4D road markings

[gonegahgah]

However, from "normal" view angles, you don't really see just an inside-out effect, you see some combination of both a "normal-looking" rotation and an "inside-out looking" rotation. (The reason for deliberately meddling with the view angle to show a purely inside-out effect is, of course, for didactic reasons, since trying to explain a mixture of two effects can be very confusing to a beginner.) Just like in 3D, when we see a rotating object, it generally isn't rotating around an axis perpendicular to our line of sight, but somewhat at a slanting angle. So we will see parts of the object trace out ellipses, which our brain interprets as circles, plus some amount of "inside-out" effects, such as when a polygonal face turns over, and so forth.

Hmm, I guess I realise after all that that is true. We do see the faces of 3D blocks appear to turn inside out when we rotate them but we just don't interpret it that way. Cool.

Well, I'm all curious about the final result of your rotated method. It is certainly a fresh perspective on the subject (no pun intended!).

Me as well, I'm curious about the final result too. Thanks for your encouragement QuickFur. Hopefully this method will grow to something useful.

If you're looking for a list of things that are unique to 4D, the dimensional features summary page on the wiki section of this site provides a partial list. Many other details can be worked out just by exploring the implications of the listed features.

Cool. Thank you for the link. Very interesting reading.

The 4D views, I drew above, are still very generalised.
Hopefully as I work out how things are constructed I can then work out how to add them directly into those views to show how it would look.

Do the two last 4D views I depicted, shown again below, make sense against their purpose?


The left view selection should hopefully allow us to easily move towards an object that is off in 4D somewhere.
The right view selection should then, once we have the object intersecting our 3D plane, make it easier to line up in whatever way we need to to utilise it.
eg. to find and then go through the open door, etc.

When I was drawing these I realised that there were these two distinct purposes. Well firstly I realised that it would be difficult to move directly to objects using the right view.
You would have to go through a series of moving towards the 4th direction and then turning back towards the object to try to get to it.
But, the left view selection should hopefully allow you to line up objects off in the 4th direction and rotate them into your 3D plane directly in front of yourself.
Then, as I say and hope, you could switch to the right view selection and more intimately align with the object, still utilising 4th direction rotation where needed.

Is there enough information yet to tell if that is correct?

And I am looking more for a game where you're playing a 4Der living in his native 4D world. That's why I focus mainly on the projections approach, as that would be the closest to experiencing 4D as a native 4Der. It would be hard to understand from a 3D-centric perspective, of course. But then again, this is 4D we're talking about.

I guess one of the elements I like about the empowered 3Der perspective is that you can then use this to generate puzzle moments in a game. If we could actually be a 4Der then we might simply be stuck having to program a 4D first person shooter (gasp!). Lol, just kidding. I do like some, of what I see as puzzle moments, that the rotation method is seeming to suggest to me.

For example, at the start of the game you could find some bits and pieces that make no sense because you can't yet see into the 4th dimension. Being a 3Der you can only manipulate them in 3D ways and they don't seem to bear any relationship. However, off in 4D they actually can fit together. By some sort of clues the player tries to put the objects against each other, and or sliding/rotating them and, without any visible reason for that happening, they seem to click into place. Once they can finally see in the 4th dimension it then makes sense to them how the objects actually joined.

Is such a scenario plausible?

[quickfur]

However, from "normal" view angles, you don't really see just an inside-out effect, you see some combination of both a "normal-looking" rotation and an "inside-out looking" rotation. (The reason for deliberately meddling with the view angle to show a purely inside-out effect is, of course, for didactic reasons, since trying to explain a mixture of two effects can be very confusing to a beginner.) Just like in 3D, when we see a rotating object, it generally isn't rotating around an axis perpendicular to our line of sight, but somewhat at a slanting angle. So we will see parts of the object trace out ellipses, which our brain interprets as circles, plus some amount of "inside-out" effects, such as when a polygonal face turns over, and so forth.

Hmm, I guess I realise after all that that is true. We do see the faces of 3D blocks appear to turn inside out when we rotate them but we just don't interpret it that way. Cool.

