4D countryside

[quickfur]

It's been a while since we had fresh speculation on living in tetraspace. So here's some.

In 3D, the horizon is a single line across the field of vision, dividing the sky (which we could regard as the "upper half" of the 2D image of the countryside) from the ground (the lower half of the image). Roads that lead far ahead will approach this horizontal line. A lateral road, running from left to right, would always intersect with roads that stretch forward to the horizon.

In 4D, the horizon is a plane, and divides the sky from the ground. The sky would project to the upper half-space of the 3D image of the countryside, and the ground to the lower half-space. Roads leading far ahead will approach this horizontal plane at some point. A lateral road, running across the field of vision, can easily avoid intersecting with any road that stretches forward to the horizon; indeed, they can spiral around the road and wander off without ever touching.

In 3D, a mountain in the distance projects to a roughly triangular shape protruding from the horizon, and buildings on its slopes project within the triangular outline.

In 4D, a mountain in the distance would project to a roughly conical shape protruding from the horizon, and buildings on its slopes to vertical columns within the conical volume.

Buildings in 3D along the roadside project to trapezoidal shapes: the front of a store facing the forward-stretching road, for example, projects to a trapezoid rising from the slanted edge of the road. The side of the building projects to an adjoining trapezoid.

Buildings in 4D along the roadside would project to frustum shapes; the front of a store facing the forward-stretching road would project to a frustum with its underside tracing the slanted face-edge of the road. The side of the building projects to an adjoining frustum.

The windows of a 3D building project to tetragonal outlines within the trapezoidal outline of the building, and a person standing behind the window has the bottom part of his body truncated by the bottom edge of the window. The edges of the window are on the left and right, bordering the tetragon.

The windows of a 4D building project to cuboidal hollows within the frustum shapes of the building's "cells"; a person standing behind the window has the bottom part of his body truncated by the bottom face of the window. The sides of the window are four faces bordering the cuboid: left, right, ana, kata, joined in a cycle.

The little lane branching off the main road and leading to the front door of the 3D building projects to a little rectangular extrusion from the trapezoidal image of the road, reaching to the open front door, and being truncated by the far edge of the doorway's trapezoidal outline.

In 4D, the little lane is a trapezoidal prism branching off the main road, and leading to the front door of the building, and being truncated by the far face of the doorway's frustum-shaped outline.

In 3D, the only way for two buildings on either side of the road to join to each other is by an overpass or an underground tunnel. The road divides the buildings into the two groups: those on the left, and those on the right.

In 4D, the road does not divide the city block; two buildings on opposite sides of the road can easily join to each other at ground level by an extension that simply wraps around the road. Buildings could even spiral around the road while remaining exactly by the roadside.

In 3D, the road has two side, where two pavements may be paved.

In 4D, the road has one side, a hollow tube (polygonal or otherwise) that completely surrounds it. A single pavement in the shape of a tube is sufficient. In fact, in 4D, you never ever need to cross the road! You can always walk around it. So crosswalks and pedestrian lights are completely unnecessary. Drivers don't need to expend road rage on jaywalkers, because there are none! (At least, no sane ones... the only jaywalkers would be those who want to get hit. Actual effort is required for them to end up in the middle of a road!)

In 4D, a single building can completely surround a road on all sides, yet the sky is still visible.

In 3D, to find the entrance to a building you simply have to walk in a circle around the building (assuming it's not attached to another building) and you're bound to reach it eventually.

In 4D, even if you know the entrance is on this side of the building, it may not be easy to find it, since you need to cover a 2D area in order to locate it!

In 3D, when viewed from a distance, the front door is usually a rectangular in the middle of the front face of the building, with windows on either side.

In 4D, the front door is a cuboid in the middle of the front cuboid of the building, with windows all around it!

This abundance of windows, of course, hints at the amount of space there is in a 4D building. A single storey 3D building with rows of 4 windows along its walls has the area of 16 rooms (assuming four walls and one window per room). A 4D building with rows of 4 windows along its walls has 16 windows per facet, and it has six walls. So that makes a total of 96 windows: 96 rooms, and only one storey with at most 4 windows in a row!

Looking down the road again, the ocean next to the mountain projects to left of the mountain, and the forest projects to the right. This completely fills up the horizon in 3D.

In 4D, the ocean next to the mountain projects to one side of the mountain, and the forest projects to the other, but they do not cover the horizon: there is still the "front" and "back" sides (in the 3D image, which I guess you could regard as ana/kata), which has room for a grassy plain and a desert on the opposite side to all fit into the same scene!

Furthermore, in 3D, the river that flows from the forest-side of the mountain to the ocean crosses the road, so a bridge is needed.

In 4D, the river flowing from the forest-side of the mountain to the ocean need not even leave the horizontal plane; it can flow "around" the mountain to the ocean, totally missing the road! And even if its tributary does flow past the vicinity of the road, no bridge is needed: the road simply winds around the river.

Hopefully by now you have a better idea of the awesomeness of living in tetraspace.

[Keiji]

It's been a while since we had fresh speculation on living in tetraspace. So here's some.

Wow, this is nostalgic

In 4D, the road has one side, a hollow tube (polygonal or otherwise) that completely surrounds it. A single pavement in the shape of a tube is sufficient. In fact, in 4D, you never ever need to cross the road! You can always walk around it. So crosswalks and pedestrian lights are completely unnecessary. Drivers don't need to expend road rage on jaywalkers, because there are none! (At least, no sane ones... the only jaywalkers would be those who want to get hit. Actual effort is required for them to end up in the middle of a road!)

This is awesome.

I'd always thought about a 4D road as having a square cross-section, not a circular one, but I guess that makes more sense. What would we do about dividing it up into lanes, though? Perhaps we should have groups of one-lane cylindrical roads, with pavement occupying all space in-between?

