixed point on the planet is a great circle and they all rotate at the same rate, so you cannot tell them apart anymore, unlike the case when there are two distinct rates of rotation, where you can identify two specific great circles orthogonal to each other at which the different rates of rotation are happening. Under a Clifford rotation, every point on the planet traces out a great circle that can equally be considered as the "equator" of the rotation as any other point. The planet thus "loses" its orientation, in the sense that you can no longer uniquely identify the two planes in which it is rotating, but any two orthogonal planes would serve that role equally well. However, the rotation still retains its chirality, so in some bizarre sense you can still distinguish between planets that rotate "forwards" vs. "retrograde", even though they no longer have a unique orientation.

When you have two different rates of rotation, the path of any fixed point on the planet would be spiral rather than circular; thus one might argue that it represents a higher energy level. The planet would also have a clear orientation where one plane rotates faster than the orthogonal plane. Because of the anisotropy of the spiralling paths, it seems logical to expect that over time some of the energy of the faster rotation would transmit to the slower rotation, until the two equalize and the planet settles into a Clifford rotation.

Eventually, once you have a Clifford rotation, you no longer need two different periods in your clock. But I'd expect that you might need a compass with special features in order to help you find your way across a landscape that has no fixed frame of reference like North/South poles.

[PatrickPowers]

According to Wendy, a rotating 4D planet, regardless of whether it started off with two different rates of rotation or not, would eventually settle into a Clifford rotation (two equal rates of rotation), because that is the state where there is no tidal tension between the two rates of rotation.

A Clifford rotation is peculiar in that it is, to borrow a term from chemistry, racemic: the spiralling motion of any given point on the planet becomes symmetric with the motion of any other point via a symmetry operation, so that you can no longer tell where the two "rotational poles" are. Every circle traced out by a fixed point on the planet is a great circle and they all rotate at the same rate, so you cannot tell them apart anymore, unlike the case when there are two distinct rates of rotation, where you can identify two specific great circles orthogonal to each other at which the different rates of rotation are happening. Under a Clifford rotation, every point on the planet traces out a great circle that can equally be considered as the "equator" of the rotation as any other point. The planet thus "loses" its orientation, in the sense that you can no longer uniquely identify the two planes in which it is rotating, but any two orthogonal planes would serve that role equally well. However, the rotation still retains its chirality, so in some bizarre sense you can still distinguish between planets that rotate "forwards" vs. "retrograde", even though they no longer have a unique orientation.

When you have two different rates of rotation, the path of any fixed point on the planet would be spiral rather than circular; thus one might argue that it represents a higher energy level. The planet would also have a clear orientation where one plane rotates faster than the orthogonal plane. Because of the anisotropy of the spiralling paths, it seems logical to expect that over time some of the energy of the faster rotation would transmit to the slower rotation, until the two equalize and the planet settles into a Clifford rotation.

Eventually, once you have a Clifford rotation, you no longer need two different periods in your clock. But I'd expect that you might need a compass with special features in order to help you find your way across a landscape that has no fixed frame of reference like North/South poles.

I think planets would have non-Clifford rotations. The debates is elsewhere in these archives. My view was that angular momentum would be conserved, and that Wendy's counter-arguments applied equally well to our 3D Earth. The rotation of our Earth does slow over time, but so little that it will still be going strong when the sun goes nova. It's impossible to do any experiments, so no one can be certain. There are even more basic issues. In a 4D Universe we don't even know that matter could exist.

[quickfur]

The thing is, this whole idea of 4D universes comes from the speculation of what-ifs: what if the universe has 4 macroscopic dimensions of space rather than just 3. The basic implicit assumption is that everything else remains unchanged, and continues to work the way they do in the familiar 3D universe, except where the 4D space dictates otherwise. It's a ripe field for interesting speculations, but it really just stops there.

It's a whole different kettle of fish when you start asking things like how electromagnetism will work, or how 4D atoms will work. Initially it was assumed that things would work out more or less the same, except maybe with a 4D twist to it, but as we got deeper, we discovered that certain things we take for granted, like stable orbits and atoms made of the well-known nucleus + electron shells must be fundamentally different, in a fundamentally incompatible way, in order to exist at all. You can try to tweak this or change that to make it somehow all work "as usual", but it all begs the question: why should a 4D universe behave anything like the 3D one at all? We're biased by 3D phenomenon that we're acclimatized to and take for granted; but nothing dictates that a hypothetical 4D universe must behave similarly. In fact, even in the realm of 3D space, there's a lot of parameters that, in theory, don't have to be the way they are. Needless to say already about a hypothetical 4D universe where certain fundamental 3D phenomena cannot possibly work even if you wanted them to.