Yep, once we recognize that, interpreting 4D rotations become a little easier.

[...] Do the two last 4D views I depicted, shown again below, make sense against their purpose?


The left view selection should hopefully allow us to easily move towards an object that is off in 4D somewhere.
The right view selection should then, once we have the object intersecting our 3D plane, make it easier to line up in whatever way we need to to utilise it.
ie. to find and then go through the open door, etc.

This is very interesting. So the left kind of rotation lets you point in 4D (as in, change your direction), and the right kind of rotation lets you "look around" in 4D. Very clever.

Incidentally, you appear to have discovered one of the funny little facts about 4D, as embodied in the right kind of rotation: a 4Der can rotate his head without tilting it sideways (i.e. without changing the vertical axis), and without taking his eyes off whatever he's looking at. In 3D, you can do one or the other, but not both. This rotation corresponds with your right kind of rotation.

Consequently, this means that the "look around" rotation is basically a way of letting you see everything that a 4Der standing in your place would see, whereas the "pointing" rotation is to change your viewpoint to look at something else.

When I was drawing these I realised that there were these two distinct purposes. Well firstly I realised that it would be difficult to move directly to objects using the right view.

Indeed, the right view does not change the direction you're pointing, so you will not be able to leave the hyperplane you're currently oriented in.

You would have to go through a series of moving towards the 4th direction and then turning back towards the object to try to get to it.
But, the left view selection should hopefully allow you to line up objects off in the 4th direction and rotate them into your 3D plane directly in front of yourself.
Then, as I say and hope, you could switch to the right view selection and more intimately align with the object, still utilising 4th direction rotation where needed.

Unfortunately, you can't line up objects just by using the right view, because you're still confined to the current hyperplane. The left view is needed to rotate the hyperplane you're in so that you can access things that lie outside. If you only had the right view, you'd be able to see nearby things outside the hyperplane you're in, but you won't be able to reach it. Some things will also not be visible just by rotating the right view; you need the left view rotation to reach outside and to see stuff that otherwise can't be seen (e.g. if something is far away in the direction perpendicular to your current hyperplane).

[...] I guess one of the elements I like about the empowered 3Der perspective is that you can then use this to generate puzzle moments in a game. If we could actually be a 4Der then we might simply be stuck having to program a 4D first person shooter (gasp!). Lol, just kidding. I do like some, of what I see as puzzle moments, that the rotation method is seeming to suggest to me.
[...]
Is such a scenario plausible?

If you design it correctly, it certainly seems workable. Have the player initially confined to what appears to be merely a 3D map, then at some point reveal the right view to him (but without saying what it is), and suddenly he seems to have acquired the ability to cause the world around him to shift in odd ways. Then after yet another bit, reveal the left view and now he gets to wander around in 4D. If you do it right, there should be a few "aha!" moments in there.

[quickfur]

[...]Unfortunately, you can't line up objects just by using the right view, because you're still confined to the current hyperplane. The left view is needed to rotate the hyperplane you're in so that you can access things that lie outside. If you only had the right view, you'd be able to see nearby things outside the hyperplane you're in, but you won't be able to reach it. Some things will also not be visible just by rotating the right view; you need the left view rotation to reach outside and to see stuff that otherwise can't be seen (e.g. if something is far away in the direction perpendicular to your current hyperplane).

Actually, what I said above is wrong. Assuming that you have gravity which pulls you on the 3D floor, either rotation is enough to let you access the entire floor, and, by extension, any stairs, etc., so you can access the entire 4D area just by the right rotation alone. The difference is that the right rotation doesn't change your forward-pointing orientation, but the left rotation does, so the left view is liable to be a lot more confusing.

It is possible, for example, for the left view rotation to flip you into your mirror image, so that the map now becomes apparently mirror-imaged. Perfect recipe to get the player completely lost.

[quickfur]

[...] It is possible, for example, for the left view rotation to flip you into your mirror image, so that the map now becomes apparently mirror-imaged. Perfect recipe to get the player completely lost.