In 4D, even if you know the entrance is on this side of the building, it may not be easy to find it, since you need to cover a 2D area in order to locate it!

Well, they better had be well signposted then!

(This makes me want to devise a 3D script for 4D people to read and write, since I'm interested into conlanging too XD)

This abundance of windows, of course, hints at the amount of space there is in a 4D building. A single storey 3D building with rows of 4 windows along its walls has the area of 16 rooms (assuming four walls and one window per room). A 4D building with rows of 4 windows along its walls has 16 windows per facet, and it has six walls. So that makes a total of 96 windows: 96 rooms, and only one storey with at most 4 windows in a row!

You mean 64 rooms, right?

Looking down the road again, the ocean next to the mountain projects to left of the mountain, and the forest projects to the right. This completely fills up the horizon in 3D.

In 4D, the river flowing from the forest-side of the mountain to the ocean need not even leave the horizontal plane; it can flow "around" the mountain to the ocean, totally missing the road! And even if its tributary does flow past the vicinity of the road, no bridge is needed: the road simply winds around the river.

Already documented, but still awesome.

[quickfur]

It's been a while since we had fresh speculation on living in tetraspace. So here's some.

Wow, this is nostalgic

A bit of nostalgia every now and then can't hurt.

[...]I'd always thought about a 4D road as having a square cross-section, not a circular one, but I guess that makes more sense. What would we do about dividing it up into lanes, though? Perhaps we should have groups of one-lane cylindrical roads, with pavement occupying all space in-between?

Well, it doesn't need to be square or circular... for a while I was speculating on hexagonal roads with triangular lanes. Triangular lanes aren't ideal, though, since you want a more convex cross section to avoid adjacent vehicles from scratching into each other (sleepy drivers overdriving their lanes, e.g.). So another way is to have cylindrical lanes in a "bundle", say a hexagonal bundle, with the central part of the road as a cylindrical "island". I was thinking of leaving the central open as a passing lane, but it seemed a bit too dangerous (since any vehicle from the adjacent 6 lanes may need to use it at the same time: formula for accidents), so I decided that changing lanes should only happen around the circumference.

For local roads, 6 lanes per road seems excessive, so it could be reduced to 2 cylindrical lanes, one in each direction. For major thoroughfares, the 6 lanes can be divided into 3 lanes per direction.

Another neat feature in 4D is that highways/freeways don't need on/off ramps; an adjacent local road can simply connect to it at ground level and the two never have to intersect. These connecting roads can twist in complicated 3D shapes to join each direction on the local road to each direction on the highway, without ever needing to leave ground level.

In 4D, even if you know the entrance is on this side of the building, it may not be easy to find it, since you need to cover a 2D area in order to locate it!

Well, they better had be well signposted then!

Indeed.

(This makes me want to devise a 3D script for 4D people to read and write, since I'm interested into conlanging too XD)

Ooh, another conlanger! I dabbled in some ideas for a 4D conlang, but never really got very far. One thing is for sure: a 4D mouth has so many more ways of making new sounds. For example, there could be several different lateral consonants, since you could either have a lateral with gaps on two opposite sides of the tongue, or you could have a laxer lateral with gaps all around the tongue.

This abundance of windows, of course, hints at the amount of space there is in a 4D building. A single storey 3D building with rows of 4 windows along its walls has the area of 16 rooms (assuming four walls and one window per room). A 4D building with rows of 4 windows along its walls has 16 windows per facet, and it has six walls. So that makes a total of 96 windows: 96 rooms, and only one storey with at most 4 windows in a row!

You mean 64 rooms, right?

No, it's 96: there are 16 per wall, and there are 6 walls, so 16*6 = 96.

Looking down the road again, the ocean next to the mountain projects to left of the mountain, and the forest projects to the right. This completely fills up the horizon in 3D.

In 4D, the river flowing from the forest-side of the mountain to the ocean need not even leave the horizontal plane; it can flow "around" the mountain to the ocean, totally missing the road! And even if its tributary does flow past the vicinity of the road, no bridge is needed: the road simply winds around the river.

Already documented, but still awesome.

Another line of speculation concerns facial expressions... I know we've speculated on the number of facial cavities in the past, as well as whether it would be most expedient to have 3 eyes or more in order to get a good amount of parallax for depth perception; but I don't know if we've considered much the far expanded possibilities of facial language in 4D.

For example, frown lines on the brow in 3D become frown plane in 4D; when you're in 4D, you can really frown! Also, the edge of your eyelids would be 2-manifolds, so a far greater variety of eye-narrowings, starings, etc., are possible. You could narrow your eyes in only 1 dimension (like we can in 3D), or you could narrow your eyes in 2 dimensions (for a much meaner look). There are also so many more ways to roll your eyes!

A 4D mouth would have lips with 2-manifold edges, so you could make a much larger variety of mouth shapes. (Think of pouting in full 3D... you could be really bitter!) A 4D mouth would also probably have a 2D area of teeth (yet in 4D they only serve as a ridge, so it's not as though the mouth is filled with teeth). You could have a really teethy smile! Of course, that also means there are so many more teeth to take care of, and much more opportunity for painful toothaches!

[Keiji]

Perhaps we should have groups of one-lane cylindrical roads, with pavement occupying all space in-between?

Did you notice this?

I think it'd be a good idea, as it'd put pedestrian space between all lanes. Not just pedestrian space, but it could also be used for lights, signs etc.

Well, it doesn't need to be square or circular... for a while I was speculating on hexagonal roads with triangular lanes. Triangular lanes aren't ideal, though, since you want a more convex cross section to avoid adjacent vehicles from scratching into each other (sleepy drivers overdriving their lanes, e.g.).

And this exact problem would be avoided with my idea =p Pairs of lanes could alternate between connecting to each other and not for switching over, so that there was only ever a maximum of 2 lanes connected widthwise at any point along the road.