IOW it's no longer a question of, what if space is 4D and everything else stays the same; now it's, what doesn't change if the 4D universe must be consistent from ground up? Note that it's possible to have an entirely self-consistent 3D universe with Aristotelian physics, even though it doesn't match the way the observable universe behaves; similarly nothing says you can't postulate a 4D universe with Newtonian physics that ignores things like quantum physics, since there isn't even a 4D universe we can observe to see whether things match up. Why should 4D quantum physics behave anything like 3D quantum physics does, or why should there even be quantum physics in 4D in the first place, not something else completely foreign to our understanding? At some point, it becomes completely arbitrary since there is no standard by which we can measure whether such a universe will work or not. You basically make up your own rules and call it a day. Nothing about physics says a 4D universe must exist, so why should a 4D universe require physical laws that even remotely looks like those in 3D?

[PatrickPowers]

You can't spin pizza dough flat in a 4D universe.

[quickfur]

And you can't tie shoelaces either.

[PatrickPowers]

Actually I think you could tie knots in a 4D Universe.

The mathematical concept of a knot is a closed curve of a N-2 dimensional object. Those can exist in any number of dimensions, though I was never able to "see" them. Too weird.

The common concept of a knot has nothing much to do with that. Real-world knots are arrangements of compressible strings held together by friction. That would work in any number of dimensions, though it would be harder to get the shoe laces into that state. As long as the freedom of movement is less than the radius of the string the knot holds. The difference would be in tying the knot. Here in 3D you just arrange the strings into a loose version of the knot and then yank on the ends to tighten them. That won't work in higher dimensions. There's no such thing as a loose knot. You can still get the strings into that compressed state but it's harder to do.

[quickfur]

In 4D, it's easier to knot sheets. But don't ask me how to practically do that, it's totally mind-bending. The mathematical analogue to this is the knotted spheres: spherical surfaces (spherical in the 3D sense) that are "knotted" by different displacements into 4D. They are very mind-bending to imagine. The Klein bottle is a closely-related example (but it isn't a sphere technically, since it's non-orientable -- it's also a misnomer: in 4D, the Klein bottle would not be able to hold any fluid).

I'd imagine practically knotting shoe-sheets (as opposed to shoelaces!) would be very complicated, because they'd have 2 degrees of freedom instead of just 1, so it'd take a lot more effort to wrap them around in the right ways that would knot them. Of course, unlike mathematical knots you'd probably be able to use open-surface 2D sheets (i.e., cut cloth) for this, rather than a closed (boundaryless) surface. But it'd still be a challenge IMO.

In 5D you'd probably have to use 3D knotting cubes or some such. Don't ask me how to do it, though. But for high-dimensional knotting manifolds the Calabi-Yau manifold would probably serve as a good idea of what something like that might look like.

1D strings in 4D would be very hard to knot in a way that won't come loose as soon as you tug on it hard enough. It's probably possible, the same way jigsaw puzzle pieces can hold together (imagine if you shrunk down the pieces small enough that they behave like points), but the slightest out-of-plane twist and it will all come apart. So probably won't work very well for shoelaces and such. But on the plus side, all those tangled-up wires behind my computer desk would no longer be such a pain to manage! You could just tug on any wire and the others would conveniently just slide off without causing a big knot. That's something I'd look forward to if I ever get the chance to live in a 4D universe.

[PatrickPowers]

In 4D, it's easier to knot sheets. But don't ask me how to practically do that, it's totally mind-bending. The mathematical analogue to this is the knotted spheres: spherical surfaces (spherical in the 3D sense) that are "knotted" by different displacements into 4D. They are very mind-bending to imagine. The Klein bottle is a closely-related example (but it isn't a sphere technically, since it's non-orientable -- it's also a misnomer: in 4D, the Klein bottle would not be able to hold any fluid).

I'd imagine practically knotting shoe-sheets (as opposed to shoelaces!) would be very complicated, because they'd have 2 degrees of freedom instead of just 1, so it'd take a lot more effort to wrap them around in the right ways that would knot them. Of course, unlike mathematical knots you'd probably be able to use open-surface 2D sheets (i.e., cut cloth) for this, rather than a closed (boundaryless) surface. But it'd still be a challenge IMO.

In 5D you'd probably have to use 3D knotting cubes or some such. Don't ask me how to do it, though. But for high-dimensional knotting manifolds the Calabi-Yau manifold would probably serve as a good idea of what something like that might look like.

1D strings in 4D would be very hard to knot in a way that won't come loose as soon as you tug on it hard enough. It's probably possible, the same way jigsaw puzzle pieces can hold together (imagine if you shrunk down the pieces small enough that they behave like points), but the slightest out-of-plane twist and it will all come apart. So probably won't work very well for shoelaces and such. But on the plus side, all those tangled-up wires behind my computer desk would no longer be such a pain to manage! You could just tug on any wire and the others would conveniently just slide off without causing a big knot. That's something I'd look forward to if I ever get the chance to live in a 4D universe.