Actually, thinking about it a bit more, you can get flipped into your mirror image with just the right view rotation too. It's just a bit harder to do it if you don't know what you're doing. In that light, perhaps the left view might actually be less confusing, in the sense that it's an easy way to flip yourself "right side right" (as opposed to left side right ) if after wandering about with the right view rotation you find yourself flipped.

Basically the effect shows up in the way that, when you return to a known location on the map, instead of seeing things the way you remember them you see a mirror-imaged arrangement of things instead. The left view rotation thus gives you an easy way of rectifying this: turn 180° and you should see what's behind you in the correct mirror-image, then turn around in 3D 180° and now everything is back to normal. Very weird, and very fun. (To do the same thing with the right view rotation requires a much more elaborate series of steps, which is easy to get wrong, leaving you even more confused than before.)

[quickfur]

[...]
Actually, thinking about it a bit more, you can get flipped into your mirror image with just the right view rotation too. It's just a bit harder to do it if you don't know what you're doing. [...]

Gah. I'm wrong again. I really should think things through before replying. Both left and right views will flip you when you turn 180°. The right view rotation is actually the simplest: you turn 180° and the world gets mirror-imaged. You don't have to do any 3D turns at all. The left view rotation will flip you and turn you around to face behind, so you have to turn 180° in 3D to look at what you were looking at before again.

The right view rotation will cause you to first see a 180° worth of slices of the 4D world. At the end of the 180° you will get back to the 3D slice you were originally looking at, except that now everything is mirror-imaged. Continue turning, and the slices you saw before will be repeated, but all of them will be mirror-imaged. When you get to 360°, you're back to the original view with everything the right way round again.

The left view rotation is a bit more complicated. You will see a 180° worth of slices of the 4D world first; when you get to 180°, you will see what was originally behind you, but in mirror-image. Continue turning, and you will see the behind of the previous 180° of slices, all in mirror-image (but you wouldn't know that 'cos in the first 180° turns you're looking in the opposite direction), and when you get to 360°, you're back to the original view with everything the right way round again.

[gonegahgah]

It's always an exercise in mental gymnastics the 4D stuff so it's very easy to go astray for a short moment...

Then I realise that the following would be fairly identical, I think now, from what you say:



Except that the left approach would still have the crowding problems which the right approach helps to alleviate by using the freer air space and underground for objects that are off in the 4th dimension. Is it correct that these two are basically the same? It's pretty much the same as the 2Der's example so it should be okay, I guess.

Incidentally, you appear to have discovered one of the funny little facts about 4D, as embodied in the right kind of rotation: a 4Der can rotate his head without tilting it sideways (i.e. without changing the vertical axis), and without taking his eyes off whatever he's looking at. In 3D, you can do one or the other, but not both. This rotation corresponds with your right kind of rotation.

Cool. I didn't think of that. That's interesting to contemplate.

Unfortunately, you can't line up objects just by using the right view, because you're still confined to the current hyperplane. The left view is needed to rotate the hyperplane you're in so that you can access things that lie outside. If you only had the right view, you'd be able to see nearby things outside the hyperplane you're in, but you won't be able to reach it. Some things will also not be visible just by rotating the right view; you need the left view rotation to reach outside and to see stuff that otherwise can't be seen (e.g. if something is far away in the direction perpendicular to your current hyperplane).

Exactly. Some of the staged elements to a game could play on them only having one view selection until they find the parts to add the second view selection.

It is possible, for example, for the left view rotation to flip you into your mirror image, so that the map now becomes apparently mirror-imaged. Perfect recipe to get the player completely lost.

Hmmm, that could be an early puzzle element in itself somehow maybe in the game? Might be able to be combined with some other things to make puzzles that are a combination of these aspects working together. I wonder.

[wendy]

When you look at the 'across' space in four dimensions, you have to think that it's like turning our L/R into a full circle.

