Questions regarding building a 4d world

[Vector_Graphics]

Is the "Duocylindrical projection" of a 3-sphere (for example, a 4d planet) attested anywhere? I.e. projecting onto a duocylinder, and then unwrapping said duocylinder into a 3d space as 2 cylinders, which allows double rotations to easily be visualized as each "equator" is the central line of one of the cylinders.

Additionally, how distinct, CONCEPTUALLY, would the two "sideways" dimensions be to 4d beings? Given that X and W constitute a plane, you can rotate on that plane while keeping forward and up facing the same way.

[gonegahgah]

"Additionally, how distinct, CONCEPTUALLY, would the two "sideways" dimensions be to 4d beings?"

You really are alluding to the answer in your question above. They are indeed not distinct.
It goes further than that as well. No sideways direction is distinct.
Sideways in 4D is the whole 360° of angles in the X-W plane and none of those infinity of angles is distinct from each other.
A 4D creature does not think left, right, ana, kata.
They only think sideways and that they may need to rotate through their sideways to align with something.
As you mentioned they can do this while standing up and while facing the same front.
They don't even think clockwise or anticlockwise because these are interchangeable in 4D.
They may think in terms of their hand which is closest to a particular finger on another hand. That is certainly valid.

I think this probably helps with your first question too.
Any surface map in 4D has to be 3D. Two distinct planes do not make a 3D surface image.

[Vector_Graphics]

Yes, I meant the volume of 2 cylinders re: the map. The "central line" being the line going down the middle of each cylinder.

[PatrickPowers]

A 4D creature does not think left, right, ana, kata.

I believe any intelligent creatures could and would. The same situation obtains here in our 3D world. The up-down dimension is distinguished by omnipresent gravity, but there is no intrinsic difference between the two horizontal directions. We usually distinguish them by reference to our bodies (left right forward back), more rarely by using the directions found on a compass. N dimensional beings could do the same. These concepts are far too useful to be missed.

[PatrickPowers]

Deleted.

[PatrickPowers]

Is the "Duocylindrical projection" of a 3-sphere (for example, a 4d planet) attested anywhere? I.e. projecting onto a duocylinder, and then unwrapping said duocylinder into a 3d space as 2 cylinders, which allows double rotations to easily be visualized as each "equator" is the central line of one of the cylinders.

I have written a book that includes maps in 4D. In 4D a map is 3D, so it's possible to make a model of such a map here in our real world.

Here's version 0.1 of the book. A year ago Researchgate said it was going to shut down this service but didn't follow through. But who knows, it might disappear at any time. The book has no math, so if you want that hi.gher.space is the place.
https://www.researchgate.net/publication/359213812_Elsewhere_Everyday_Life_On_A_Hypergeometric_Earth

I'm up to version 0.8 now. It's about twice as long. I could finish it off easily but can't get motivated. If you are willing to give feedback I will send you a copy. That would get me going.

Even here in 3D we have all sorts of maps. There is no perfect solution. The most popular was is the Mercator projection because it was the most useful for sailing ships. The Mercator projection appears to be one of those things that is impossible in 4D. My favorite 4D map is the Cucumber Zeppelin. It sounds like what you are suggesting is two solid cylinders, each of which is a complete map but from different perspectives. Seems like quite a reasonable thing to do. The Cucumber Zeppelin is similar to a cylinder but less distorted.

Once I made a wooden model of a map of a 4D planet that looked and felt quite nice. It turned out I'd made a math error. I then made a corrected model but it looked like a cow pie so I threw it away.

Hmm, come to think of it you can make a Mercator projection in 4D, you just have to have two maps, each distorted in a different way.

[PatrickPowers]

Delete.

[Vector_Graphics]

A 4D creature does not think left, right, ana, kata.

I believe any intelligent creatures could and would. The same situation obtains here in our 3D world. The up-down dimension is distinguished by omnipresent gravity, but there is no distinguishing the two sideways directions. We usually distinguish them by reference to our bodies (left right forward back), more rarely by using the directions found on a compass. N dimensional beings could do the same. These concepts are far too useful to be missed.

Yeah. Clearly they'd have words and concepts for it, but like, I'm mainly wondering about rotation on XW keeping forward and up the same. However, now that I'm modeling the surface of a 4d world as a 3d volume, (and yes, I'm using [surface], deal with it) I'm starting to understand more.

[Vector_Graphics]

Is the "Duocylindrical projection" of a 3-sphere (for example, a 4d planet) attested anywhere? I.e. projecting onto a duocylinder, and then unwrapping said duocylinder into a 3d space as 2 cylinders, which allows double rotations to easily be visualized as each "equator" is the central line of one of the cylinders.