For local roads, 6 lanes per road seems excessive, so it could be reduced to 2 cylindrical lanes, one in each direction. For major thoroughfares, the 6 lanes can be divided into 3 lanes per direction.

With all the ridiculous amount of space for things, including people, in 4D, do you really think 3 lanes per direction would be sufficient?

I've seen five lanes per direction in Singapore and that was still crowded! In 4D you'd probably need to square that number for the busiest roads!

Another neat feature in 4D is that highways/freeways don't need on/off ramps; an adjacent local road can simply connect to it at ground level and the two never have to intersect. These connecting roads can twist in complicated 3D shapes to join each direction on the local road to each direction on the highway, without ever needing to leave ground level.

Well, yes, this is kind of obvious and easy to think about if you just picture the footprint of the road layout in a realm, similarly to drawing a 3D road layout on paper.

Ooh, another conlanger! I dabbled in some ideas for a 4D conlang, but never really got very far. One thing is for sure: a 4D mouth has so many more ways of making new sounds. For example, there could be several different lateral consonants, since you could either have a lateral with gaps on two opposite sides of the tongue, or you could have a laxer lateral with gaps all around the tongue.

I think we would have to invent a hypothetical 4D anatomy before we could even start making a 4D conlang.

You mean 64 rooms, right?

No, it's 96: there are 16 per wall, and there are 6 walls, so 16*6 = 96.

There are 96 windows. I'm trying to correct that you said there are 96 rooms when there should be just 64.

[quickfur]

Perhaps we should have groups of one-lane cylindrical roads, with pavement occupying all space in-between?

Did you notice this?

I think it'd be a good idea, as it'd put pedestrian space between all lanes. Not just pedestrian space, but it could also be used for lights, signs etc.

But what about switching lanes?

Well, it doesn't need to be square or circular... for a while I was speculating on hexagonal roads with triangular lanes. Triangular lanes aren't ideal, though, since you want a more convex cross section to avoid adjacent vehicles from scratching into each other (sleepy drivers overdriving their lanes, e.g.).

And this exact problem would be avoided with my idea =p Pairs of lanes could alternate between connecting to each other and not for switching over, so that there was only ever a maximum of 2 lanes connected widthwise at any point along the road.

Well, OK, I was just thinking of the most trivial extensions of 3D road layout; if you want to design an efficient road system, that's a different kettle o' fish.

For starters, you'd be dealing with dividing the area of a given road cross-section, say a circle or some other shape, and then study different ways of laying out lanes to maximize usefulness. "Usefulness", of course, would be measured by throughput and ease of use, ease of use being to accomodate commonly needed maneuvers such as changing lanes, changing direction, making turns, etc..

As far as throughput is concerned, it is just a matter of expanding the number of lanes, but one can imagine that there would be economic constraints here, since one does have to consider the amount of ashphalt (or whatever the 4D equivalent would be) needed to build the road, keeping in mind that since this is 4D, this amount, per unit road length, is proportional to the square of the road radius, whereas in 3D, it is only a linear dependence on the road width.

As for changing lanes, it seem desirable to take advantage of the additional degree of freedom in 4D to have maximal overlap, so that one can get from one lane to any other lane (going in the same direction), as far as is possible.

One also has to consider road safety rules for intersections; keeping in mind that intersections aren't really necessary since we can simply designate a particular lane as the turning lane, and have it fork off the road and join with a lane in an adjacent road. In other words, traffic lights are probably superfluous. (Indeed, having any intersection at all seems to be the recipe for accidents, since we have n lanes intersecting with m lanes on both roads, and there are so many ways to turn---better just join the roads with arterial turning lanes than having an actual intersection. No bridges are necessary so the cost of building the road system this way seems minimal.)

For local roads, 6 lanes per road seems excessive, so it could be reduced to 2 cylindrical lanes, one in each direction. For major thoroughfares, the 6 lanes can be divided into 3 lanes per direction.

With all the ridiculous amount of space for things, including people, in 4D, do you really think 3 lanes per direction would be sufficient?

True, with the population density potentially growing to the fourth power, we probably have a lot of traffic congestion. Having said that, though, if this is really that pressing an issue, we could resort to planar pavements (extending in two directions). I expect that pedestrain traffic would be best handled this way, with vehicular traffic still confined to linear roads to avoid hazardous situations of vehicles trying to go in all 360° directions on the same road surface.

I've seen five lanes per direction in Singapore and that was still crowded! In 4D you'd probably need to square that number for the busiest roads!

Keeping in mind that the number of possible lanes (assuming roughly equal lane cross-section area) increases quadratically as road radius increases, this problem shouldn't be that hard to solve. If you have, say, a hexagonal road with 6 lanes, if you expand it with one lane-width on each side, you have an additional 12 lanes. Expand it by another lane-width on each side, and you have another 18 lanes, making a total of 36 lanes. Is that enough for you?

Another neat feature in 4D is that highways/freeways don't need on/off ramps; an adjacent local road can simply connect to it at ground level and the two never have to intersect. These connecting roads can twist in complicated 3D shapes to join each direction on the local road to each direction on the highway, without ever needing to leave ground level.

Well, yes, this is kind of obvious and easy to think about if you just picture the footprint of the road layout in a realm, similarly to drawing a 3D road layout on paper.

Yes, which is the best way to think about this.

Of course, this will be slightly more complicated when you're talking about downtown Tetronia, where traffic congestion is so bad they need bridges...

Ooh, another conlanger! I dabbled in some ideas for a 4D conlang, but never really got very far. One thing is for sure: a 4D mouth has so many more ways of making new sounds. For example, there could be several different lateral consonants, since you could either have a lateral with gaps on two opposite sides of the tongue, or you could have a laxer lateral with gaps all around the tongue.