Suppose you're standing at y=5, with forward in 'y' and height in 'z'. Now the across-space is the x-axis. You can put a line in the x-axis, say from +3 to -3. This is waht we call 'left' and 'right'. In four dimensions, you can be standing on the y-axis, with height in z, but the across-space is now not just x space, but the 2-space made of w and x. Any point on a diameter-6 circle is as readily got to as our 'turn left' and 'turn right'. More-over, there is no implying condition that says that w has to point this way, and x that way. The parity conditions just tell us that when we put w at a position, then x is 90 degrees clockwise in the same space.

This means that if we suppose writing on trafic signs implicitly require wxz (recall, it's perpendicular to the traffic), then we can't assume that w or x are going to be pointing in the same direction for all obserbers. The problem is akin to putting a cylinder on the table, does tell us the round end bits point at +/- z, but we have no means to stick the label of the tin pointing to any part of the room.

[gonegahgah]

Our 3Der on a 3D slice of the 4Der's road... Where did that road go? What is that meaningless yellow flat thing, with a black border, that is floating up in the air?

[quickfur]

When you look at the 'across' space in four dimensions, you have to think that it's like turning our L/R into a full circle.

Suppose you're standing at y=5, with forward in 'y' and height in 'z'. Now the across-space is the x-axis. You can put a line in the x-axis, say from +3 to -3. This is waht we call 'left' and 'right'. In four dimensions, you can be standing on the y-axis, with height in z, but the across-space is now not just x space, but the 2-space made of w and x. Any point on a diameter-6 circle is as readily got to as our 'turn left' and 'turn right'. More-over, there is no implying condition that says that w has to point this way, and x that way. The parity conditions just tell us that when we put w at a position, then x is 90 degrees clockwise in the same space.

This means that if we suppose writing on trafic signs implicitly require wxz (recall, it's perpendicular to the traffic), then we can't assume that w or x are going to be pointing in the same direction for all obserbers. The problem is akin to putting a cylinder on the table, does tell us the round end bits point at +/- z, but we have no means to stick the label of the tin pointing to any part of the room.

So traffic signs would have to use pictorial symbols for indicating directions, and it seems that writing in general would need to be vertical (like ancient Chinese writing), since that's the only orientation that's fixed. Otherwise one would have to be able to read left-to-right, right-to-left, and, indeed, along any of a full circle of directions. Furthermore, glyphs would need to be symmetric around the vertical axis, or mostly so, for ease of reading. Perhaps it's OK to use asymmetric glyphs, but every such glyph would have to exclude any of its rotated counterparts, otherwise it would be ambiguous.

[quickfur]

It's always an exercise in mental gymnastics the 4D stuff so it's very easy to go astray for a short moment...

Then I realise that the following would be fairly identical, I think now, from what you say:



Except that the left approach would still have the crowding problems which the right approach helps to alleviate by using the freer air space and underground for objects that are off in the 4th dimension. Is it correct that these two are basically the same? It's pretty much the same as the 2Der's example so it should be okay, I guess.

I suppose they should be the same. Though I do still prefer the left approach, even if it suffers from crowding problems; it seems a bit counterintuitive that objects laid out in the lateral dimensions would be represented by a vertical rotation.

[...]

Unfortunately, you can't line up objects just by using the right view, because you're still confined to the current hyperplane. The left view is needed to rotate the hyperplane you're in so that you can access things that lie outside. If you only had the right view, you'd be able to see nearby things outside the hyperplane you're in, but you won't be able to reach it. Some things will also not be visible just by rotating the right view; you need the left view rotation to reach outside and to see stuff that otherwise can't be seen (e.g. if something is far away in the direction perpendicular to your current hyperplane).

Exactly. Some of the staged elements to a game could play on them only having one view selection until they find the parts to add the second view selection.

Actually I was wrong. Either one of the rotations will give you full access to the entire 3D floor plan. It's just a matter of how it changes your 4D orientation. One changes what you're looking at, and the other only changes your orientation.

Last night as I was going to bed, I had an idea of how to visualize these rotations in terms of how they allow you access to the entire 3D floor. Again, think of an overhead map projection: you have a 3D floor plan (for simplicity, let's temporarily ignore stairs and the like, which lets you access full 4D space). Let's say the floor plan is drawn on a cube, with some subdivisions where walls are, etc..

