Interesting Notions - starting with 'other side'

[gonegahgah]

Correct me if you think otherwise to any of these notions...

Firstly: 'other side'
In a 4D world they would not have such a thing as 'other side'.
For example: A road would have 360° of roadside - not just two sides; nor just the four being assigned to 4D.
Instead of parking their cars at the 'side' of the road they would tend to think of it as parking 'outside' the road.

[Keiji]

Indeed, a lot of things would go in a circle with a notion of clockwise/anticlockwise instead of left/right.

Mind you, it's possible to retain the simplicity of left/right if 4Ders used something akin to my planar rail idea

[gonegahgah]

Hmmm, interesting one to think about Keiji? Would clockwise form a simple circle in a 4D world?
I'll have to think about that...
In 2D clockwise is like a pendulum moving back and forth.
In 3D clockwise is a circular motion.
I would assume that clockwise in 4D would be a more complicated motion?

Another interesting thing is that they probably wouldn't have 'anti-clockwise' in 2D at all.
The motions would be exactly the same except out of sync time-wise.

In 4D they would probably not have 'anti-clockwise' either instead having a whole range of motions that aren't clockwise.
If they needed such a term they may call it 'non-clockwise'?

I'll have to revisit your planar rails Keiji.

[wendy]

With space in a+b dimensions, it is possible to suppose a link that transfers a sense in one space to that in another.

For example, in three dimensions, there is the 'right-hand-rule', which says that if you make an anticlockwise rotation, say like the directions the fingers point in the right hand, the the plane's vertical points in the direction of the thumb. It can be applied in both directions: a direction of a thumb can be associated with a rotation in the plane.

In four dimensions, the equivalants is that a chiral 3d-space gives rise to a direction in the normal space, and that a chiral 2d-space (like a rotation), gives rise to a similar chiral 2d-space, (like the rotation in the orthogonal). This corresponds to things like swirlybobs.

[Keiji]

In 2D clockwise is like a pendulum moving back and forth.

No, clockwise and anticlockwise refer to which direction something is moving in a circle. A circle is 2D, so clockwise and anticlockwise are 2D concepts.

In 2D, if a disc is spinning clockwise, it can't change to be spinning anticlockwise unless you slow it down, stop it and start it up again the other way.

In 3D, if a disc is spinning clockwise, you don't have to stop it. You could just remove it from whatever it is resting on, turn it upside down (while it is still spinning) and put it back, and now it is spinning anticlockwise.

In 4D and above, if a disc is spinning clockwise, you could rotate it to be spinning clockwise without even removing it from its stand, as the stand could be perpendicular to the direction you use to rotate it.

[gonegahgah]

Swirlybobs sound cool.

What I was meaning Keiji was that people in the difference D's would have different concepts of the word 'clockwise'.
Our 'clockwise' would have the same motion in all D worlds; I agree.

However, in a 2D world their clocks - if flat faced - would move up and down to show the time of day.
In the 4D world (which is where I guess swirlybobs come in?) their 'flat' clock faces would show a more intricate motion than simple circularity.

So I imagine that 'clockwise' would mean different motions in each of the D's. Is that right?

[Keiji]

I think 4D clocks would still move in circular motion, with one dimension restricted, simply because any more complicated motion wouldn't be useful.

It would be like, taking progress bars and curling them up into spirals or other squiggly shapes, you could do it, but nobody does because it's hard to read a curly progress bar, whereas it's easy and practical to read a straight one.

[Klitzing]

You differ in extension of some 2D display within 3D space (our watches) into other dimensions.

In fact you either could read "2D within 3D" as something, what is "2D within some D-dimensional space" with just D=3, or you might read it as "(D-1)-dimensional within D-dimensional". Both extensions would be perfectly allowed.

So Keiji just added a different point of "usefullness". Linear progressbars are easy to read. In case of cyclical motions also circular displays (as watches are) do serve quite well. But I cannot think about any use for spherical surface display - out of my pocket. - Might be for some pathways on a sphere?

--- rk

[gonegahgah]

Hi Keiji, worlds in a 4D world would probably have 2 axis of rotation wouldn't they?
So a clock representing time would have to take this motion into account wouldn't it?
To my mind a spherical 'surface' would better serve the purpose wouldn't it?
It might be interesting to work out time pieces for a 4D creature?

Hi Klitzing, a spherical surface in 4D occupies the whole sphere. It's not the same as our world where a spherical surface is just its outside.
It's like in a 2D world a square's surface(s) is a line whereas for us a squares surface is its square face.
So a clock motion on a 'flat' 4D spherical 'face' would travel throughout the sphere's front surface; and not just on its 'outside' surface as we think of it...