I have written a book that includes maps in 4D. In 4D a map is 3D, so it's possible to make a model of such a map here in our real world.

Here's version 0.1 of the book. A year ago Researchgate said it was going to shut down this service but didn't follow through. But who knows, it might disappear at any time. The book has no math, so if you want that hi.gher.space is the place.
https://www.researchgate.net/publication/359213812_Elsewhere_Everyday_Life_On_A_Hypergeometric_Earth

I'm up to version 0.8 now. It's about twice as long. I could finish it off easily but can't get motivated. If you are willing to give feedback I will send you a copy. That would get me going.

Even here in 3D we have all sorts of maps. There is no perfect solution. The most popular was is the Mercator projection because it was the most useful for sailing ships. The Mercator projection appears to be one of those things that is impossible in 4D. My favorite 4D map is the Cucumber Zeppelin. It sounds like what you are suggesting is two solid cylinders, each of which is a complete map but from different perspectives. Seems like quite a reasonable thing to do. The Cucumber Zeppelin is similar to a cylinder but less distorted.

Once I made a wooden model of a 4D map that looked and felt quite nice. It turned out I'd made a math error. I then made a corrected model but it looked like a cow pie so I threw it away.

Hmm, come to think of it you can make a Mercator projection in 4D, you just have to have two maps, each distorted in a different way.

That book is awesome, holy heck.

[gonegahgah]

I believe any intelligent creatures could and would. The same situation obtains here in our 3D world. The up-down dimension is distinguished by omnipresent gravity, but there is no intrinsic difference between the two horizontal directions. We usually distinguish them by reference to our bodies (left right forward back), more rarely by using the directions found on a compass. N dimensional beings could do the same. These concepts are far too useful to be missed.

They will not distinguish a left and a right as there is no such distinction in 4D.
They could, for purposes of mathematics, assign an arbitrary sideways perpendicular 1 and a sideways perpendicular 2 and even perhaps create an alignment to some real world marker.
These would obviously be perpendicular to each other, but, apart from their alignment to a possible marker they would not otherwise have an overtly distinguishable directionality relative to their forward and up directions.
Left is definitely left for us but this is not so for a 4D creature. There is nothing to distinguish one sideways from another 360° of sideways except to align it with some other third party object.
We, in 3D, don't need reference to any third party object to say our left is this side and our right is that side (whether we even had hands or not).
A 4D creature does need some reference to a third party object to signify any sideways distinguishment (even if that distinguishment is to some point on their own body's 360° of sideway-ness).
They could even assign w,x,y,z; where we assign x,y,z.
So, like us they will have up/down and forward/back, but they are extremely unlikely, compared to us with our labels of left/right, to ever have four distinct sideways names.
They can't even say, "Well this is our major hand, and this is our to clockwise hand, and this is our to anti-clockwise hand, and this is our opposite hand", simply because they cannot cement or distinguish clockwise from anti-clockwise.
They will always think in terms of 360° of sideway-ness despite our 3D attempts to infuse 4 sideways cardinal directions upon them.

[Vector_Graphics]

simply because they cannot cement or distinguish clockwise from anti-clockwise.

I mean... wouldn't they distinguish clockwise and anticlockwise relative to their "forward" and "up" (since, mirroring the two requires a rotation along one sideways direction and one non-sideways direction, so necessarily WY, WZ, XY, or XZ, changing their facing direction?) Like, I agree that the sideways directions couldn't be distinguished relative to forward and up alone, and would need a directional marker along the sideways plane, but your example of handedness COULD very well be that directional marker in most cases (though, multidextrous people would have a hard time with that)

[Vector_Graphics]

Though what does make this interesting is how they would point out specific sideways directions without handedness (or, say, pointing). Because there'd still need to be a way to specify turning on a road, say - perhaps road intersections could have signs pointing "right" for each way you enter the intersection?

[PatrickPowers]

Suppose you are a 4D person. You could have right/left legs and ana/kata eyes. That is, these directions are defined relative to your body. This is how we do things here in 3D. Children have to be taught left and right. I can even remember learning it. I have a callus on my right hand from holding a pencil. I'd feel for that to distinguish left from right.

It is true that such a 4D person can rotate in the right-ana plane. But I expect people would have customary preferences just as we do in 3D where most people are right handed. I can rotate so that my right hand is to my left. But it's unnatural and strained.

Mathematically there is no difference between ana-kata and left-right but I don't see anything that stop some arbitrary standard from being applied. And the distinction is so useful that I'm certain such a standard would be.

