4D road markings

[quickfur]

The recent 4D game thread brought up an interesting topic. What kind of road markings would there be on a 4D road?

In my quick-n-dirty hand-made povray sketch of a 4D road scene, I inserted a dotted line down the middle of the road, just because it's a nice analogy to 3D roads. However, this is obviously unnecessary (and rather silly) from a native 4D perspective. First of all, a line down the middle of a cylindrical road doesn't serve any purpose; it can't divide the road into lanes (you need 2D sheets for that).

So that makes me wonder, what kind of markings would be on a 4D road? Perhaps none -- if we have enough material to build two lanes, we could just build them as two cylinders side-by-side, with no overlap between them at all. So no markers are necessary; only single-direction traffic is allowed on each lane. But what if the road needs to have multiple lanes? Perhaps it's too costly to build two separate lanes -- it might be a rural road with limited space or building materials; how would one divide the road into two directions? Or what if we're building a highway with multiple lane traffic in a single direction: how would we delineate lane boundaries on the road surface?

The simplest solution to dividing a two-lane road into two directions is to paint a 2D divider down the middle to split the cylindrical road into 2 sections. But this would require a lot of paint, which might offset the savings of not building two separate roads in the first place. So perhaps a better solution would be to have some kind of pattern that indicates the boundary. Say a lattice like pattern, or maybe some kind of criss-crossing pattern, with different patterns corresponding to different conventions (like the difference between dotted white lines and solid double lines in our 3D world).

However, regardless of how you do it, a 2D divider still requires lots of paint, and lots of effort painting it. So another idea is to actually paint two dotted lines, but instead of indicating lane divisions, like in 3D, the lines indicate lane positions. That is, you're meant to drive on the lines, so the lines serve as a kind of guide to where your vehicle should be. This saves a lot of paint, and also is much easier to paint, so I propose this as a superior solution.

This solution also generalizes better to multi-lane traffic in a single direction: generally, on highways you don't want to mix traffic in two directions, so you'd have two road surfaces, one for each direction. But you need lanes within a single direction, so trying to paint 2D boundaries to divide the cylindrical road surface into, say, 6 lanes in a hexagonal arrangement, is a lot of work and a lot of paint. Much easier to just paint 6 lines (maybe dotted, maybe just an unbroken line) as guides for each lane. Cars will just stick to their line instead of driving between them.

This also makes it more economical to have large numbers of lanes on the road surface: just paint lines arranged in concentric bundles; the outer lanes (lines) will have highway entrance/exit lanes forking off/merging in, while inner lanes can be dedicated for faster travel.

[wendy]

I suppose centre-line stripes make more sense then edge markings. The issue at hand is that there is much less paint to create a stripe then to paint the whole walls of the lane.

[gonegahgah]

Hi quickfur. Cool pictures. Would the following work?

[quickfur]

Hi quickfur. Cool pictures. Would the following work?

I suppose that could work in this case... though it's not scalable. What if you have a highway with, say, 16 lanes? I think my idea of using guide lines (har har) instead of lane dividers would work better.

Although, if you restrict the lines to a 1D arrangement, then it could work: say you have a circle of cylindrical lanes with markers between them, and the middle is an island (i.e. can't drive on it). But if there are too many lanes, the circle gets too big and you end up with lots of wasted space in the middle of the road.

[gonegahgah]

One of the tricks with guidelines may be how far your car is over the guideline.
For example if we had guidelines on our roads then how much would it be acceptable to place our car's centre to the left or right of the guideline.
Too far and perhaps you might have cars clipping from opposite directions. It might require the lanes to be made wider perhaps?

[gonegahgah]

What about utilising inside and outside lines like we do? Would that work? ie.



And this would be scaleable wouldn't it?

[Keiji]

Well now, if we went with my idea of the planar rail, you have a nice 2D surface to paint lanes on anyway so it would be no different to 3D

The main reason I keep going on about that isn't inherently because of its similarity to 3D, but because I imagine that the more "common" 4D road ideas would be very difficult to drive on and a lot more attention would be necessary. Whereas with planar rail, you only steer left or right to select your lane, and all actual turning is done by selecting a lane which then bends in that direction in a "ribbon" fashion, so you don't need to steer for the bends.

[quickfur]

Well now, if we went with my idea of the planar rail, you have a nice 2D surface to paint lanes on anyway so it would be no different to 3D

The main reason I keep going on about that isn't inherently because of its similarity to 3D, but because I imagine that the more "common" 4D road ideas would be very difficult to drive on and a lot more attention would be necessary. Whereas with planar rail, you only steer left or right to select your lane, and all actual turning is done by selecting a lane which then bends in that direction in a "ribbon" fashion, so you don't need to steer for the bends.

True. Though I'd expect that roads that arise from old commonly-trodden trails would not tend to have such a nice structure.