[Klitzing]

Hi Keiji, worlds in a 4D world would probably have 2 axis of rotation wouldn't they?
...

Wrong. A rotation within 4D is still 2D, thus the perpendicular space would be 2D as well. Sure you could use 2 axis to span that subspace. But any other such 2 axis in that space would serve as well. So it is wrong that there are 2 axis for a rotation in 4D. But there is a 2D subspace, orthogonal to which any rotation in 4D would act.

Hi Klitzing, a spherical surface in 4D occupies the whole sphere. It's not the same as our world where a spherical surface is just its outside.
It's like in a 2D world a square's surface(s) is a line whereas for us a squares surface is its square face.
So a clock motion on a 'flat' 4D spherical 'face' would travel throughout the sphere's front surface; and not just on its 'outside' surface as we think of it...

That's wrong too. Within our 3D space a watch (display) is just 2D. And the pathway then is locally 1D, circling around the border of that watch display. Thus it uses that border line. Accordingly that one is dissected by minute marks.

Moving that all up 1 dimension (whithout pondering its sense) then would lead to a setup in 4D with a 3D-realmic watch display, in fact a 3D ball, and the pointers would follow some path on the border of that ball, its 2D surface... - You surely could lift up the pointers 1 dimension too. Then those become 2D halfplanes, circling around the surface of that 3D ball (within a 4D setup). Thus that halfplanes intersect that ball-surface in meridians. - So perhaps, in contrast to our 12 h display here a 24 h display would serve better: then it just provides a visual display of actual time zone (of noon, say). - But then again, that 3D ball clearly just displays the equatorial section of the 4D glome, which the relevant planet would be...

--- rk

[gonegahgah]

Wrong is such a harsh word but you are correct Klitzing.
Still it would be interesting to work out a 4D clock.
I imagine our time is a representation of where a line through the centre of our rotation in the plane of our orbit is.
This covers places like the poles that see 6 months of day and 6 months of night...
Would orbits in a 4D world - if they were possible - be planar also?

[quickfur]

Wrong is such a harsh word but you are correct Klitzing.
Still it would be interesting to work out a 4D clock.
I imagine our time is a representation of where a line through the centre of our rotation in the plane of our orbit is.
This covers places like the poles that see 6 months of day and 6 months of night...
Would orbits in a 4D world - if they were possible - be planar also?

First of all, a clock's purpose is to show cyclic time: we start from midnight at 00:00 (or 12:00), and count up to 11:59, then wrap around back to 00:00. (Or, if you're using a 24-hour clock, count up to 23:59 then wrap back to 00:00). The fact that in 3D we use circular clocks for this purpose is really incidental, it's not a necessary consequence of cyclic time. Digital clocks, for example, show exactly the same cyclic time as a circular clock; they are not any less accurate just because they aren't circular. It just happens that in 3D, a circle happens to be an (n-1)-dimensional sphere, which just happens to map nicely onto the task of a repeating cycle of time.

In 4D, one would therefore choose a clock shape based on the same repeating cycles, rather than using, say, a (4-1)D sphere just by pure analogy with 3D clocks. So I'd say that spherical clocks, while compelling from our 3D-centric POV, are probably impractical in 4D.

Nevertheless, that doesn't mean there is no fun to be had in 4D! For example, we keep time by a 24-cycle of hours (or 12-cycle, if you use am/pm) coupled with a 60-cycle of minutes (which doubles as another 60-cycle of seconds). Ignoring the seconds part for now, consider how, giving the amount of freedom we have in 4D, we might represent a clock having a 24-cycle and a 60-cycle. Since in 4D simultaneous rotations in two orthogonal planes are possible, why not have it conveniently mapped to the 24-cycle and the 60-cycle? So we could have some kind of rotating device where rotation in the XY plane represents the hour, and rotation in the ZW plane represents the minute. This allows us to represent *both* hour and minute with a single clock hand, and the way we tell what time it is, would be by looking at the projection of this hand onto the XY and ZW planes, which can be implemented as two plates attached to the rotational mechanism with hours (respectively, minutes) marked on each plate. Then we can read the hour by looking through the first plate, and the minute by looking through the second plate. The overall shape of such a device, then, would be something like a duocylinder.

So there you have it: a native 4D way of constructing a clock, that's both functional, and a lot more interesting than a 3D-spherical clock.