Mathematical Lord Kelvin style knots (mknots) and everyday friction knots(fknots) are not the same thing at all. Except for simple rings, mknots have no practical use that I can think of. I believe I have never seen one in real life. They occur as religious symbols, that seems to be it.

In 4D, a sphere would be the simplest mknot.

I tried to figure out how 4D people would have cloth or paper. I eventually gave up. I'm too dumb to imagine 2D "thread."

You could tie an fknot, but it would require more skill than our knots. Shoelaces would work, you just need spherical 3D eyelets.

4D has 1D lines, like string, spaghetti, etc. It also has 2D lines, which serve as partitions of surfaces.

"1D strings in 4D would be very hard to knot in a way that won't come loose as soon as you tug on it hard enough" Isn't it the other way around? The tighter the more compression of the string, the more compression the more friction, the more friction the less motion, the less motion the firmer the knot, but as soon as they start coming loose they become a tangle instead of knot. 4D is a no more tangles Universe so the knot falls apart. As long as the strings can't move, you are OK. If the strings can't move it doesn't matter how many degrees of freedom there are.

[quickfur]

If the strings are fixed in place and can't move, then it's not a knot anymore, it's a lock. Or a clasp. A lock-and-key or plug-and-socket sort of deal.

We've talked about this before on this forum. It relates to the 2 kinds of holes that are possible in 4D: a hole of the first kind is what you get if you take a solid 3D cube and punch out a ball-shaped hole in the center. This makes it possible to pass through the middle of the cube if you travel along the 4th direction. A hole of the second kind is what you get if you take a long tube and loop it around and attach the two ends together, like a donut. In 3D, this produces the same kind of hole as punching out a disk in a paper, but in 4D, the two kinds of holes are distinct. The donut hole is "bigger" in some sense, in that you can pass through it from 2 perpendicular directions, whereas to pass through the hole of the first kind you must travel in a single specific direction.

Now here's the interesting thing: the two kinds of holes are complementary to each other's inversion, i.e., if you insert the long tube into the hole-in-a-cube, then glue the tube into a loop, then the two pieces will interlock like chain links and will no longer fall apart. In contrast, two cubes with holes in the middle couldn't be interlocked at all (there's not enough "hole" to fit into each other), and two donuts won't interlock because they behave like 1D knots in 4D: they are equivalent to the unknot, so they are trivially pulled apart (there is "too much hole" between the two of them to hold anything in place). For a chain link with the first kind of hole, you need a hole of the second kind in order to accomodate the chain; for a chain link with the second kind of hole (a donut chain link), you need a hole of the first kind in order to hold it in place. So two chain links with the same kind of hole won't work together, but two chain links of two different kinds of holes will interlock.

So, long story short, you can make something easily tie-able by having a buckle in the same topology as a cube with a hole of the first kind, and a belt with a hook-shaped head that can be closed into a loop (making a hole of the second kind). Then it's just a matter of inserting the hook head into the buckle and closing it into a loop, and now you have a secure knot.

This basic mechanism can then be extended into larger knotting structures, like a "teeth and holes" kind of clasp consisting of a piece with a row of holes of the first kind, and a second piece with loopable hooks that make a row of holes of the second kind. Like a comb. Then it's just a matter of fitting the hooks into the holes and locking them. Note that the hooks don't actually have to make closed loops; all you need is U-shaped hooks; as long as outward pressure is applied the hold will be secure. To make an extra-firm grip, you could have a 2D array of holes of the 1st kind and a 2D array of hooks; fit the latter into the first, and now you have a 2D area of locked structures to hold the two parts together, much stronger than a linear row of hooks.

Though this is really kinda of cheating, since we're bypassing making actual knots in the sense of tying 2D sheet-laces together.

[PatrickPowers]

If the strings are fixed in place and can't move, then it's not a knot anymore, it's a lock. Or a clasp. A lock-and-key or plug-and-socket sort of deal.

The strings can move all they want. The parts that are held in place by friction can't move relative to one another. That's the nature or maybe even the definition of a friction knot.

I have more to say about the rest, but it appears I'm first going to have to learn the calculus of variations or maybe topology in order to prove what I have in mind.

[PatrickPowers]

In math a sphere is always a surface. A solid sphere is a ball.

In general, an an ND world, the (N-2)-sphere is a ring. It has pretty much the same properties as the rings were are used to.

In our 3D world, the 1-sphere a.k.a. circle is a ring. Rings can interlock. Adding some thickness to that 1-sphere produces an annulus. Two annuli may interlock as long as the thickness is less than the inner diameter.

In the 4D world, the 2-sphere is a ring. These rings can interlock. A point passing through a ring can move in almost any direction. Adding some thickness to that ring produces an annulus. In 4D that could serve as the steering wheel of an automobile. It is weird to think of grabbing a 2-sphere, but in 4D it is natural to do that. In 4D it is also possible to grab an infinite 2-D plane.