But now that I think of it, I suppose trying to lump everything into a single tubular (or some such shape) length of road is unnecessarily troublesome. With all that space available to lay roads on, conceivably the 4Ders would just have single-lane single-direction roads with the occasional ramps for merging/diverging. Unlike in 3D, there's no real need to cram everything into a single road surface; besides, unless you have a square grid subdivision of the road cross section (which is impractical for other reasons, like the complexity of lane changing), it's not a very efficient use of the road surface. Given that the road surface must cover a 3D area, this is wasteful and costly. Might as well make use of the greater availability of space to build multiple narrower (and disjoint) lanes.

Like wendy has said, the road system would likely more resemble our blood circulation system than any direct analogue of multi-lane roads in 3D. Major roads would just have a (disjoint) bundle of say 6 or 8 lanes with the periodical off-ramp to either switch to a different lane or merge with the lanes in another road nearby. For smaller roads, it probably suffices to have 1 or 2 lanes per direction, with only binary forks/merges.

Of course, well-trodden paths for on-foot travel would have a different structure, as presumably they would have arisen from commonly-used trails initially forged by some pioneers. These would likely tend to have a net-like structure with hubs where many paths converge, probably with population centers situated on these hubs. For foot-trodden roads, there's really no need for any elaborate lane system nor any concern for complexity of maneuvering, as is the case with vehicular traffic. Given the amount of floor space available, on-foot roads and vehicular roads could very well be completely disjoint road systems that still cover the same region (by interpenetrating in a 3D way). Sidewalks would be superfluous, except perhaps for road maintenance crew -- there's no reason to walk along a vehicular road when there's already a well-established system of foot trails that more-or-less cover the same geographical region.

Furthermore, within the city, one could conceivably have alleyways in the form of 2D sheets, since they would naturally arise as the boundary between buildings. One could pave specific sections of it, I suppose, but unless there's a good reason to, I don't see why material should be wasted on such things. One could just leave it as bare ground to walk on, perhaps with the occasional linear path paved (or just trodden) through it where the most common routes are.

[Keiji]

Major roads would just have a (disjoint) bundle of say 6 or 8 lanes with the periodical off-ramp to either switch to a different lane or merge with the lanes in another road nearby.

I would still expect multi-lane roads to have the lanes going in the same direction connected so that you can change lanes whenever you want, rather than having a complex system of ramps to switch at discrete points. If you've ever driven on a motorway (and don't fancy being the slowest vehicle around) you'll know why this is important.

[wendy]

In four dimensions, it is probably quite easy to separate the carriageways, since nothing is lost in the 'nature-strip'. Makes U-turns a bugger though. I normally visualise the roads being separate, right down to suburban-street level. You could use a variety of one-way roads, but there would be no need to signal them as such, since the whole road system is constructed as one-way (like the blood system), with little room for U-turns (save slip-lanes etc). But then i don't steer a motor-car, and rarely even go in them. I think sunday was the first time this year i'd been in a motor-car.

What you have to recall too is that many of these things are built on the edges of cities with lesser cost, and in town at greater cost. In four dimensions, there is considerably greater cost associated with creating lane-edging (which is generally marginic or N-2 D), against lane-centres (which is latric or 1D). In 3d, the two are the same, so lane-edges and lane-middles are the same cost. In four dimensions, the lane-edges are a higher dimension to lane-edges.

With road-direction signs, like 'toowong left next turn', it's more complex. The across-space of a travelling vehicle is 2d, and one can diverge out at any point of the circle therein. What's more, all one can assume, is that for all observers, the circle is clockwise, but individual motor-cars can be pointing to different directions of the circle. So 'left' and 'right' really don't have a meaning. You have to point in the correct direction to get the desired results. Also wordy signs would not make the cut, for exactly the same reason.

[quickfur]

Major roads would just have a (disjoint) bundle of say 6 or 8 lanes with the periodical off-ramp to either switch to a different lane or merge with the lanes in another road nearby.

I would still expect multi-lane roads to have the lanes going in the same direction connected so that you can change lanes whenever you want, rather than having a complex system of ramps to switch at discrete points. If you've ever driven on a motorway (and don't fancy being the slowest vehicle around) you'll know why this is important.

True. So one would adopt a cylindrical arrangement of lanes (which wastes a lot of space for the island in the middle) or perhaps a tessellated sort of arrangement (extrusion of a square or hexagonal grid, with lanes placed along the vertices of the original grid). Your planar rails idea works too.

[...] With road-direction signs, like 'toowong left next turn', it's more complex. The across-space of a travelling vehicle is 2d, and one can diverge out at any point of the circle therein. What's more, all one can assume, is that for all observers, the circle is clockwise, but individual motor-cars can be pointing to different directions of the circle. So 'left' and 'right' really don't have a meaning. You have to point in the correct direction to get the desired results. Also wordy signs would not make the cut, for exactly the same reason.