[quickfur]

Here's another way of building a native 4D clock: since both minute and second are 60-cycles, and as wendy has stated on various occasions, any freely-rotating object in 4D would tend to settle into a Clifford double rotation (i.e. two equal simultaneous rotations), we can take advantage of this fact by having a single hand represent the minute and second using a 60-cycle rotation in both planes. Since such a motion would be dynamically stable, it would seem to be preferable from an engineering POV (you can save energy by simply allowing the object to rotate freely, instead of forcing it to a non-stable rotation by some mechanical means). This then leaves the 24-hour cycle to a second hand, which could perhaps be represented as a boring ole 2D circle.

Alternatively, one could go back to an even more basic level, and consider the sundial. Place a stick vertically in the ground, centered on a spherical plate. At sunrise, mark where the shadow of the stick intersects the boundary of the sphere: this is the sunrise point. At a certain point in the morning, the length of the shadow will be exactly the radius of the base plate, so this can be marked as a second reference point. Note that by now, the shadow may have shifted sideways a bit, depending on the season. Call this second mark the morning point. From here on until noon, at regular intervals, mark the tip of the stick's shadow on the base plate, thus tracing out a kind of curve. At noon, place a special mark as the noon point. If you live on the equator, then the shadow at noon will be perpendicular to the sunrise shadow; but generally speaking, it will be at some angle depending on the season. Continuing through the afternoon, mark the tip of the shadow until it reaches the edge of the base plate again. Call this the evening point. Then at nightfall, mark the intersection of the shadow with the edge of the plate -- this will be close to the evening point but probably somewhat displaced from it. Now, trace the curve that passes through the morning point to the evening point, interpolating the points in between. This curve (probably some kind of quadratic curve) serves as the "clock" for that day, if subdivided into even intervals.

This isn't all, though. The curve we've obtained only works for that particular day (or maybe several days before/after), but as the planet rotates (assuming an orbital motion, that is), the curve will change its shape. If we trace these curves at regular intervals throughout the year, we'll get a bunch of concentric quadratic curves on the base plate. If you live near the tropics, roughly half the curves will be of opposite convexity; that tells you which half of the year you're in. Now if you pick a specific time of each day on each curve, it will trace out an almost-perpendicular curve cutting across these curves, that represents a fixed time of the day throughout the year. By drawing these "hour curves" at regular intervals throughout each day curve, you'll now have a bunch of lines that lets you tell what time it is on any day of the year, just by looking at the shadow on the sundial. These hour curves may have spiralling shapes; it depends on where you are on the planet, and the planet's tilt w.r.t. to the sun, etc.. And by looking at which day curve the tip of the shadow falls on, you can tell what day of the year it is.

So we've essentially collapsed minute/hour/day into a single "hand" (the shadow on the sundial). The same can be said of sundials on Earth, of course, except that the day curves and hour curves on Earth are much simpler 'cos they only lie on a 2D surface (the curves of the 4D sundial pass through the 3D surface of the spherical base plate).

Now, this only works during the daytime, obviously. If we now use, say, hourglasses or other mechanical means to keep track of time through the night, we can extrapolate where the sundial shadow might fall if it were day. This then gives us a bunch of night-curves, which are virtual curves extrapolated from the day-time hour curves via indirect means (they are "virtual" curves because the sundial stick doesn't actually make any shadow unless the planet is transparent, but mathematically you could trace the "virtual shadow" in the same way it is physically traced during the day). These night curves connect the hour curves of the previous day with the next, so if you trace night curves for each night of the year, all of the day curves and night curves together will connect into a single heavily-spiralling "year curve". I'm not sure what shape this big curve will trace out, but it will certainly be quite an interesting 3D spiralling curve! (And it will differ based on where you are on the planet.)

It would be interesting to figure out exactly what this spiralling shape is, and what kind of 3D volume it encloses (if it encloses any 3D volume at all -- it may not). I'm thinking perhaps by marking certain reference points on this curve, say the peak of each season, we may be able to define a 4D analogue of clockwise/counterclockwise by using 3D enantiomerism, but I'm not sure if that will actually be possible. In any case, it would not have a direct connection with rotation as it does in 3D; it would only make sense with periodic spiralling curves, which have a complex, non-obvious relationship with pure and simple plane rotations.

[quickfur]

Hi Keiji, worlds in a 4D world would probably have 2 axis of rotation wouldn't they?
...

Wrong. A rotation within 4D is still 2D, thus the perpendicular space would be 2D as well. Sure you could use 2 axis to span that subspace. But any other such 2 axis in that space would serve as well. So it is wrong that there are 2 axis for a rotation in 4D. But there is a 2D subspace, orthogonal to which any rotation in 4D would act.[...]

I always tell people that rotation is best understood not as motion "around an axis", but rather "circular motion in a 2D plane".