A related question is, how many batters boxes in 4D baseball? One, two, or four? I don't know. I'm inclined toward a single cylindrical batters' box that encircles home plate. We are inclined to think of a cube as the natural extension of a square into 4D but a cylinder could be more common.

[PatrickPowers]

Though what does make this interesting is how they would point out specific sideways directions without handedness (or, say, pointing). Because there'd still need to be a way to specify turning on a road, say - perhaps road intersections could have signs pointing "right" for each way you enter the intersection?

Let's say you have a two-lane road in 4D. It could be shaped sort of like a double barreled shotgun.

Any arrows would point wherever they point. So you are asking how the direction the arrow is pointing would be described. It would be relative to the center line of the "shotgun."

[mr_e_man]

Previous discussions of 4D roads:

viewtopic.php?p=28423#p28423
http://hi.gher.space/wiki/Planar_rail
viewtopic.php?f=27&t=1443
viewtopic.php?f=27&t=1667
viewtopic.php?f=27&t=1706
viewtopic.php?f=27&t=2123

[gonegahgah]

I mean... wouldn't they distinguish clockwise and anticlockwise relative to their "forward" and "up" (since, mirroring the two requires a rotation along one sideways direction and one non-sideways direction, so necessarily WY, WZ, XY, or XZ, changing their facing direction?) Like, I agree that the sideways directions couldn't be distinguished relative to forward and up alone, and would need a directional marker along the sideways plane, but your example of handedness COULD very well be that directional marker in most cases (though, multidextrous people would have a hard time with that)

Let's consider that a 4D being has four eyes (though I would default to three) then let's say that the 4D being is standing in front of a spinning corkscrew that is pointing towards them.
The 4Der could rotate around sideways (without doing so through the forward direction) and the corkscrew will change orientation towards each of their eyes.
We 3D creatures can't rotate sideways and still keep looking forward at the corkscrew (unless we somersault sideways but for us the corkscrew orientation direction of clockwise will still not change).
Let's say the 4Der has only 3D vision in each eye (which they don't) but that each eye is still located at 4 equidistant points in 4D space (we'll call it left-right-ana-kata).
For the 4Der, one of those eyes would see the corkscrew spinning "clockwise". The two eyes adjacent on either side would not see it spinning. The opposite eye would see it spinning "anticlockwise".
So the corkscrew is spinning both ways relative to the 4D creature.
I mentioned that the 4Der would have to have 3D eyes because with their actual 4D eyes they will observe all of those 4 mentioned orientations with every single eye.
Additionally they will actually see the full 360° of orientation with every eye.
So the corkscrew will not appear to be turning clockwise nor anticlockwise to the 4Der to any of their eyes.
It will just appear for each eye to have a full sphere of observations (sphere because they are looking forward) of the corkscrew spinning.
It's as if they can look at the corkscrew spinning towards them with a full 360° of orientations around the sideways circle all at once.

I hope that helps? I'll try to see if I can come up with some 2D/3D analogy (if possible) if the above doesn't fully help explain?

[gonegahgah]

I guess another way to think about it is to put a one eyed worm in space.

Beginning with 3D space and a 3D worm, can that worm differentiate a sideways, or even an up and down?
To them every direction around them is, well not even 'sideways', but around.
There is now nothing to differentiate left from right from up from down.

Now, make that 4D space and the worm 4D as well still with a single eye.
Now they have a complete sphere of 'around' or perhaps we should call it 'aroundside' (because 'beside' in our thinking inherantly implies left or right).
So our poor worm can no longer differentiate left from right from up from down from ana from kata.

But the situation remains that the 3D worm in 3D space would look at an end on corkscrew in front of themselves and it would either look clockwise or anticlockwise.
And again, this is not the case for the 4D worm in 4D space who would see this corkscrew both as clockwise, and anticlockwise, and a whole lot of partial orientations in between all at the same time.

[gonegahgah]

Let's say you have a two-lane road in 4D. It could be shaped sort of like a double barreled shotgun.
Any arrows would point wherever they point. So you are asking how the direction the arrow is pointing would be described. It would be relative to the center line of the "shotgun."

The arrows would be on the side of the road (the aroundside) like for us and in an aroundside angle that is close to where the turn will head towards.
In our 3D world we have (in Australia) both the left side and the opposite side (and in the US this is opposite) but in a 4D world we have an entire 360° of roadside.
If the turn direction happens to be in the direction of the opposing lane, with traffic travelling the other way, then we will have to go around their "side".
However this can be done without the need of a bridge or an intersection.
That is one of the great benefits of 4D.

