City blocks in 4-D

[pat]

Most cities are laid out in a rectangular grid. There is a set of streets running north-south and a set of streets running east-west.


Moving up to four dimensions, one natural extension would be a rectangular grid where every point with integer coordinates is an intersection. But, this has an unfortunate side effect. In the grid above, there are two roads meeting at each intersection. In this four-dimensional version of the same, there are three roads meeting at each intersection. Waiting for traffic lights would be far more annoying. The analogous case for us would be if we had one set of north-south streets, one set of east-west streets, and one set of northeast-southwest streets.


It seems to me, that it would be better (though not quite as efficient in terms of minimal travel distances) to do something to try to keep the intersections down to two-streets at a time. One could accomplish this by having north-south and east-west streets meet at points where the x and y coordinates are even integers whilst the up-down streets meet the other streets at points where the x and y coordinates are odd integers. The analogous thing for us would be the same as the last grid with every second street removed.


Of course, in 4-d the situation is better. Triangles aren't very efficient for most buildings, you waste too much in the acute corners. In 4-d, however, buildings could still have rectangular (prism) foundations.

Any thoughts about other trade-offs that I could be missing in traffic patterns?

[swirl gyro]

You don't need traffic lights. The streets don't need to intersect at all. But they do need to "touch", so you can get from one to the other. Or you could use onramps/offramps to connect them.

[Keiji]

swirl gyro: that way, you could only move realmar vehicles around the streets, not flunar vehicles.

[jinydu]

What does "flunar" mean?

[Keiji]

Flunar means it has four dimensions, just like realmar means it has three.

[pat]

swirl gyro: that way, you could only move realmar vehicles around the streets, not flunar vehicles.

I don't see why...

[pat]

You don't need traffic lights. The streets don't need to intersect at all. But they do need to "touch", so you can get from one to the other. Or you could use onramps/offramps to connect them.

I suppose so. I didn't go that route though figuring that would require lots more space to accomplish. It may be reasonable in cities where space isn't at a premium. On the other hand, there is a lot more space on which to build in a 4-D city as opposed to a 3-D one. And, these onramps/offramps wouldn't require huge support structures, they can all be at "ground level".

[Keiji]

swirl gyro: that way, you could only move realmar vehicles around the streets, not flunar vehicles.

I don't see why...

You said, they don't have to intersect, but only touch. I am assuming you meant touching on a realm. A flunar object cannot move through a realmspace.

[pat]

Edit by BobXP: There is no need to quote the whole of the last post in a topic.

Ah, when swirl said "touch", I took that to mean "overlap a little bit". Also, I was assuming that the road would be flat (3-dimensional) and the cars would sit atop the road. The cars wheels would only have to contact the road over a 3-dimensional area. So, even though the car is 4-dimensional, the roads only need to have adjacent 3-dimensional areas for a car to drive from one to another.

[Keiji]

Yes, that would be correct. I can't imagine right now how axles and wheels etc would work in 4D so I wouldn't be able to work out how much space they would take up, so I will have to leave a bit of a "gray area" on this one.

[PWrong]

What about dividing the road into lanes?

You need at least two lanes in any dimension, to avoid head on collisions.

With just two lanes, adjacent to each other, turning left or right will be different to turning ana or kata. I can't see any inherent problem with this, but it's non-intuitive, and it would be confusing to tetronians.

A rectangular prism with 4 lanes might be best then. But which lanes do you drive in?

[Keiji]

Duh, whichever is closer to your exit at the next junction...

[PWrong]

Ok, you got me. I had already assumed that, and I was wondering something else, but the question came out wrong somehow.

I was thinking about whether the forward cars would drive on the west-side lanes when driving on a north-south road, on the garp-side lanes when driving on an east-west road, and on the north-side lanes when driving on a garp-marp road. If that makes any sense. :? This would mean the shadow of the car would appear to rotate on its side. So to us, living on the realmar surface of the road, it looks like the shadows going backwards are upside down, and the shadows turning left and right are rotating 90 degrees.

Here's the 3D analogy. You're an ant sitting on the median strip in the middle of the road. Half of the cars are moving towards you, and you notice the front of the car, and the driver on one side of the car. The other half of the cars are moving away from you, on the opposite side. You see the back of the car, and also the driver appears to be on the opposite side (still closest to the middle). So there are two asymmetries. In 4D, there are 3 asymmetries, so some cars will appear upside down.
When a car is moving away from you and it turns left, you will suddenly see the side of the car rather than the back.

I can't help imagining that the shadow of the 4D car looks like an ordinary 3D car, so I get this image of a hovering car turning onto its side to turn left.

[Keiji]

I can't help imagining that the shadow of the 4D car looks like an ordinary 3D car, so I get this image of a hovering car turning onto its side to turn left.