Road signs wouldn't be a problem if one used pictorial representations of direction, say we draw an arrow pointing at one of the circle of 360° possible directions. The driver then can simply follow the direction of the arrow -- no matter what his original orientation was, he just has to align his vehicle to the direction of the arrow. Since we have 3D surfaces to work with, the sign could just be a vertical stack of these arrows with the respective place names inscribed beside them.

[Keiji]

With road-direction signs, like 'toowong left next turn', it's more complex. The across-space of a travelling vehicle is 2d, and one can diverge out at any point of the circle therein. What's more, all one can assume, is that for all observers, the circle is clockwise, but individual motor-cars can be pointing to different directions of the circle. So 'left' and 'right' really don't have a meaning. You have to point in the correct direction to get the desired results. Also wordy signs would not make the cut, for exactly the same reason.

Another problem avoided by planar rail, for vehicles travelling on a plane have just the same left and right as we do.

[quickfur]

With road-direction signs, like 'toowong left next turn', it's more complex. The across-space of a travelling vehicle is 2d, and one can diverge out at any point of the circle therein. What's more, all one can assume, is that for all observers, the circle is clockwise, but individual motor-cars can be pointing to different directions of the circle. So 'left' and 'right' really don't have a meaning. You have to point in the correct direction to get the desired results. Also wordy signs would not make the cut, for exactly the same reason.

Another problem avoided by planar rail, for vehicles travelling on a plane have just the same left and right as we do.

I don't really see that as a problem. I mean, 4Ders walking will already have to deal with 360° of lateral periphery, so they will by no means be unfamiliar with the concept.

Further, a 4D vehicle can be easily equipped with a single side-view mirror that extends around its lateral sides (forming a kind of flat torus). It's not hard to scan it by turning one's head/eyes in a circle. (In 3D you have to look left and right separately because the left and right sides are disconnected; but in 4D, left and right are connected in a continuous circle, so only a single mirror is needed, and you just scan it in a circle.) Any potential hazards are thus easily spotted when one wishes to change lanes, etc..

One may object that it takes time to scan the entire circle of the sideview mirror, but remember that in order for the vehicle to turn in the desired direction, one already has to realign the wheel mechanism to be able to turn in that plane anyway. (I.e., "spin" the vehicle in-place so that the turning plane lines up with the plane that the new lane and the current lane lies in. See this post and the associated discussion on 4D wheels for the rationale.) While realigning the wheels in this way, one might as well scan the mirror simultaneously.

(A separate rear-view mirror is needed for checking what's behind, of course, but for lane-changing it's not really that crucial. Though I do highly recommend checking one's rear-view mirror after merging into the new lane, just in case the vehicle behind you is threatening to rear-end you. )

[Keiji]

I decided to just post all the grand advantages of the planar rail system over on the wiki, rather than cluttering this up more. Feel free to add any nifty ideas you think of.

[quickfur]

Your planar rails idea sounds like it would be something developed later on in the hypothetical history of the 4Ders. What with the recent discussions about planets having toroidal climate zones, and Wendy's observations that the sun moves in the opposite apparent direcetion in the arctic vs. the tropic (equator), this makes the perfect setting for some 4D comedy. A northerner (arctic dweller) travels to the tropics, and comes back reporting how those strange tropic dwellers live under a sun that moves the "wrong" way, and drive weird vehicles confined to rails instead of "normal" 3D road surfaces, etc.. And correspondingly a tropic dweller visiting the arctic zone returns with stories of the primitive arctic dwellers still using the unsafe and complicated 3D road surfaces, and their sun moves in the wrong direction, etc..

[Keiji]

Yes, it would be developed later on, it's more innovative 3D roads and 1D rails might come first, until some bright spark gets the idea to merge them as I have.

[quickfur]

Yes, it would be developed later on, it's more innovative 3D roads and 1D rails might come first, until some bright spark gets the idea to merge them as I have.

Hmm. Smells like another Girl Genius fan...

One of these days I need to sit down and work out a native 4D world in more detail. Maybe draw some sample maps, etc., to show how trails/roads converge on population centers, how buildings might be laid out in a primitive village vs. a more advanced and well-planned metropolis, etc..

A primitive village, for example, would probably have a more-or-less random distribution of buildings in a roughly spherical region with foot trails emanating off from it in various directions. Population centers also tend to spring up around river branches, so one can expect a main river forking within the confines of the village, with buildings and roads laid around the branches of the river. One would expect a road or two to follow the river, since that would be an ideal way to find one's directions prior to the development of more advanced navigational techniques. Unlike 3D, such roads can spiral around the river, or perhaps even flatten into a cylindrical area around the river where it has been trodden down a lot, breaking up into spirals or incomplete cylindrical fragments as one moves away from the village, and perhaps ending with a single narrow trail spiralling around when one gets farther away from civilisation. One would also expect trails emanating from the river banks to give easy access to the water.