I think we would have to invent a hypothetical 4D anatomy before we could even start making a 4D conlang.

Yeah, here we run into the problem that human anatomy really is quite specific to 3D, and there's really no unique generalization to 4D. So there's a lot of leeway in 4D anatomy, and everyone is bound to have their own ideas.

Having said that, though, sometimes I find myself wishing to invent an English-like language which has features more conducive to talking about things in 4D. The problem with using an existing natlang is that spatial terms are specific to 3D and hard to generalize to 4D; but on the other hand, if all we want is to talk about 4D layouts, it seems to be an overkill to have to deal with anatomy, culture, and history as well.

You mean 64 rooms, right?

No, it's 96: there are 16 per wall, and there are 6 walls, so 16*6 = 96.

There are 96 windows. I'm trying to correct that you said there are 96 rooms when there should be just 64.

Right, there are only 64 rooms, not 96. (I hate arithmetic!)

It would be interesting to think about corridor layout, since to reach these 64 rooms with merely linear corridors may not be as good as using some 2D "wide" corridors as well. One can imagine an efficient layout might be to use a 2D "wide" corridor slicing across the floor, with 1D "narrow" corridors branching off to reach the rooms. This allows one to get from one room to another without excessively winding routes.

Having said that, a 4D castle might be deliberately designed with a difficult-to-navigate corridor system (or lack of system thereof), to allow for obscure passages and other such things that 3D castles of old are renowned for.

[Keiji]

Perhaps we should have groups of one-lane cylindrical roads, with pavement occupying all space in-between?

Did you notice this?

I think it'd be a good idea, as it'd put pedestrian space between all lanes. Not just pedestrian space, but it could also be used for lights, signs etc.

But what about switching lanes?

Well, as I said further on in that post, pairs of lanes would connect periodically.

Well, it doesn't need to be square or circular... for a while I was speculating on hexagonal roads with triangular lanes. Triangular lanes aren't ideal, though, since you want a more convex cross section to avoid adjacent vehicles from scratching into each other (sleepy drivers overdriving their lanes, e.g.).

And this exact problem would be avoided with my idea =p Pairs of lanes could alternate between connecting to each other and not for switching over, so that there was only ever a maximum of 2 lanes connected widthwise at any point along the road.

Well, OK, I was just thinking of the most trivial extensions of 3D road layout; if you want to design an efficient road system, that's a different kettle o' fish.

For starters, you'd be dealing with dividing the area of a given road cross-section, say a circle or some other shape, and then study different ways of laying out lanes to maximize usefulness.

Not necessarily, I still like my "bundle of cylinders" method

"Usefulness", of course, would be measured by throughput and ease of use, ease of use being to accomodate commonly needed maneuvers such as changing lanes, changing direction, making turns, etc..

As far as throughput is concerned, it is just a matter of expanding the number of lanes, but one can imagine that there would be economic constraints here, since one does have to consider the amount of ashphalt (or whatever the 4D equivalent would be) needed to build the road, keeping in mind that since this is 4D, this amount, per unit road length, is proportional to the square of the road radius, whereas in 3D, it is only a linear dependence on the road width.

But you also need to notice that in 4D, there is a ridiculous amount of material available too.

As for changing lanes, it seem desirable to take advantage of the additional degree of freedom in 4D to have maximal overlap, so that one can get from one lane to any other lane (going in the same direction), as far as is possible.

That's also an idea, but this would be very dangerous for drivers - you'd have to watch for traffic turning into your lane from SIX other lanes, not two.

One also has to consider road safety rules for intersections; keeping in mind that intersections aren't really necessary since we can simply designate a particular lane as the turning lane, and have it fork off the road and join with a lane in an adjacent road. In other words, traffic lights are probably superfluous. (Indeed, having any intersection at all seems to be the recipe for accidents, since we have n lanes intersecting with m lanes on both roads, and there are so many ways to turn---better just join the roads with arterial turning lanes than having an actual intersection. No bridges are necessary so the cost of building the road system this way seems minimal.)

Yes, it is much better to avoid having intersections. Even in tight spaces, you can quite easily have two perpendicular roads offset from each other in the third horizontal and add eight tiny sliproads without taking up very much space at all.

For local roads, 6 lanes per road seems excessive, so it could be reduced to 2 cylindrical lanes, one in each direction. For major thoroughfares, the 6 lanes can be divided into 3 lanes per direction.

With all the ridiculous amount of space for things, including people, in 4D, do you really think 3 lanes per direction would be sufficient?

True, with the population density potentially growing to the fourth power, we probably have a lot of traffic congestion. Having said that, though, if this is really that pressing an issue, we could resort to planar pavements (extending in two directions). I expect that pedestrain traffic would be best handled this way, with vehicular traffic still confined to linear roads to avoid hazardous situations of vehicles trying to go in all 360° directions on the same road surface.

Hmm...

In 3D, things are obvious because the three dimensions are allocated frontal, lateral, vertical, with vertical discrete (thus we usually ignore it). Frontal dimensions allow movement and lateral dimensions allow placement. If you have no frontal dimensions you can't move because you are trapped where you are, but if you have no lateral dimensions you have nowhere to go because the entire world is one big walkway.

Now in 4D, there are two ways to do this: you can either have two frontal and one lateral dimension, or you can have one frontal and two lateral dimensions.

What you suggest is the first case. You can move in 360 degrees for as long as you like without bumping into something (unless you get to a dead end!).

What I suggest (for whole roads) is the second case. You only move forwards or backwards, and buildings are all around you, making more use of space (not like we're short of it in 4D, though!).

However, within my "bundle of cylinders" model, the lanes are 1f2l and the pavements are 2f1l. It's impossible to have both be 2f1l, as then the pavement and lanes would occupy the same space, and if both were 1f2l, there would be empty space, though this could be occupied with buildings too.