[gonegahgah]

Mathematically there is no difference between ana-kata and left-right but I don't see anything that stop some arbitrary standard from being applied. And the distinction is so useful that I'm certain such a standard would be.

I tend to think that 4D people are going to be more sidewards agnostic.
There is no real reason to have a prefence along any sideways angle even if they have four hands (to fit the notion of left-right-ana-kata) or three hands.
They are not going to make roads that only turn left, right, ana or kata which may even be arbitrary and out of alignment to the 4D driver's current rotation anyway.
Roads will turn in whatever direction seems most suited to the situation at hand.

[gonegahgah]

Graphically there may be some incentive to have a 'sideways major' and a 'sideways minor' for whatever reason...
However graphically we don't tend to say x and anti-x and instead simplify this to x and -x instead.
My feeling is that the same is likely to occur and that the 4D people are unlikely to refer to an expansive four cardinal sideways directions.
For us when we do graphing we don't tend to think in terms of left and right.
This is despite that for us in our 3D world that generally z is up, -z is down, x is generally right, -x is generally left, y is generally backwards, and -y is generally forwards.
Or the programmer's deformation of these with z being into the screen and y being up (or down in memory addressing terms).
So I would propose that people in a 4D world are unlikely going to have enough in their world to which they would anchor four distinct sideways terms.
But... maybe they would introduce two related sideways perpendicular terms for whatever reason. Perhaps: "apollo" and "artemis".

[Embi]

I haven't come across the "Duocylindrical projection" for 3-spheres before, but it sounds like a cool idea for visualizing 4D stuff. As for how 4D beings would perceive those "sideways" dimensions, it's hard to say since we're stuck in 3D.

[gonegahgah]

As for how 4D beings would perceive those "sideways" dimensions, it's hard to say since we're stuck in 3D.

Hi Embi, welcome to the forum!

It is hard to say but it is also hard for a 2D person to say anything about sideways. For them it just does not exist.
Being able to look backward, without standing on their head, or tilting their head backward, with everything then upside-down, seems unfathomable to them.
To them there is no "this side", or "that side", only up and down (and forward of course, and somersaults I guess for fun).
For them there is no way for the world to rotate where it doesn't rotate toward the sun and then on around to the night sky.
The same thing appears to be the case for us but if we were on the planet Uranus, with a tilt of approximately 98° we could have a situation where we are rotating but continually facing a similar angle to the sun at all times.
How would we explain that to them?

The chief principle is that we have broader information here than they do. And we have to assume that the 4D person has even broader information than us.
So we can't assume that they think the same way that we think at all. The only thing we can attempt to do is look at the factors that will shape their thinking.

In the case of the 4D person there is nothing to tie particular left, right, ana, or kata directions too.
For us it is easy. Left is that side, and right is the other side, and those sides always have the same orientation to things in front of us.
But, that is not the case for someone in 4D. Sure some things will, but most things will not, align to four perfect perpendicular directions, or even six perfect perpendicular directions if we count forward and back.
It is very questionable that they would even have 4 arms, or in-row legs, whereas we have to have left and right legs, even if we are multi-legged creatures.
Side by side legs work best in our two sided world but there will be no such thing as distinct left or right legs in 4D.
There is just no overarching reason in 4D for things to align to four cardinal directions of side-by-side.

Even if they did why would there be any reason to have four co-ordinate directions when left and right can swap, by the same through-forward rotation that we would use, while ana and kata remain the same to the forward direction.
The creatures eyes would also do the same swap so they themselves would look no different to themself but things in front of them would now be flipped in their view.
And which flip should they consider that to be? “Left and right flip” when they don't have left or right sides. All they have is an aroundside.

So, this is why we have to take extra care not to burden our 4D friends with out limited 3D way of thinking.
We have to empathise more and really think about how the world exists for them.
Only then do we have a chance of nailing their thinking and terminology hopefully more accurately...

[gonegahgah]

I'll just add one other thing...

In some of our courts around the world people are asked to put their right hand on the bible.
We can easily tell if they are accidentally putting their left hand instead...

But, how do you ask a 4D person the same thing?
If we ask them to put their kata hand on the bible, which hand is that?

[quickfur]

Is the "Duocylindrical projection" of a 3-sphere (for example, a 4d planet) attested anywhere? I.e. projecting onto a duocylinder, and then unwrapping said duocylinder into a 3d space as 2 cylinders, which allows double rotations to easily be visualized as each "equator" is the central line of one of the cylinders.

This topic has been discussed here before.