Why would 4D cars be an extension of 3D cars anyway?

[PWrong]

They wouldn't be, but they would have a vague black 3D shadow. I just don't like visualising shadowy objects floating around, so it's easier to pretend the shadow looks like a car.

[wendy]

You no more need city blocks in 4D, then you need boom-gates in the blood-system. In fact, the whole point of city blocks, is that a road both unites (ie connects A from B), and divides (ie you have to cross it) in 3D.

In 4D, the road unites (still from A to B), but a 2D space divides. The whole of the 4D city can be set up so that cars go one way (like blood does), and that you can walk from the middle of town to the very edge without having to cross a street.

[quickfur]

You no more need city blocks in 4D, then you need boom-gates in the blood-system. In fact, the whole point of city blocks, is that a road both unites (ie connects A from B), and divides (ie you have to cross it) in 3D.

In 4D, the road unites (still from A to B), but a 2D space divides. The whole of the 4D city can be set up so that cars go one way (like blood does), and that you can walk from the middle of town to the very edge without having to cross a street.

Funny, I was thinking the same thing for the past few days or so. A 4D city would cover a 3D volume on the surface of the ground, so if you think of a cubical city (imagine a Rubik's cube where each "subcube" is a "city block"), the "grid lines" correspond to streets. Note that each "city block" touch 6 adjacent city blocks with a 2D area, and in fact can be joined to each other since the streets don't cross sub-cube faces. So there is no need of city blocks at all. You can just walk around a street without crossing it.

Also, I've been trying to visualize a 4D city as a 4D being would see it. This, of course, is by perspective projection into a 3D volume. Assuming that you're on a road with a square cross-section, you'd see the horizon on the XY plane and the road as a square pyramid whose apex touches the origin. The interesting part is that you can have buildings on 4 sides of the road. In fact, you can have buildings completely surround the road. There are two ways side streets can connect to the road: either north/south or east/west. It can be both, too, but as pat pointed out, that would be rather inconvenient.

I haven't figured out how to handle 2-way traffic yet, but I suppose you could divide the road into two rectangular-cylindrical parts (cut the square cross-section into 2 rectangles), or perhaps into 4 cuboidal parts with 2 "lanes" going one way and the other 2 lanes the other way (cut the square cross-section into 4 squares).

Buildings have 6 perpendicular walls, and from the correct viewpoint you can see 3 of them at a time. Windows would also be laid out in a 3D lattice; that makes for a LOT of windows per building. :-) Supposing you have a 2-storey building with 2 windows along a line, that gives you 8 windows per wall, and 8*6 = 48 windows for the entire house. Supposing each window to correspond to one room, that's a 48-room house, even though it's only 2-storey and only has 2 rooms per line! And this is not even counting internal rooms (without windows). 4D certainly makes for very compact space.

[wendy]

Cubes are stupid. Look instead at the notion that solid 3d space is little more than the ground-plan of four dimensions. So if you have water and electricity and waste going in and out of a building, you don't need crossings for them.

So in 4d, you don't need crossings for the things.

While the number and size of things like windows and fences etc may be big, the thing is a matter of depth, not substance. For example, the cost of building a barrier of size X must ultimately come from the interior of X, and since we have volume = depth * surface, we see that the surface isn't going to change, and we get the effective cost as cost = price * surface is the same in 3d as in 4d.

So while you have bigger surfaces, it is the size of the enclosed volume that makes the thing costy.

W

[pat]

The whole of the 4D city can be set up so that cars go one way (like blood does), and that you can walk from the middle of town to the very edge without having to cross a street.

Sure, but who want to go through the toes to get from one side of the heart to the other? And, who can deal with a roadmap that looks like our circulatory system?

I would think you'd still want roads that run generally perpendicular for ease of navigation. Planar roads would just be asking for accidents. And, if there are linear roads, then there would have to be intersections or on/off ramps.

[wendy]

The point about blood is that it runs on one-way streets. There is no need to mimmick the cycling of all trafic through the heart of town, so to speak.

It would be more like a denser system of cross roads, so you can do a u-turn anywhere, more or less.

[pat]

Okay... I suppose, I'm still trying to see how to efficiently place buildings then between the streets. There's a lot more room to do it in.

In SimCity 2000 (and, I assume in 2004), citizens will only build a new building within two grid-squares of a road. IRL, we rarely build further than one grid-square from a road (in urban areas).

In 4-D, there have to be similar considerations. You don't want to park your car at the "curb" and still have to walk a long distance to the front door. Though, if you're willing to have a somewhat lengthy driveway (or foyer), you can fudge this more easily in 4-D than 3-D.