A large city would probably have a more systematic layout of roads and buildings. One might find a city on the shores of a sea, built around a sea port, for example. The buildings close to the shore would follow the irregular 2D boundary of the sea, but farther away from the shore it would probably settle into some kind of systematic tiling of 3-space. A cubical tiling is probably most suitable, since it is easiest to decompose into smaller cubes/cuboids to accomodate different-sized buildings, and it gives the most obvious way of arranging roads, which more exotic tilings like rhombic dodecahedra don't lend themselves very well to. Direction-finding is also simplest in a cubical tiling. Now since there is no need to break the city up into blocks, one would expect that actual buildings would tend to span multiple cubical cells, perhaps even adopting complicated voxellated floor plans -- any necessary roads going through the building can be accomodated by simply sectioning off a linear portion of the floor plan.

In fact, for high-traffic buildings like hotels and convention centers, one would expect roads that do pass through the building: having the buliding completely surround the road makes loading/unloading much more efficient. Train/bus stations have the advantage that no elevated walkways are necessary, since the floor can just wrap around the platforms and tracks. This arrangement also make boarding/disembarking a very efficient business: the train/bus can load/unload on all lateral sides (this doesn't affect the lateral rails idea: the rails will just be located in a groove lower than the platform).

It will also be very easy to merge existing buildings into one without needing to build overhead walkways, etc.. One would expect that large enterprises would buy up a few adjacent cubical plots of land, and merge the buildings into one (or build a larger one in its place -- but rebuilding is a costly affair in 4D due to the sheer amount of material that must be handled). This may be how buildings may eventually turn into irregular voxellated affairs, the result of the merging of smaller buildings. A successful business can expand its offices by acquiring neighbouring buildings and merging them into the growing voxellated whole. Neighbouring competing businesses may eventually end up with interlocked buildings -- an ironic situation, if you think about it.

[gonegahgah]

I hope you do QuickFur. I really think the whole 4D thing just cries out to be realistically (as possible) realised in a computer game. I think you will most probably be vital to that realisation one day. I personally feel that all the current 4D games fall short of that reality at the present.

I was just wondering if the following would make any sense. Say you had a 4-plank that the player in a 3D slice was orientated so that they saw it as wide when they came to it and it was wide all around even into the 4th direction. Then they came to a really narrow plank that was actually wide in the 4th direction but narrow in their current orientation.

Would the 2nd plank be fairly easy to traverse as well? Say the 1st plank has a square entry and the 2nd plank has a triangle entry. At the right rotation the plank could seem really narrow while still being very easy to traverse; would it?

[quickfur]

I hope you do QuickFur. I really think the whole 4D thing just cries out to be realistically (as possible) realised in a computer game. I think you will most probably be vital to that realisation one day. I personally feel that all the current 4D games fall short of that reality at the present.

Which current 4D games are you referring to? I'm only aware of a scanty few -- John McIntosh's 4D maze, bo198214's 4D building blocks, MagicCube4D, Magic120Cell, and that's about it. McIntosh's 4D maze AFAIK is the closest it comes to simulating a virtual 4D environment, though it doesn't have gravity so you kinda have to fake it by refraining from making turns involving the vertical axis, so that you get a better idea of what a 4D world with gravity would feel like.

The main stumbling block with 4D games that simulate a virtual world is the difficulty of presentation. Even with the 4D maze's simplistic line-oriented display, the projected image of the maze is already intensely complicated -- and it's just a maze made of simple tesseractic blocks! A virtual world with any amount of non-trivial detail would be a bear to render, and even harder to present in a way that the player can make sense of. Doesn't mean we shouldn't try, of course, but it is a difficult task!

I was just wondering if the following would make any sense. Say you had a 4-plank that the player in a 3D slice was orientated so that they saw it as wide when they came to it and it was wide all around even into the 4th direction. Then they came to a really narrow plank that was actually wide in the 4th direction but narrow in their current orientation.

Would the 2nd plank be fairly easy to traverse as well? Say the 1st plank has a square entry and the 2nd plank has a triangle entry. At the right rotation the plank could seem really narrow while still being very easy to traverse; would it?

The easiest way to understand this is by floor plans, or maps (i.e. projection along the vertical axis). Under a map projection from 4D->3D, the footprint of a 4Der may perhaps show up as a pair of 3D blobs (or an n-tuplet of blobs, depending on what number of 4Der legs you subscribe to). The 4Der would cast a shadow that falls around the footprint blobs, perhaps as a roughly spherical shape (assume a spherical shadow for simplicity). The footprint of a 3Der would show up as 2D patches, and the 3Der himself would cast a 2D shadow, say, for simplicity, a circular shadow covering the footprint patches.