The whole concept of a rotational axis only works in 3D. In 2D, there's no such thing as a rotational axis: there are only stationary points ("centers of rotations", if you will). In 4D, there aren't any rotational axes either; rather, there are stationary planes around which the rotation happens. Due to the difficulty of us 3D-centric beings comprehending such a foreign concept as rotation "around a plane", I think it's better to understand rotation in N dimensions as happening in a 2D plane. So instead of speaking of rotational axes, we should speak of rotational planes (that is, 2D planes, not (n-1)D, which is something else in general). This approach to understanding rotations generalizes easily across all dimensions.

To be precise, though, rotations in 2D planes are only "plane rotations"; in 4D and above, you can have other types of more complicated rotations. However, the good news is that all of them can be decomposed into some number of simultaneous plane rotations, so by grasping the concept of a plane rotation, we can easily deal with these more complex rotations. The 4D clifford rotation is a classic example: two plane rotations that happen simultaneously in mutually-orthogonal planes (e.g. XY and ZW). The rates of rotation are independent of each other, and can freely vary (unlike in 3D, where trying to apply a second plane rotation to a rotating object merely combines the two into another (oblique) plane rotation).

So coming back to planetary motion: in 4D, planets would be able to rotate simultaneously in two orthogonal planes. But as wendy has mentioned many times, inertia / tidal forces will eventually cause the two rotations to equalize into a double-rotation, in which both rates of rotation are equal. So one would expect that after the initial formation of a planet, its initial rotation (which will most likely be two simultaneous rotations at two different rates) will settle into a double-rotation.

Additionally, relative to a star that the planet is orbiting, the planet would tend to orbit in a boring ole circular orbit (if we pretend that such orbits are stable -- unfortunately they aren't, so 4D planetary systems do not last, but since nobody has found an alternative stable planetary system yet, we can for the time being pretend that we have a perfect circular orbit which is at least stable in theory). Why not more interesting orbital paths? Consider: if 4D physics look anything like 3D physics, then in a planetary system one would have, in the simplest case, a central star and an orbiting planet. This planet would have some kind of initial direction of motion, it doesn't matter what the direction is. First, it should be clear that this direction should be perpendicular from the star-to-planet vector, since if it's not, the planet would be in an unstable path that will either spiral inwards to the star, ending in a fiery collision, or spiral outwards away from the star, causing the planet eventually to escape the gravity of the star and fly out into cold, dead outer space. So we have a single star-to-planet vector, and another vector, the planet's initial motion, perpendicular to this vector. These two vectors define a 2D plane in which the planet will move -- there's nothing in the system that will cause the planet to move outside of this plane, no matter what its initial direction is. Choosing a different initial direction merely causes the motion to be in a different 2D plane, but it's still a 2D plane.

So, sadly, if planetary motion exists in 4D it can only be a boring ole circle, in spite of the additional degrees of freedom conferred by the ambient 4D space. Even more sadly, it can't even deviate from this circle -- unlike 3D where elliptical orbits are the norm, in 4D anything that isn't a perfectly circular orbit is unstable, and will either cause the planet to collide with the star or fly off into outer space. In this sense, 3D is more interesting than 4D, 'cos it allows for interesting orbital motions that are non-circular yet stable across vast periods of time.

Of course, all of this assumes that orbital systems in 4D are adequately modelled by the 2-body system of 1 star and 1 planet. The calculations are too complex to work with if more than 1 planet is present (even in the 3D case, there is no analytical solution for the 3-body problem that converges fast enough to be practically useful -- adding a 3rd orbiting body into the system causes a quantum leap in its complexity), and, lacking any observable 4D planetary systems, we really don't have any clue as to where to even begin to look for possible stable orbiting structures. It's clear, though, that in the vast space of possibilities, most of them are unstable -- the sensitivity of stable orbits in 4D to perturbations is basically infinite: a single speck of dust falling onto the planet from space could mean the difference between a planet that survives long enough for life to exist and a planet that dies a fiery death by colliding with the star, or a cold dead planet that escaped its parent star and thus receives no energy input to sustain any form of life. Given the vastly greater number of parameters in 4D to tweak, there are just so many more ways for things to go wrong, that it's doubtful whether any stable planetary system exists in 4D (besides the perfect circular orbit, which is impractical, as a single speck of dust would knock the planet into an unstable path -- and as our own Earth's history (or the surface of the Moon, for that matter) tells us, far larger things than specks of dust collide with orbiting bodies all the time).

[gonegahgah]

Sorry people. I've always been a fan of Amiga computers and I just got myself a new AmigaOne X1000. So that is going to tie me up a bit for awhile.
I am very interested in your notes quickfur so I will try to read them very soon...