Now in 3D, we sometimes have enclosed areas of 2f0l, such as car parks, usually in the city centres, with residential parking just on the side of the road or in garages (these are equivalent though). In 4D, I'd thus expect residential areas to use the 1f2l model, with city centres using the 2f1l model. Since there's a 1l in there, it means the entire car park can be on the side of the road! 4D may go into 3f0l packing if it really needs the space (such as seats in a theater), but this would probably be rare.

I've seen five lanes per direction in Singapore and that was still crowded! In 4D you'd probably need to square that number for the busiest roads!

Keeping in mind that the number of possible lanes (assuming roughly equal lane cross-section area) increases quadratically as road radius increases, this problem shouldn't be that hard to solve. If you have, say, a hexagonal road with 6 lanes, if you expand it with one lane-width on each side, you have an additional 12 lanes. Expand it by another lane-width on each side, and you have another 18 lanes, making a total of 36 lanes. Is that enough for you?

Definitely, thanks.

Another neat feature in 4D is that highways/freeways don't need on/off ramps; an adjacent local road can simply connect to it at ground level and the two never have to intersect. These connecting roads can twist in complicated 3D shapes to join each direction on the local road to each direction on the highway, without ever needing to leave ground level.

Well, yes, this is kind of obvious and easy to think about if you just picture the footprint of the road layout in a realm, similarly to drawing a 3D road layout on paper.

Yes, which is the best way to think about this.

Of course, this will be slightly more complicated when you're talking about downtown Tetronia, where traffic congestion is so bad they need bridges...

Would bridges really be necessary, even in such places?

Ooh, another conlanger! I dabbled in some ideas for a 4D conlang, but never really got very far. One thing is for sure: a 4D mouth has so many more ways of making new sounds. For example, there could be several different lateral consonants, since you could either have a lateral with gaps on two opposite sides of the tongue, or you could have a laxer lateral with gaps all around the tongue.

I think we would have to invent a hypothetical 4D anatomy before we could even start making a 4D conlang.

Yeah, here we run into the problem that human anatomy really is quite specific to 3D, and there's really no unique generalization to 4D. So there's a lot of leeway in 4D anatomy, and everyone is bound to have their own ideas.

Having said that, though, sometimes I find myself wishing to invent an English-like language which has features more conducive to talking about things in 4D. The problem with using an existing natlang is that spatial terms are specific to 3D and hard to generalize to 4D; but on the other hand, if all we want is to talk about 4D layouts, it seems to be an overkill to have to deal with anatomy, culture, and history as well.

Still, it makes for another topic of interest, right?

You mean 64 rooms, right?

No, it's 96: there are 16 per wall, and there are 6 walls, so 16*6 = 96.

There are 96 windows. I'm trying to correct that you said there are 96 rooms when there should be just 64.

Right, there are only 64 rooms, not 96. (I hate arithmetic!)

It would be interesting to think about corridor layout, since to reach these 64 rooms with merely linear corridors may not be as good as using some 2D "wide" corridors as well. One can imagine an efficient layout might be to use a 2D "wide" corridor slicing across the floor, with 1D "narrow" corridors branching off to reach the rooms. This allows one to get from one room to another without excessively winding routes.

Having said that, a 4D castle might be deliberately designed with a difficult-to-navigate corridor system (or lack of system thereof), to allow for obscure passages and other such things that 3D castles of old are renowned for.

Hmm, yes...

What about a multi-storey building?

[quickfur]

[...]

[...]
Well, OK, I was just thinking of the most trivial extensions of 3D road layout; if you want to design an efficient road system, that's a different kettle o' fish.

For starters, you'd be dealing with dividing the area of a given road cross-section, say a circle or some other shape, and then study different ways of laying out lanes to maximize usefulness.

Not necessarily, I still like my "bundle of cylinders" method

In the end, I think we're really talking about the same thing; the hexagonal layout I had in mind has 6 lanes surrounding a central island, which is pretty much isomorphic to a bundle of 7 cylinders with the middle cylinder as an island.

[...]As far as throughput is concerned, it is just a matter of expanding the number of lanes, but one can imagine that there would be economic constraints here, since one does have to consider the amount of ashphalt (or whatever the 4D equivalent would be) needed to build the road, keeping in mind that since this is 4D, this amount, per unit road length, is proportional to the square of the road radius, whereas in 3D, it is only a linear dependence on the road width.

But you also need to notice that in 4D, there is a ridiculous amount of material available too.

Yes, but it also takes the same ridiculous amount of work to quarry that material and process it into something useful for building roads. One could argue that we also have a ridiculous amount of people in 4D, but somebody's gotta pay them to do the job!

As for changing lanes, it seem desirable to take advantage of the additional degree of freedom in 4D to have maximal overlap, so that one can get from one lane to any other lane (going in the same direction), as far as is possible.

That's also an idea, but this would be very dangerous for drivers - you'd have to watch for traffic turning into your lane from SIX other lanes, not two.

Well, the practicality of this really depends on how many eyes we want Tetronians to have.

[...]
True, with the population density potentially growing to the fourth power, we probably have a lot of traffic congestion. Having said that, though, if this is really that pressing an issue, we could resort to planar pavements (extending in two directions). I expect that pedestrain traffic would be best handled this way, with vehicular traffic still confined to linear roads to avoid hazardous situations of vehicles trying to go in all 360° directions on the same road surface.

Hmm...

In 3D, things are obvious because the three dimensions are allocated frontal, lateral, vertical, with vertical discrete (thus we usually ignore it). Frontal dimensions allow movement and lateral dimensions allow placement. If you have no frontal dimensions you can't move because you are trapped where you are, but if you have no lateral dimensions you have nowhere to go because the entire world is one big walkway.