Now, a 3Der standing somewhere on the floor will have a forward-pointing direction, and a left/right direction. We may represent this by a T-shaped symbol somewhere in the cube (it's kinda like a "you are here" symbol on the floor plan). The long leg of the T points in the direction the 3Der is looking at. The top bar of the T represents the left/right direction. We may mark one end of the bar as L (left) and the other end as R (right). Notice that this T shape lies on a plane that cuts across the cube. This plane represents all the places the 3Der can reach using only 3D directions. Turning left or right in 3D is equivalent to rotating this T symbol within this plane. So without executing any 4D turns, the 3Der will be unable to reach all parts of the 3D floor.

What is the effect of your left view/right view rotations? First, let's take the easy case: the right view rotation. This rotation spins the T symbol around the axis parallel to its long leg. So imagine if you make a 90° turn with the right view rotation. Now the long leg of the T symbol is still pointing in the same direction as before, and still lies in the original plane. However, the bars of the T are now outside the original plane, at 90°. The T symbol therefore now lies in a new plane, which is a rotated version of the previous plane. This new plane now cuts across the cube along a different slice, and so if the 3Der now moves around using 3D movements only, he will be able to reach places on the 3D floor previously not reachable.

Now suppose that, after making that initial 90° right view turn, the 3Der makes a 30° turn in 3D. This has the effect of rotating the T by 30° in the new plane. Notice that now the long leg of the T is no longer pointing in a direction parallel to the original plane. If the 3Der were to make another right view turn now, it would rotate the plane into a third orientation that cuts through both the first and second planes. This allows him to reach yet more parts of the 3D floor. If you think about it, by combining right view turns with 3D turns, the 3Der will eventually be able to reach all parts of the 3D floor.

What about the left view rotation? This rotation spins the T symbol around the axis parallel to its top bar. So it points the long leg outside of the original plane. Let's say we make a 45° turn with the left view rotation. Now the plane of reachability is at a 45° angle from before. Let's say the 3Der then makes a 3D turn by 90°. Now the long bar of the T is pointing in a direction parallel to the original left-right axis in the first plane. But the current left-right axis is now at a 45° angle to the original plane, so if the 3Der now makes another left-view turn, it will rotate the plane of reachability into a brand now orientation not previously reachable. So again, by combining 3D turns with the left-view turn, the 3Der will eventually be able to access the entire 3D floor.

So I was wrong... only one of the rotations is enough to give full access to the 3D floor. But it might be nice to still have both options.

[gonegahgah]

I suppose they should be the same. Though I do still prefer the left approach, even if it suffers from crowding problems; it seems a bit counterintuitive that objects laid out in the lateral dimensions would be represented by a vertical rotation.

Just like the inverted glasses you described I feel it would become intuitive quite quickly. It's like the equivalent representation for the 2Der's rotated slices:

But the 2Der doesn't have the option to represent the the left option; they can only represent the right option; so they are stuck with that.
Or, alternatively, they could place everything, that is off in our 3rd dimensional, behind and in front of what they have in their 2D plane; which would be very crowded.
The same goes for us which is why the sky and underground hopefully afford the more intuitive space to place objects off in the 4th direction.
We do, unlike the 2Der have the option to use either representation but utilising the emptier up-down direction eliminates some of the crowding as for the 2Der's example.
The intro, and other examples, here use that same trick at times of using vertical to represent the ana-kata directions.
It's always a juggle to understand statically but I feel that immersion would make it easier and soon very intuitive.

As I've mentioned previously, a 2Der doesn't really get a sense of left or right as we do; only that they are opposite. You could transpose them and they wouldn't know.
The same goes for us with ana-kata. Ana could be rotated skywards or groundwards; it really makes no difference. Maybe it would even be useful to be able to switch?

Actually I was wrong. Either one of the rotations will give you full access to the entire 3D floor plan. It's just a matter of how it changes your 4D orientation. One changes what you're looking at, and the other only changes your orientation.