The amount of floor area that is visible to the 3Der, then, is just a 2D slice of the 3D floor plan, oriented in whatever way the 3Der is currently pointing. The floor itself, being a 4D floor, maps to various 3D volumes on the map. Let us say that the lower half of the 3D volume of the map represents a solid 3D floor, and that the upper half represents a deep chasm of unspecified depth. Projecting from somewhere on the boundary of the lower half is, say, a long cuboid shape, representing a cubical plank.

For the 4Der, as long as the plank is of a wide-enough radius, there is no problem walking on it -- we may say that as long as the 4Der's shadow falls within the volume of the plank, it is relatively easy to walk on.

For the 3Der, though, how the plank appears depends on his orientation. For example, if he is oriented such that the 2D slice of his field of vision cuts through the diagonal of the plank's square cross-section, and does so through the length of the plank, then he would see a very wide, square plank. But if his field of vision slices through, say, just a tiny corner of the square cross-section of the plank, then he would see a very narrow plank. Of course, that also means that he is standing close to the edge of the plank, so the narrowness of its appearance does indicate a danger of falling off (though it doesn't represent the actual width of the plank).

It's also possible that the 3Der is oriented such that the 2D slice of his field of vision intersects the plank at an angle, such that it makes a rhombus cross-section with the plank. In this case, the plank would appear as a strange floating rhombus-shaped island hovering over the chasm, much shorter than its actual length. It would not be obvious at all that this strange thing is actually part of a 4D plank that's connected to the main floor. It's also possible for the 2D slice to cut through where the plank joins the main floor, but non-parallel to the plank's length, so that it appears to be shorter than it really is. Perhaps there's a platform on the far side of the chasm, joined to the far end of the plank; but from this orientation, the 3Der would not see this joint, and it will appear that the plank doesn't reach the far side at all.

You don't need two different planks for these effects to happen; you just have to be oriented wrongly.

All in all, the poor 3Der's deficient vision gives him all sorts of strange and confusing illusions, which makes navigation across the 4D floor a rather difficult task.

[Keiji]

I imagine a 4D game ought to project the floor space - just like earlier video games would display what was essentially a floor plan.

The material making up the floor itself would be transparent, and walls and objects would be rendered opaquely.

You would of course have the issue that you would not be able to see e.g. a door the other side of the room from you with a table in the middle, unlike a tetronian who could just look over the table. However, if you walk around the table, it comes into view.

Not sure where to go with this, really...

[quickfur]

I imagine a 4D game ought to project the floor space - just like earlier video games would display what was essentially a floor plan.

The material making up the floor itself would be transparent, and walls and objects would be rendered opaquely.

You would of course have the issue that you would not be able to see e.g. a door the other side of the room from you with a table in the middle, unlike a tetronian who could just look over the table. However, if you walk around the table, it comes into view.
[...]

Well, we don't need to render the floor at all, just like most earlier video games basically just left it blank and only drew the walls & objects of interest. But I'd imagine you'd want to render the objects transparently so that you can see what's currently in view.

You'd still do 4D visibility culling, for example if there's an object in the room that's supposed to be, say, a mid-height stone wall, then some of the objects behind it would be hidden or only partly visible. If you climb on a platform, then more things will come into view; if you step into a depression in the ground, then some things currently in view will pop out of view.

Then there's the problem of representation: how do you represent non-trivial 4D objects on a "flat" 3D surface? A cubical desk will show up as a cube, and so will a tesseractic wall. How to indicate the difference? Maybe a "fake 4D" representation can be used, like a cube-within-a-cube projection of a tesseract for the wall, and a cube with 8 dots near its corners to represent a cubical desk with 8 legs. Trees may be represented as a kind of fractal-like ball perhaps.

One can add a limited amount of 4D depth by analogy with "fake 3D" that older games used to have, like bevelling surfaces to indicate height differences, staircase objects that "transported" you into what is essentially a new map (conceptually, the upper/lower floor), etc..

Hmm, this is an interesting idea indeed! It may be within grasp with current 3D game technology! It's certainly easier than dealing with full 4D->3D projection, that's for sure.

[Keiji]

Maybe a "fake 4D" representation can be used, like a cube-within-a-cube projection of a tesseract for the wall, and a cube with 8 dots near its corners to represent a cubical desk with 8 legs. Trees may be represented as a kind of fractal-like ball perhaps.

One can add a limited amount of 4D depth by analogy with "fake 3D" that older games used to have, like bevelling surfaces to indicate height differences, staircase objects that "transported" you into what is essentially a new map (conceptually, the upper/lower floor), etc..

Yes, this is exactly what I've been getting at. Some 2D top down games do incredibly well at presenting 3D, so we should be able to present 4D using a 3D "top down" format.