So, typically one prefers one's business easily accessible from a main traffic thoroughfare, one's residence just slightly removed from a main traffic thoroughfare, as few intersections as necessary to get you where you want to go, an efficient use of space within your building/home (not many narrow passages or weird angles), and other such things....

What sorts of cities does this result in?

Certainly, there can be much more that's "just on the outskirts of downtown". But, does this result in too costly a network of roads (too low of a taxpayer/kilometer-of-road ratio)?

Certainly, there can be fewer intersections. But, would people still make some intersections where more than two roads meet? (We don't ever need to do this, but there are plenty of instances of it.)

Would it be just that much easier for each store to have its own parking lot? or would we still have malls?

Would there still be a utility toward a centralized downtown?

Thanks for your thoughts so far...

[quickfur]

In SimCity 2000 (and, I assume in 2004), citizens will only build a new building within two grid-squares of a road. IRL, we rarely build further than one grid-square from a road (in urban areas).

In 4-D, there have to be similar considerations. You don't want to park your car at the "curb" and still have to walk a long distance to the front door. Though, if you're willing to have a somewhat lengthy driveway (or foyer), you can fudge this more easily in 4-D than 3-D.

I think considerations of convenience and traffic flow apply just as much in 4D as in 3D (actually, in any finite number of dimensions). It's just that in higher dimensions, we can pack more within a certain distance, and have more non-intersecting flow layouts. Ideally, every building is preferably within some convenient distance D from the closest road. So you can think of it as an N-hypersphere centered on the access point on the road (say, where you parked your vehicle) with radius D. All buildings that touch this hypersphere would be "conveniently accessible". So the higher the dimension, the larger the number of buildings you can fit within this space.

So, typically one prefers one's business easily accessible from a main traffic thoroughfare, one's residence just slightly removed from a main traffic thoroughfare, as few intersections as necessary to get you where you want to go, an efficient use of space within your building/home (not many narrow passages or weird angles), and other such things....

Right, and I think these considerations apply regardless of which dimension you're in.

[....]Certainly, there can be fewer intersections. But, would people still make some intersections where more than two roads meet? (We don't ever need to do this, but there are plenty of instances of it.)

It depends. I think having 6 roads meet at an intersection is somewhat at the max possible. Any more than that and you start to get impracticably long traffic light waits and/or overly elaborate rules of traffic. So at some point, you have to space out intersections to keep things practical. One could, of course, build on/off ramps for such multi-way intersections; but I think the reasons we don't have on/off ramps for every intersection in our 3D cities also apply to 4D cities. Ramps take up lots of space that could otherwise be put to better use, and they are costly to build. Of course, given enough resources, anything can be done, but then that's independent of how many dimensions you're working in.

Would it be just that much easier for each store to have its own parking lot? or would we still have malls?

Would there still be a utility toward a centralized downtown?

I don't think these questions depend very much on the number of dimensions you're in. Keep in mind the only difference between the dimensions is that higher dimensions get to pack more into a space of a given radius. Considerations of cost, convenience, etc., is IMHO invariant w.r.t. the number of dimensions you have. And judging by how modern 3D cities are laid out, it seems that people have a preference for putting as much as possible in as little space as possible---probably because it reduces the time needed to get from A to B. So I'd expect 4D cities to be a lot more packed than 3D cities, but still have more or less analogous density distributions.

[wendy]

A road network in four dimensions is relatively easy to imagine.

Houses could still be layed out on blocks like they are now, but the frontage of the street would be a relatively tiny affair. A two-directional street would run as a pipe in the corner of the room. Three quarters of the garden would be back-yard stuff.

Even so, one does not have to cross the street. The postie could go along a spiral around the road (unaffected by traffic), dropping his letters off.

One could then also beach the cars somewhere off the sprialing or straight footpath, so that people can move along footpaths without having to climb over the beached cars, or crat-slips (driveways).

The average distance to the front might be slightly longer (eg 1,414*), but you can reduce this to one by hexagonal-prism blocks.

Main roads would be typically seven or nineteen-lane affairs, these in each direction. This is partly because we are talking with delivering volume from a 3D space, not from a 2D space, and the ground-horizon is 2D, not 1D. So 7 and 19 lane highways are like our four and six lane things in this world.

None the same, a main road would not greatly impact surburbia. A great road would dangle through the suburbs like a string in so much wheat.

The density of roads and predestrian traffic would be heavier in the city, but we could filter out, easily, busses and taxies into their own networks.

One can easily effect "four-leaf-clover" style turnarounds without chewing up much space. A given road, for example, may act as a common segment to a several four-leaf-clover style arrangements, so one can leave in any one of eight or ten different directions, including where you came from.