Sorry, I had thought you had already made that correction. Either view gives you full access to left and right still. This allows you, in the positioning view selection, to wander around an object, that intersects your 3D plane, to try to find the opening. Also, once you have an object in you view, you can still move left and right and, together with rotation, adjust how far you are ana/kata-wards. The location view just allows you to more easily bring objects off in the 4th dimension into your 3D plane in the first place.

So I was wrong... only one of the rotations is enough to give full access to the 3D floor. But it might be nice to still have both options.

The usefulness of having both options was a surprising but pleasant discovery for me when I realised it. It sort of made some kind of sense, only after this emerged, that this would be the case for us 3Ders when navigating a 4D world. I wouldn't have realised it until I drew up the alternate view selections. It was one of those 'aha' moments for me.

I like what one prominent indie game developer, Jonathon Blow the creator of Braid, said and that was that a good way to design a game is to design the laws of the game's universe and then discover what these laws mean. He said it much better than that. I'll try to find and quote Jonathon more correctly. Here we have a ready made universe, by extrapolation, and what we are doing is discovering what it's laws mean in terms of our interaction with it and a 4Ders interaction with it.

[gonegahgah]

Here's an example of how I'm thinking a particular road sign might approximately look using the rotation method. Any guesses what the road sign is?



The rotational method used is the following:



I'm rotating the 3D plane kata-wards only in 1° intervals for these images.

Here's a little animation of the rotation. Should really add more frames to make it more connected.

[Keiji]

Warning: streetlights?

It doesn't help that we don't have yellow diamonds over in England...

[gonegahgah]

Warning: streetlights?

Thanks Keiji, that gave me a lovely chuckle. It does look a bit like that frame by frame..
It is a warning type of sign.

It doesn't help that we don't have yellow diamonds over in England...

What do you guys have?

[Keiji]

Red-bordered triangles.

[gonegahgah]

Red-bordered triangles.

Kool, it's probably closest to one of your blue circle signs.
Here are some of our signs that might help in identifying what the one is that I'm depicting:

Ignore the red circling.
The example again:

[Keiji]

Ah, I see - it's the sign one from right, two down, the arrow curving to the right

Our equivalent would be this one:

[quickfur]

Here's an example of how I'm thinking a particular road sign might approximately look using the rotation method. Any guesses what the road sign is?


[...]

I have to say this completely baffles me. It looks like a series of slices, but since this is your rotational method, they're rotated slices. I have trouble putting them together in my head.

[Keiji]

I can't see how they could be rotational slices at all. Now that I realised it was the bend sign, I can't see it being anything else, and they're quite clearly parallel slices.

[gonegahgah]

I can't see how they could be rotational slices at all. Now that I realised it was the bend sign, I can't see it being anything else, and they're quite clearly parallel slices.

That's it Keiji; almost.
The rotation is very slight - hence 1° for each frame - across a space less than 30cm across that is several metres away.
You're right, it is a 'bend ahead' sign. The question then is: which type of bend? It isn't a 'road bends to the right' ahead sign though.
Just as a clue: each following slice is slightly more ana-wards - via the rotation - than the last while still being almost in front of us - along the direction of the road.

PS. The drawing is not anywhere near as precise as a 4D modeller would draw the 3D slices. Hopefully it's a close representation. We can discuss that further.

[Keiji]

You're right, it is a 'bend ahead' sign. The question then is: which type of bend? It isn't a 'road bends to the right' ahead sign though.
Just as a clue: each following slice is slightly more ana-wards - via the rotation - than the last while still being almost in front of us - along the direction of the road.

Well, kata. It doesn't really matter though since you can just turn the sign round into whatever direction is appropriate!

[gonegahgah]

Well, kata. It doesn't really matter though since you can just turn the sign round into whatever direction is appropriate!

That's correct Keiji; the sign says 'road bends to the kata ahead'.

I've taken the plunge now and started looking at Google Sketch-up which is a 3D modeller.
I've spent an hour working some of it out and have now drawn an octahedron with transparency; which is the shape I am after.
I hope to show shortly, more precisely, how we can think of a 3Der rotating their view through the 4th dimension, in this case for our road sign...