[quickfur]

[...] Yes, this is exactly what I've been getting at. Some 2D top down games do incredibly well at presenting 3D, so we should be able to present 4D using a 3D "top down" format.

The easiest proof-of-concept game would be a tile-based game where each tile is a "3D sprite" made by projecting the 4D object into a 3D model. This would let you get the hang of navigating a 3D floor plan, and once we get that working, we can look into more sophisticated approaches like representing height with diagonal displacement (a faked isometric projection), continuous movements, etc..

Some time ago I experimented with automatic terrain generation (in 3D), where you start by generating a random height field, then run a simulation of wind/water erosion to smooth it out for N cycles. I have managed to get some surprisingly realistic-looking maps sporting rivers and mountain ranges (though the latter are biased towards the coordinate axes so don't look as realistic). Something like that can also be used to generate a tile-based terrain in 4D, which might give us a better feel for how 4D terrain would work.

[gonegahgah]

Interesting idea. We'll have to depict a top down 4D view one day.

The current "4D" game that is being worked on by Marc ten Bosch is Miegakure (mie gakure is japenese meaning "hidden and seen").
If you haven't seen it before you can find it at http://marctenbosch.com/miegakure/.

But, my impression is that it isn't a precise representation of 4D. Buildings should not have entire missing walls in one direction; just as ours don't; but they do in the game.
I am keen to play Marc's game but I see it as more of a blend of 3D and 4D rather than what I'm truly hoping for which is a 3Der exploring a real (as possible) 4D world.
Also, rather than rightangle dimensional shifting, I'm looking for more finer and intuitive rotation, via psuedo-location landmarks, through the full extra available 360°.
I've used the term psuedo-landmarks as the objects are not seen directly where they are because they are off in the extra sideways available 4-space.
They are depicted in coloured shadow form and, by their tilt and their location in the sky or underground, we can tell how much rotation is required to rotate our 3D slice of view into the extra-sideways to see them directly.

I've been working on some pictures to depict this so I'll add them here soon and also show them in the Baby Steps thread.

[gonegahgah]

So to explain what I was saying...

First the 2Der to give us context:



The 2Der lives in a single 2D plane. They only have forward and backward progression.


Using the rotation approach we can think of our 2Der being able to rotate their 2D plane into our third dimension. Then they can take in all our space albeit a slice at a time.


Unfortunately their 2D computer screens can not depict sideways so the only practical place to put landmarks, that are unseen off in 3D somewhere, is in the sky or underground.
In this respect they rotate our sideways dimensions downwards or upwards into their view aiming for those landmarks as a way to know where they are and steer towards them.
The landmarks are shown as coloured shadows and are tilted to tell how much rotation is needed as well as their placement telling the 2Der their distance away.


Us 3Ders already enjoy having sideways so their is none of the 2Der's conundrum when we want to draw a 3 dimensional world.


But we may have asperations, as our 2Ders might, to explore a higher dimensional world. So we might do something similar to our 2Der.
However we could directly rotate into those extra sideways dimensions just using the manner in this picture; like our 2Der.
The problem with this approach is that the world swings around wildly and objects off in 4D space would fight for space with objects that are already in our 3D plane.


The solution, like for our 2Der, is to place the objects up in the sky or underground, make them coloured shadows, and tilt them to give us much the same information.
This approach makes it very easy for us to steer towards objects that are off in the 4th dimension somewhere.


Fortunately, our 3D world gives us more options for seeing a 4Der's world - than does the 2Der have to see our world - and this is another way too.
The advantage with this view is that once we have lined up a 4D object before us and in our 3D plane we can start to align it in a way that will make sense to us using this view.
This particular view gives us a unique 4D view where things like the archer's target will look the same no matter what extra side angle you rotate too.
If the target is right in the middle it will remain unchanged. If it is slightly off middle it should (I'm guessing) remain unchanged but move in a circle while staying upright.

One of the cool things I realised, and I know I would have read it in the intro but we forget things, is that these diagrams made me fully understand that rotating our 3D plane 180°into the 4th dimension would return us to our original plane with everything back to front. That is cool and I'm pleased to relearn and understand it.

These are the pictures in the intro I am referring to:



But, it is pleasing that the rotational method produces the same result.

I figure in a game that the player would eventually gain the ability to select between the available views for the uses I mentioned.
But, I wouldn't give them the ability to select views from the start.
Actually from the start they would have no 4D abilities at all and would have to gain those by working out the bits they locate...

[quickfur]

[...]
The current "4D" game that is being worked on by Marc ten Bosch is Miegakure (mie gakure is japenese meaning "hidden and seen").
If you haven't seen it before you can find it at http://marctenbosch.com/miegakure/.

But, my impression is that it isn't a precise representation of 4D. Buildings should not have entire missing walls in one direction; just as ours don't; but they do in the game.