Now in 4D, there are two ways to do this: you can either have two frontal and one lateral dimension, or you can have one frontal and two lateral dimensions.

What you suggest is the first case. You can move in 360 degrees for as long as you like without bumping into something (unless you get to a dead end!).

What I suggest (for whole roads) is the second case. You only move forwards or backwards, and buildings are all around you, making more use of space (not like we're short of it in 4D, though!).

This makes sense. Generally, I'd assume vehicles are more prone to accidents, and thus it is advantageous to constrain their possible directions of motion. For pedestrians, you really don't have that problem (you could accidentally bump into somebody I suppose, but it's hardly fatal), and furthermore 2D surfaces in 4D occupy very little room, so it seem advantageous to allow 2 degrees of freedom on pavements and walkways.

However, within my "bundle of cylinders" model, the lanes are 1f2l and the pavements are 2f1l. It's impossible to have both be 2f1l, as then the pavement and lanes would occupy the same space, and if both were 1f2l, there would be empty space, though this could be occupied with buildings too.

It's not really confined to your bundle of cylinders model; what I envisioned originally was to have the pavement surround the road, which essentially makes it a cylindrical sheet wrapping around the roadside. One of the 2 dimensions wraps around, of course, but I don't think that's a problem, since it can easily connect with other walkways that go around buildings and the like.

Now in 3D, we sometimes have enclosed areas of 2f0l, such as car parks, usually in the city centres, with residential parking just on the side of the road or in garages (these are equivalent though). In 4D, I'd thus expect residential areas to use the 1f2l model, with city centres using the 2f1l model. Since there's a 1l in there, it means the entire car park can be on the side of the road! 4D may go into 3f0l packing if it really needs the space (such as seats in a theater), but this would probably be rare.

I would expect 3f0l packing for parkades, actually, because it reduces the maximal extent of the parkade (think, 10x10x10 cubic meter parking lot as opposed to 100x10 square meter; the former only extends 10m in each direction, whereas the latter requires 10 times that size in one of its directions, but their effective capacity is the same). I'd expect land to be allocated in more or less cubic units of hyper-area, so making maximal use of all 3 dimensions of one's lot seems most reasonable.

Furthermore, even though there is a lot of space available, one does have to keep in mind that the hyper-area of building floor space is roughly proportional to the number of occupants in the building, and that the number of occupants is roughly proportional to the number of vehicles, so as the size of the building increases, the size of the parking lot must necessarily increase with the same power as well. Since building capacity increases to the fourth power of the size along each dimension, I doubt a parking lot whose area only increases to the second power would be sufficient to meet the needs of the occupants.

[...]Of course, this will be slightly more complicated when you're talking about downtown Tetronia, where traffic congestion is so bad they need bridges...

Would bridges really be necessary, even in such places?

Well, if downtown Tetronia is filled with skyscrapers whose floor hyper-area is proportional to H*W^3 (where H is the number of floors and W is the average size in the remaining 3 dimensions), then you might need multi-storey roads just to cope with the sheer amount of traffic.

[...]Having said that, though, sometimes I find myself wishing to invent an English-like language which has features more conducive to talking about things in 4D. The problem with using an existing natlang is that spatial terms are specific to 3D and hard to generalize to 4D; but on the other hand, if all we want is to talk about 4D layouts, it seems to be an overkill to have to deal with anatomy, culture, and history as well.

Still, it makes for another topic of interest, right?

The thing about conlangs is that a significant part of a language's vocabulary and grammar is not specific to 4D, but only to the culture/philosophy of its speakers. The fact that they live in 4D space mainly affects only spatial terms and expressions, and indirectly small parts of culture such as knotting sheets and the like. As far as other aspects of language are concerned, such as abstract concepts and perceptions, they are really independent of 4D space. Any decision on the language in these areas are arbitrary; you pretty much have as much freedom as creating an Earth-based conlang in these aspects. Which means that two people creating a 4D conlang are likely to come up with completely different languages.

[...]Having said that, a 4D castle might be deliberately designed with a difficult-to-navigate corridor system (or lack of system thereof), to allow for obscure passages and other such things that 3D castles of old are renowned for.

Hmm, yes...

What about a multi-storey building?

Castles are multi-storeyed! Now imagine a 4D castle deliberately designed with labyrinthian corridors that connect between floors... If the Minoans had known of 4D, Perseus probably would never have found the Minotaur at all!

[Keiji]

Not necessarily, I still like my "bundle of cylinders" method

In the end, I think we're really talking about the same thing; the hexagonal layout I had in mind has 6 lanes surrounding a central island, which is pretty much isomorphic to a bundle of 7 cylinders with the middle cylinder as an island.

Well, the difference is my method has pavement between every lane, whereas your method has a central island (which cannot be used as pavement as it is essentially a 4D central res).

Although, come to think of it, I'm thinking your method may actually work well with some slight modifications - if we separate the hexagon into halves, the island is now accessible and can be used as pavement, and pavement also splits the two carriageways. Here's a crude mspaint diagram.

In this situation, which lane should we designate fastest? I'm thinking of using the middle lane of each triplet as the fastest lane, as that's the furthest from both the opposing carriageway and the pavement, but that would mean that the fast lane has traffic merging on both sides, whereas in 3D, we tend to structure our lanes from slowest to fastest so that the fastest lane has only traffic merging from left.

[...]As far as throughput is concerned, it is just a matter of expanding the number of lanes, but one can imagine that there would be economic constraints here, since one does have to consider the amount of ashphalt (or whatever the 4D equivalent would be) needed to build the road, keeping in mind that since this is 4D, this amount, per unit road length, is proportional to the square of the road radius, whereas in 3D, it is only a linear dependence on the road width.

But you also need to notice that in 4D, there is a ridiculous amount of material available too.