Here's what I've drawn so far. My first 3D picture. Woot!


I can also sort of picture, in my head, connecting the frames vertically via a combination of rotation, projection and coloured shadowing to give us a fuller image of the sign.
I'll be interested to see if that works as I'm thinking.

I should mention that I didn't use a 'road bends to the right ahead' nor 'left' as it would have been fairly obvious.
Having kata - or I could have used ana - gave us a bit more of a puzzle to work out; which was a bit of a fun diversion for me after all the brain strain.
Hopefully you didn't mind me throwing it at you as a puzzle?

[quickfur]

[...] I've taken the plunge now and started looking at Google Sketch-up which is a 3D modeller.
I've spent an hour working some of it out and have now drawn an octahedron with transparency; which is the shape I am after.
I hope to show shortly, more precisely, how we can think of a 3Der rotating their view through the 4th dimension, in this case for our road sign..

Yay! Will you be using the shadowy representation for the parts of the sign outside the current 3D hyperplane?

I have to admit I found your previous sequence of rotating slices really hard to wrap my brain around. I found it really difficult to not take the slices as separate signs with a sequence of squiggles that one has to just memorize the meaning of (as opposed to actually visualizing it with 4D depth). I guess I'm just not used to thinking of 4D in that way. But I'm willing to try it out with the shadowy representations thrown into the mix. I'm kinda expecting those to be easier for me to "see" than the separated slices that you had previously.

[Keiji]

Yay! Will you be using the shadowy representation for the parts of the sign outside the current 3D hyperplane?

It's a sign, there wouldn't be any parts outside the realm. Just like our signs have no parts outside the plane.

[gonegahgah]

Yay! Will you be using the shadowy representation for the parts of the sign outside the current 3D hyperplane?

Once, I've worked out how to use SketchUp, I'll certainly give that a try.

I have to admit I found your previous sequence of rotating slices really hard to wrap my brain around. I found it really difficult to not take the slices as separate signs with a sequence of squiggles that one has to just memorize the meaning of (as opposed to actually visualizing it with 4D depth). I guess I'm just not used to thinking of 4D in that way. But I'm willing to try it out with the shadowy representations thrown into the mix. I'm kinda expecting those to be easier for me to "see" than the separated slices that you had previously.

In my mind it should be. Here are some pictures of what I have been able to produce so far with SketchUp:


With these it might be easier to see how the slices make up the image.
I'll add more details to the diagrams, to make this clearer, as I progress with SketchUp.

These show different views of the same sign. Interestingly, I am only depicting the surface detail of the sign.
Regarding that, the picture only shows ana-kata, left-right and all the angles that comprise sideways, but is not showing foward-back at all.
Please note that this is not a depiction of the rotation method. It is only a full surface depiction; which is different.
In this depiction we would not have a pole because the pole would be behind the sign; just as the poles are behind our street signs.
Because I'm only showing the sideways directions I can not show the pole in the same picture; as it is not sideways but behind.

When I add more details that will become a little clearer.

[gonegahgah]

I should also mention that none of the above surface images can be considered to be pointing in any particular direction by themselves.
It depends on where the backplane is in relation to the surface.

For comparison in our world:



If you peeled the surface off the above sign, flipped it from left to right, then stuck it back on, you would instead have a 'road bends to the right ahead' sign.
The surface detail can point either way; but where is the back defines the direction.

Same goes for the surfaces I have drawn. If you stick it on the backplane one way it will be a 'road bends to the right ahead' sign. If you stick it on another way it will be a 'road bends to the ana ahead' sign. If you stick it on another way it will be a 'road bends to the left ahead' sign. If you stick it on the final way it will be a 'road bends to the kata ahead' sign.

I would even, hesitantly, say that you could even make the sign indicate some directions in-between because their is no preferred sideways direction that is foremost.
Whereas we have only two sideways possibilities to turn towards a 4Der has a whole 360° of sideways to turn towards.
So we should be able to not just flip the sign to four orientations only; but be able to rotate it any of the full 360° of sideways and stick it on the same back plane - I'm guessing at least. Is that right?