Yeah a quick look at the game seems to indicate that it's a "multiple 3D parallel universes" setting rather than a truly native 4D world. I don't think the game lets you rotate your viewpoint so that you look into the 4th dimension instead of just looking across one of many parallel 3D hyperplanes at a time.

I am keen to play Marc's game but I see it as more of a blend of 3D and 4D rather than what I'm truly hoping for which is a 3Der exploring a real (as possible) 4D world.

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.

Also, rather than rightangle dimensional shifting, I'm looking for more finer and intuitive rotation, via psuedo-location landmarks, through the full extra available 360°.

I've used the term psuedo-landmarks as the objects are not seen directly where they are because they are off in the extra sideways available 4-space.
They are depicted in coloured shadow form and, by their tilt and their location in the sky or underground, we can tell how much rotation is required to rotate our 3D slice of view into the extra-sideways to see them directly.

It looks like Marc's game also uses that "shadowy" representation to indicate the presence of objects in the 4th direction. Of course, your approach is slightly different, in that you're not constrained to 90° movements in relation to the 4th axis, so you can shift your viewpoint to look diagonally across into the 4th direction.

I've been working on some pictures to depict this so I'll add them here soon and also show them in the Baby Steps thread.

Looking forward to those.

[gonegahgah]

[quote="quickfur"]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.[quote]
You might have to wait for that direct to brain technology we discussed then
It does truly amaze, and delight me, that our brain could perhaps actually deal with 4D 'reality' if we had some form of technology to directly control stimuli to our brains...
Maybe the Matrix could be 4D instead of 3D? Then perhaps the simulated 4Der could build a 5D world and so on...

[quickfur]

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.

You might have to wait for that direct to brain technology we discussed then
It does truly amaze, and delight me, that our brain could perhaps actually deal with 4D 'reality' if we had some form of technology to directly control stimuli to our brains...

Well, before we develop that technology, we can get a little glimpse into 4D perception by looking at projection images. I know the projection approach does suffer from some limitations, like the "inside-out" effect of rotations, but I suspect we will still have to deal with that in a direct-to-brain approach anyway. I don't think our brains would automatically understand that as a rotation, unless we can train it by direct manipulation of 4D objects. (I remember spending much time puzzling over how distant objects appear when I was young: mountains appear to move backwards when driving forwards, due to the different rates which the perspective of objects at different distances change. IOW, interpreting projection images is only partly instinctive, there's also a learned component to it.) Direct manipulation of 4D objects would require (simulated) 4D limbs, which is probably a lot harder than just sending visual information to the brain.

Maybe the Matrix could be 4D instead of 3D? Then perhaps the simulated 4Der could build a 5D world and so on...

I've thought about a story idea, where the protagonists discover that their 3D world is actually a Matrix-like simulated reality, and when they "wake up" they realize they're actually 4D beings.

[gonegahgah]

Well, before we develop that technology, we can get a little glimpse into 4D perception by looking at projection images. I know the projection approach does suffer from some limitations, like the "inside-out" effect of rotations, but I suspect we will still have to deal with that in a direct-to-brain approach anyway. I don't think our brains would automatically understand that as a rotation, unless we can train it by direct manipulation of 4D objects.

Our brains would take a lot of time to adjust, if they can, but a new born mind may find the task much easier.
If the 4D world is correctly put together then I suspect the new born would not experience those 'inside-out' effects; just as we don't experience them and the 2Der might think that we should. I feel the 'inside-out' effect is only due to the limitations of our 3D world canvas.

I feel that even in 4D things maintain a constant relationship and it is only our experience that makes it seem otherwise.
If a 4D environment could be simulated correctly then turning things around in 4D may have no more visual conflict to a 4Der than us turning things around in our world.

...I know the projection approach does suffer from some limitations...

I agree that it certainly has a place and I'm wondering if I would find it useful to include projection, but through the rotated planes, in the rotated method.
That and/or intelligent slices to help give the roamer the ability to discern alignments within the 4 dimensions.

One of the things that I've realised with our discussions is that 4D objects still serve purposes and that many of these will overlap with the same purposes that we have too.
And in the same manner they tend to share similar forms as well; though I have come to realise that 4D tyres can have any rim shape as you said all along.
So they certainly do have more latitude than we do (or maybe that should be longitude or some new mix).

For example the archery target is not a whole lot different to ours.
I guess the long term trick will be to work out what would naturally occur that wouldn't for us.
And, people, including yourself, have been addressing that question for roads in this very thread and for other things in other threads.

It's great to see these discussions on what can be done in 4D and in, or not, in 3D.