A railway station could have its various platforms accessable from street level, one could wonder around all of the tracks without using foot-brideges or subways. One would not have to replicate every service on every platform, one could have a newsagent on the platform, and people waiting on any track could still visit it. Such services would be more, rather than less, likely.

While one could still build underground, the railways in the suburbs, like the city, would not divide the surrounding comminities, nor would they have to be bundled with the roads.

[quickfur]

[...]Even so, one does not have to cross the street. The postie could go along a spiral around the road (unaffected by traffic), dropping his letters off.

Yeah, 4D is rather convenient that way, in that intricate networks of 1D lines never have to intersect (be it roads or computer cables--I wish I were in 4D so that I don't have to deal with the tangle of wires under my desk all the time).

One could then also beach the cars somewhere off the sprialing or straight footpath, so that people can move along footpaths without having to climb over the beached cars, or crat-slips (driveways).

Which brings up an interesting point. With pedestrians, we don't have the problem of dangerous collisions, so why not use the entire 2D surface surrounding the road as a walkway? (Approximating the road as a tube.) It could be a pavement that surrounds the road, and cars can park along its inner surface so people can walk about freely with no need to climb over anything.

The average distance to the front might be slightly longer (eg 1,414*), but you can reduce this to one by hexagonal-prism blocks.

Could you elaborate? I'm not sure I see how hexagonal prisms shorten the distance.

Main roads would be typically seven or nineteen-lane affairs, these in each direction. This is partly because we are talking with delivering volume from a 3D space, not from a 2D space, and the ground-horizon is 2D, not 1D. So 7 and 19 lane highways are like our four and six lane things in this world.

Any particular reason for the hexagonal layout?

None the same, a main road would not greatly impact surburbia. A great road would dangle through the suburbs like a string in so much wheat.

Yep. In fact, assuming that rural towns would spring up around intersections of cross-country roads, you could have communities in a roughly spherical layout around the intersection. You won't even need to build bridges or anything since you could just walk around the roads.

The density of roads and predestrian traffic would be heavier in the city, but we could filter out, easily, busses and taxies into their own networks.

Now that's an interesting idea. Assuming that the cost of building and maintaining separate networks is irrelevant, this could be very convenient indeed. You could have parallel roads, one for the private cars, one for the public buses, one for railway trains, etc., and they can be completely parallel to each other and never intersect. So you could hop on a bus, get off, walk around the bus road to the private car road, get in a friend's car, and continue your way, get off and hop on a train, etc., all this without any bridges or extra walkways.

I can see how train stations and bus terminals no longer have to be separate services. It would be rather attractive practically and economically to build interchanges where buses and trains stop at the same place, and passengers can simply walk around the road or rail to get on another vehicle.

[wendy]

Could you elaborate? I'm not sure I see how hexagonal prisms shorten the distance.

Look at a square and hexagonal tiling. The roads would go up through some of the vertices, the houses would sit in the middle of a tile.

Because the angles are bigger in the hexagon, you can get up really close to the corner where the road goes along.

It could be a pavement that surrounds the road, and cars can park along its inner surface so people can walk about freely with no need to climb over anything.

The pavement is the road over here. One walks on the footpath. The idea is to beach cars into a parking lane, rather than have them scattered around the cylinder.

Any particular reason for the hexagonal layout?

No. It just looks different. Just trying to get away from cubes.

[Rkyeun]

Road layout is actually pretty easy to imagine in 3D. The fact that we assume one of their four dimensions is a gravity makes it simple. We design a road that magnetic cars could use like in Extreme-G or F-Zero, where the track can fly and bend and twist in odd ways. And in their world, they're not flying at all, they're laying flat on the ground.

Let us imagine our cars flying to the 3D up (we will assume that gravity in 4D pulls cars kata for the purpose of simplifying our model, keeping them stuck on the ana surface of the roadway) through the center of a hollow cylindrical road. They can be on the inside or the outside, but for fun let's pretend it's inside. You can lanechange to the right and go all the way around the cylinder. When you reach the intersection, smaller tubes branch off from the larger tube, bending to the north, south, east, and west, probably with a few lanes between them to allow motorists to continue to move to the up if they don't want to take that exit.

Remember though that each of these roads, being roughly linear in shape, is servicing a much larger number of cars. An office building which would contain 27 people to use the road in 3D contains in 4D 81 people wanting to use that road. An office building holding 1000 people in 3D sends out 10000 people to 4D lunch. The roads would likely need additional interior lanes, service roads for the service roads, or some form of planar interchange I fail to fathom currently due to some limitations of my model. And for fun, we can have people walking on the sidewalk that is on the outside of the tubes.

I'm not sure how accurate this particular model is, I can visualize most 4D structures clearly but this one gives me a headache.