[Keiji]

If you wanted to use your octahedral signs, you would only be able to rotate it four ways (assuming it's the right way up) and still have it align with the octahedral backing.

However, get this - there's no need to remove the surface from the backing at all. Just by mounting the sign on the pole at a different angle - or equivalently concreting the pole into the ground at a different angle - you can get all 360 degrees.

To show this I like to use my technique of "dimension ignoring". Starting from your 3D octahedron renders, the dimension to ignore is the vertical, so we end up with a horizontal diamond. Now we place the backing, which is also a horizontal diamond, on top of the original one - this "on top" is really mapped to the 4th dimension. Therefore, with this orientation, the sign points "downwards" from our viewpoint - i.e. traffic is going from the bottom of our image to the top. We can rotate the two horizontal diamonds to any angle in 360 degrees in the horizontal plane, without removing the surface from the backing. Yet, remember how we started, before we ignored the vertical dimension, and you'll see that the octahedron is rotating around the vertical axis, allowing the arrow to point in any direction.

Hope this makes sense!

[gonegahgah]

Ah, thank you Keiji.
It was baffling me how the surface could be placed in anyway but the four directions with the backing plate. Thanks for clearing that it can't.
I thought that it had to be possible to have it pointing any sideways but I couldn't see how until you explained the added element.

That is that there are a whole 360° of the front of the pole - without also including the also present up or down angles.
So the sign can be attached to that 360° of the front of the pole at any rotation that is desired to give us a sign that points to any angle of sideways that is required.

Our 3D poles only have left-right and up-down. If we rotated our 3D flat sign 180° it would end up upside-down; so not give us the opposite sign as we know them.
This is because we would be also rotating through the up but in 4D you can rotate sideways without rotating in the up or the front and back directions.
So you can rotate a 4D sign through a whole 360° of sideways (not including front and back) and it will still have the start of the arrow at the bottom where it should be.

Cool. That does make sense now.

[gonegahgah]

The other interesting thing to make note of is that: even when you rotate into the 4th direction with one of our directions you will still see something happening in our two non-rotated into directions.

You can see this, in simpler form, even in 3D. If you rotate an object through the sideways/upwards directions then you will still see the object seem to turn to 90° right, to upside down, to 270° so that its up now points left, and then finally all the way until it is upright again. So even though it is not rotating forward it still changes appearance.

Just like our objects, an object in 4D maintains a constant relationship between its atoms. So if an object in 4D only rotates through two of the directions then you will still see changes in the other two directions. And this would look different again to when it is rotating along both twin axis at once before your eyes.

The 4D atoms of a 4-solid always maintain the same orientation to each other just as the 3D atoms of our 3-solids.
But it's still cool that their planets can have two simultaneous spins occuring at once; whereas ours can only have one spin happening.

We can have an effect in our 3D world called tumbling - I guess; correct me if I'm wrong - where spin is trying to occur across two axis at once where one direction of rotation is being shared. Is that correct? A very dizzying effect. In 4D the simultaneous separate spins, across different pairs of directions, wouldn't be dizzying; would it?

[quickfur]

[...]
Our 3D poles only have left-right and up-down. If we rotated our 3D flat sign 180° it would end up upside-down; so not give us the opposite sign as we know them.
This is because we would be also rotating through the up but in 4D you can rotate sideways without rotating in the up or the front and back directions.
So you can rotate a 4D sign through a whole 360° of sideways (not including front and back) and it will still have the start of the arrow at the bottom where it should be.

Cool. That does make sense now.

One interesting effect of 4D having 360° of sideways is that a sign pointing to, say, the left, can be rotated so that it points to the right instead, without changing the direction it's facing. So a "curve right" sign can be rotated to be a "curve left" sign or a "curve ana" or "curve kata" sign, and you can still attach it to the pole in the same way as before.

Now the interesting question is, if you paint a 3D left shoe on the sign, can you rotate it so that it becomes a right shoe?