(I remember spending much time puzzling over how distant objects appear when I was young: mountains appear to move backwards when driving forwards, due to the different rates which the perspective of objects at different distances change. IOW, interpreting projection images is only partly instinctive, there's also a learned component to it.) Direct manipulation of 4D objects would require (simulated) 4D limbs, which is probably a lot harder than just sending visual information to the brain.

Well, they haven't got that worked out either. I suspect replacing any type of normal stimuli directly with artificial stimuli is going to challenge humanity for some time :/

I've thought about a story idea, where the protagonists discover that their 3D world is actually a Matrix-like simulated reality, and when they "wake up" they realize they're actually 4D beings.

That would be cool.

[quickfur]

Well, before we develop that technology, we can get a little glimpse into 4D perception by looking at projection images. I know the projection approach does suffer from some limitations, like the "inside-out" effect of rotations, but I suspect we will still have to deal with that in a direct-to-brain approach anyway. I don't think our brains would automatically understand that as a rotation, unless we can train it by direct manipulation of 4D objects.

Our brains would take a lot of time to adjust, if they can, but a new born mind may find the task much easier.
If the 4D world is correctly put together then I suspect the new born would not experience those 'inside-out' effects; just as we don't experience them and the 2Der might think that we should. I feel the 'inside-out' effect is only due to the limitations of our 3D world canvas.

Having direct manipulation of 4D objects would do a lot to dispel the wrong interpretations of the apparent inside-out effects of 4D rotation. For one thing, most (all?) of the animations I did that show the inside-out effect is aligned in such a way that the line of sight lies in the plane of rotation. It's like when you look at a rotating 3D object at an angle perpendicular to the rotational axis, right through the middle of the object. So from a purely 2D point of view, it appears to turn "neatly" inside out.

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.

Furthermore, when we are able to directly manipulate objects, like a baby playing with blocks of various 3D shapes, we don't just see a pure rotation, but we get to see how the image of the object changes in different ways when we turn it in different ways. When it falls on the floor and rolls around, that's not a pure rotation either, but a complex series of bouncing motions, with some rotation thrown into the mix, as well as translational motion, perhaps moving away from our eye so its image becomes smaller. Faced with such a complicated series of changing images, our brain has no choice but to eventually settle upon a 3D model of the world, since nothing else would adequately explain such effects.

(Well, OK, our brain could have settled on a 4D model which would also explain these effects, but then there would be a redundant 4th axis which is not necessary to rationalize the 3D visual images, so it would just be discarded for efficiency's sake and we end up in 3D again.)

In theory, then, one might expect that playing around with 4D projection images in an application that allows (kindof) direct manipulation of the 4D object in question, our brain might eventually start forming a 4D model of the object. But there's a big discrepancy here, in that we're not really getting the full 3D images unless we have the direct-to-brain technology available, and we're also not immersed in that environment 24/7, so we're liable to just stop at a partial and perhaps inaccurate model of the 4D object. A fully immersive environment would be necessary to condition our brain to "think 4D".

I feel that even in 4D things maintain a constant relationship and it is only our experience that makes it seem otherwise.
If a 4D environment could be simulated correctly then turning things around in 4D may have no more visual conflict to a 4Der than us turning things around in our world.

Certainly, 4D things must maintain a constant relationship in spite of all appearances; that's a basic premise of higher-dimensional geometry, which would collapse on its face if it were violated. We only think of rotating 4D objects as "morphing" or otherwise changing because our habit of thinking in 3D and unfamiliarity with 4D misleads us to interpret the projections this way.

It has always been my dream to create a 4D simulator that allows one to not only manipulate objects in a 4D way, but to have a multitude of 4D objects in the environment as well as background objects like floors, walls, ceilings, ground, sky, etc., to provide a truly immersive 4D environment. Then one can learn about how objects can obscure each other, how they bounce off 3D surfaces, and other such things that would give us a much better understanding of how 4D works.

...I know the projection approach does suffer from some limitations...

I agree that it certainly has a place and I'm wondering if I would find it useful to include projection, but through the rotated planes, in the rotated method.
That and/or intelligent slices to help give the roamer the ability to discern alignments within the 4 dimensions.

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!).

One of the things that I've realised with our discussions is that 4D objects still serve purposes and that many of these will overlap with the same purposes that we have too.

Yep, that's what makes dimensional analogy so useful.

[...] I guess the long term trick will be to work out what would naturally occur that wouldn't for us. [...]

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.

[...]

I've thought about a story idea, where the protagonists discover that their 3D world is actually a Matrix-like simulated reality, and when they "wake up" they realize they're actually 4D beings.

That would be cool.

It's just a somewhat different variation on the story idea I mentioned in the Baby Steps thread about the 4D boy who has a 3D dream. I think I like the 3D dream variation better, because it can be written in such a way that it's apparently talking about a 2D dream, but halfway into the story you suddenly realize that the protagonist is actually 4D, and is talking about 3D "flatland", not 2D flatland.