Yes, but it also takes the same ridiculous amount of work to quarry that material and process it into something useful for building roads. One could argue that we also have a ridiculous amount of people in 4D, but somebody's gotta pay them to do the job!

Communist, utilitarian, despotist state?

I guess you have a point, but 4D technology would also be much more advanced so factories could easily produce materials faster, probably in a higher supply:demand ratio than our 3D ones.

As for changing lanes, it seem desirable to take advantage of the additional degree of freedom in 4D to have maximal overlap, so that one can get from one lane to any other lane (going in the same direction), as far as is possible.

That's also an idea, but this would be very dangerous for drivers - you'd have to watch for traffic turning into your lane from SIX other lanes, not two.

Well, the practicality of this really depends on how many eyes we want Tetronians to have.

Not just eyes, but also the brain's ability to cope with multiple things going on at once.

Now in 3D, we sometimes have enclosed areas of 2f0l, such as car parks, usually in the city centres, with residential parking just on the side of the road or in garages (these are equivalent though). In 4D, I'd thus expect residential areas to use the 1f2l model, with city centres using the 2f1l model. Since there's a 1l in there, it means the entire car park can be on the side of the road! 4D may go into 3f0l packing if it really needs the space (such as seats in a theater), but this would probably be rare.

I would expect 3f0l packing for parkades, actually, because it reduces the maximal extent of the parkade (think, 10x10x10 cubic meter parking lot as opposed to 100x10 square meter; the former only extends 10m in each direction, whereas the latter requires 10 times that size in one of its directions, but their effective capacity is the same).

Well, as I said, the idea of using 2f1l is for fitting the entire car park on the roadside. You could have a gigantic car park next to a shopping centre, and park your car and take 20 seconds to walk to the shop without having to navigate round a parking lot first.

I'd expect land to be allocated in more or less cubic units of hyper-area, so making maximal use of all 3 dimensions of one's lot seems most reasonable.

That would be volume

Furthermore, even though there is a lot of space available, one does have to keep in mind that the hyper-area of building floor space is roughly proportional to the number of occupants in the building, and that the number of occupants is roughly proportional to the number of vehicles, so as the size of the building increases, the size of the parking lot must necessarily increase with the same power as well. Since building capacity increases to the fourth power of the size along each dimension, I doubt a parking lot whose area only increases to the second power would be sufficient to meet the needs of the occupants.

Hmm, true. But, it would be perhaps a better ratio than our one-dimensional parking next to two-dimensional buildings (assuming they don't go deep behind the road).

[...]Of course, this will be slightly more complicated when you're talking about downtown Tetronia, where traffic congestion is so bad they need bridges...

Would bridges really be necessary, even in such places?

Well, if downtown Tetronia is filled with skyscrapers whose floor hyper-area is proportional to H*W^3 (where H is the number of floors and W is the average size in the remaining 3 dimensions), then you might need multi-storey roads just to cope with the sheer amount of traffic.

...Wow.

You mean Chris Sawyer's Locomotion has some sense to it?

[...]Having said that, though, sometimes I find myself wishing to invent an English-like language which has features more conducive to talking about things in 4D. The problem with using an existing natlang is that spatial terms are specific to 3D and hard to generalize to 4D; but on the other hand, if all we want is to talk about 4D layouts, it seems to be an overkill to have to deal with anatomy, culture, and history as well.

Still, it makes for another topic of interest, right?

The thing about conlangs is that a significant part of a language's vocabulary and grammar is not specific to 4D, but only to the culture/philosophy of its speakers. The fact that they live in 4D space mainly affects only spatial terms and expressions, and indirectly small parts of culture such as knotting sheets and the like. As far as other aspects of language are concerned, such as abstract concepts and perceptions, they are really independent of 4D space. Any decision on the language in these areas are arbitrary; you pretty much have as much freedom as creating an Earth-based conlang in these aspects. Which means that two people creating a 4D conlang are likely to come up with completely different languages.

I think we should work together on a 4D conlang!

[quickfur]

Not necessarily, I still like my "bundle of cylinders" method

In the end, I think we're really talking about the same thing; the hexagonal layout I had in mind has 6 lanes surrounding a central island, which is pretty much isomorphic to a bundle of 7 cylinders with the middle cylinder as an island.

Well, the difference is my method has pavement between every lane, whereas your method has a central island (which cannot be used as pavement as it is essentially a 4D central res).

Although, come to think of it, I'm thinking your method may actually work well with some slight modifications - if we separate the hexagon into halves, the island is now accessible and can be used as pavement, and pavement also splits the two carriageways. Here's a crude mspaint diagram.

That could work, yes. It does have the nice property of also serving the purpose of dividing lanes of opposing traffic.

In this situation, which lane should we designate fastest? I'm thinking of using the middle lane of each triplet as the fastest lane, as that's the furthest from both the opposing carriageway and the pavement, but that would mean that the fast lane has traffic merging on both sides, whereas in 3D, we tend to structure our lanes from slowest to fastest so that the fastest lane has only traffic merging from left.

I suppose that shouldn't be a big problem, although we could augment the structure by adding a fourth lane only attached to the middle lane, which is used exclusively for very fast travel. It could be sorta like a "cruise lane" which you work yourself into when you first get on the road, and then after cruising at high speed, you slowly work your way back out (we could say exits only connect to the slower lanes).

[...]But you also need to notice that in 4D, there is a ridiculous amount of material available too.

Yes, but it also takes the same ridiculous amount of work to quarry that material and process it into something useful for building roads. One could argue that we also have a ridiculous amount of people in 4D, but somebody's gotta pay them to do the job!

Communist, utilitarian, despotist state?

I guess you have a point, but 4D technology would also be much more advanced so factories could easily produce materials faster, probably in a higher supply:demand ratio than our 3D ones.