If Wendy is right about the rotation of 4D planets eventually settling into Clifford rotations, i.e., double rotation where the two rates of rotation are equal, then the Clifford rotation would cause every point on the planet's surface to be rotating in a great circle. Thus they would not be distinguishible from each other in terms of motion (since the great circles will be fibers in the Hopf fibration of the planet's surface, so they are equivalent to each other via some double rotation), and the only distinguishing factor would be how much light/warmth they receive from their host star (assuming a stable orbit -- big assumption, but simplifies the discussion).

Therefore, there would be one great circle that receives the most sunlight, which we can call the "equator". The toroidal zone of within some fixed radius r of this equator would be the zone of maximal sunlight, thus we may understand it as the 4D analogue of the tropics. The great circle that's orthogonal to the equator would be the one that receives the least sunlight; this corresponds to the north/south polar regions in 3D, but here in 4D it's one continuous zone wrapping around the planet, in the same shape as the tropical zone. We may call this the "arctic zone", and the great circle the "arctic circle" (an abuse of terminology, but it's a good pun ). These two zones roughly correspond with the two toroids wrapping around the duocylinder, with the toroidal region between them serving as the "temperate" zone (roughly corresponding to the 2D ridge of the duocylinder, except widened into a 3D manifold).

A useful way to draw a map of a 4D planet, therefore, would be like you said, unfold it into two cylinders and render it on a sheet of 4D paper (a 3D hypersurface). There would be minimal distortion around the equator and arctic circle, but as you move towards the curved surface of the cylinder things would stretch a bit, because the duocylinder has a 90° angle between its two bounding surfaces, but a 3-sphere would be "rounded" there, i.e., as if you squashed the duocylinder's ridge inwards radially until it turns into a 3-sphere. So in the unfolding process the parts of the map near the duocylinder's ridge would have to be stretched in order to fit the unfolded cylinder in the map. I didn't calculate exactly how much this distortion would be, but I'd imagine it would be similar to the distortion in the Mercator projection we use in maps of 3D planets. Or maybe a bit less, since you only need to cover only up to a 45° angle from the reference great circle.

The nice thing about this projection is that one cylinder would represent the part of the planet's surface that receives the most light from the host star (again, assuming stable orbits), so it'd be the "tropical cylinder", and the other would represent the part receiving the least light (the "arctic cylinder"). There'd be a zone in the arctic cylinder, around the arctic circle, where from the POV of an inhabitant on the surface the sun would never rise above the horizon, but merely circle around it (in 4D it's possible to move in a circle while remaining on the horizon ). The "twilight zone", anybody?

Of course, all of this rests on the assumption that one of the initial planes of rotation of the planet is relatively close to the orbital plane (the "ecliptic"), such that over time, as the planet settles into a Clifford rotation, one of the great circles would be parallel to the ecliptic and become the equator. If this is not true, then the great circle closest to the sun may change over time as the planet orbits the sun, which would lead to rather bizarre (and interesting) behaviours. Would be somewhat like seasons on Uranus. Haven't really thought through what that would look like, but it would be very interesting, I think.

[quickfur]

Hmm OK, I think the solution is quite simple, for the case where the equatorial great circle is perpendicular to the orbital plane. We're assuming that the diurnal cycle, i.e., the planet's rotation, is much faster than its orbital period, of course. So in that case, the equator/arctic circle analysis still applies, as long as we stay within a small timeframe of the year.

As the planet orbits the sun, though, this equator would migrate around the planet, resulting in the following: after 1/4 of the year, the equator would have migrated to a great circle on the planet's surface that's 45° away from the equator 1/4 of the year before, i.e., what used to be in the "temperate zone". Then another 1/4 of a year later, it would have migrated to what used to be the "arctic circle". And similarly for the 2nd half of the year as it migrates back to the original equator.

The result is that every part of the planet's surface will have 4 seasons: summer, when the sun is directly overhead (or close); autumn/fall when the sun has migrated halfway to the orthogonal circle; winter, when the sun does not rise above the horizon because it's over the orthogonal great circle; spring, when the sun is migrating back overhead, and eventually back to summer.

So in this case, the entire planet would experience 4 seasons. Unlike the original case in my first post, where some parts of the planet are in a permanent winter, some in a permanent tropical summer, and the rest of the planet, in the temperate zone, would have moderate weather year-round. Of course, in the more likely in-between case (the planet's closest great circle is at some angle between 0° and 90° to its orbital plane), you'd have something more similar to what we have on earth: the sun migrates over the year, but only up to a certain angle from the tropical circle, so the tropics remain the tropics, the temperate zone experiences 4 seasons, and the arctic zone experiences a permanent winter.