Ah, but you're dealing with polynomial powers here. Efficiency in supply/demand ratio really only buys you a constant factor, but a difference in degree between two polynomials (in this case, x^4 vs. x^3) quickly grows out of control (i.e., K*x^3 grows significantly slower than K*x^4, no matter how small K is). So at best, an efficient supply/demand ratio can only hold out temporarily; it doesn't scale.

As for changing lanes, it seem desirable to take advantage of the additional degree of freedom in 4D to have maximal overlap, so that one can get from one lane to any other lane (going in the same direction), as far as is possible.

That's also an idea, but this would be very dangerous for drivers - you'd have to watch for traffic turning into your lane from SIX other lanes, not two.

Well, the practicality of this really depends on how many eyes we want Tetronians to have.

Not just eyes, but also the brain's ability to cope with multiple things going on at once.

I'd expect Tetronians to have a much keener ability to cope with multiple things going on, seeing, as they do, a 2D horizon with so much more variety than we trionians are used to, and being immersed from youth in an environment where building floor plans are fully 3D with all its complexity.

But you do have a point that if there are too many things going on at the same time, it becomes impracticably dangerous, such as 6 lanes merging into one. Trying to shoulder check six lanes before making a move is just recipe for pure disaster.

[...]
I would expect 3f0l packing for parkades, actually, because it reduces the maximal extent of the parkade (think, 10x10x10 cubic meter parking lot as opposed to 100x10 square meter; the former only extends 10m in each direction, whereas the latter requires 10 times that size in one of its directions, but their effective capacity is the same).

Well, as I said, the idea of using 2f1l is for fitting the entire car park on the roadside. You could have a gigantic car park next to a shopping centre, and park your car and take 20 seconds to walk to the shop without having to navigate round a parking lot first.

Thinking about this again, I note that the ratio of a square's perimeter to its area is smaller than the ratio of a cube's area to its volume. This means that you can fit a lot more parked vehicles around the base of the building before you start needing a dedicated parking block.

As for the time to walk from car to building, I think 3f0l is actually better: most buildings would probably have only a few entrances (otherwise you have an excessive number of doors, which could be a security concern), so a parking lot confined to a cubical hyper-area (yes, yes, I'm avoiding the term "volume" because it could be misconstrued to mean 4D bulk), say 10x10x10, has a shorter average walk time than, say, a 100x10 parking area (you may end up walking 50m to get to the entrance).

I'd expect land to be allocated in more or less cubic units of hyper-area, so making maximal use of all 3 dimensions of one's lot seems most reasonable.

That would be volume

See why I wish we could just extend English with 4D spatial vocabulary?

[...]You mean Chris Sawyer's Locomotion has some sense to it?

Did I hear someone say "4D Locomotion"?

[...]I think we should work together on a 4D conlang!

Here's a better idea: we both come up with a 4D conlang, then put them into the same 4D world and have them interact with each other. (Here's to eliminating that boring single-climate, single-race, single-culture, single-facial-appearance, single-language trope in too many alien planet/fantasy scenarios! Heck, even good ole Earth has more variety than that, and we're talking about the infinite variety in space here.)

[Keiji]

In this situation, which lane should we designate fastest? I'm thinking of using the middle lane of each triplet as the fastest lane, as that's the furthest from both the opposing carriageway and the pavement, but that would mean that the fast lane has traffic merging on both sides, whereas in 3D, we tend to structure our lanes from slowest to fastest so that the fastest lane has only traffic merging from left.

I suppose that shouldn't be a big problem, although we could augment the structure by adding a fourth lane only attached to the middle lane, which is used exclusively for very fast travel. It could be sorta like a "cruise lane" which you work yourself into when you first get on the road, and then after cruising at high speed, you slowly work your way back out (we could say exits only connect to the slower lanes).

Ah, that would be a really good idea. So, it'd look like this slightly better crude kolourpaint diagram then.

Ah, but you're dealing with polynomial powers here. Efficiency in supply/demand ratio really only buys you a constant factor, but a difference in degree between two polynomials (in this case, x^4 vs. x^3) quickly grows out of control (i.e., K*x^3 grows significantly slower than K*x^4, no matter how small K is). So at best, an efficient supply/demand ratio can only hold out temporarily; it doesn't scale.

Well, in any case, the point is that their ratio would grow while ours is pretty much constant.

As for the time to walk from car to building, I think 3f0l is actually better: most buildings would probably have only a few entrances (otherwise you have an excessive number of doors, which could be a security concern), so a parking lot confined to a cubical hyper-area (yes, yes, I'm avoiding the term "volume" because it could be misconstrued to mean 4D bulk), say 10x10x10, has a shorter average walk time than, say, a 100x10 parking area (you may end up walking 50m to get to the entrance).

I'm thinking of town center streets which are composed of a lot of small shops in a row. Sure, for enormous supermarkets, there wouldn't be many doors, and there would be a 3f0l car park, but 2f1l parking in high streets would remove or at least reduce the need for multi-storey carparks around town.

[...]I think we should work together on a 4D conlang!

Here's a better idea: we both come up with a 4D conlang, then put them into the same 4D world and have them interact with each other. (Here's to eliminating that boring single-climate, single-race, single-culture, single-facial-appearance, single-language trope in too many alien planet/fantasy scenarios! Heck, even good ole Earth has more variety than that, and we're talking about the infinite variety in space here.)

Well, I wouldn't attempt this on my own - there are a lot of practical effects ( ) that we have to take into account and it's easy for one person to ignore the differences sometimes.

[quickfur]

[...]
I suppose that shouldn't be a big problem, although we could augment the structure by adding a fourth lane only attached to the middle lane, which is used exclusively for very fast travel. It could be sorta like a "cruise lane" which you work yourself into when you first get on the road, and then after cruising at high speed, you slowly work your way back out (we could say exits only connect to the slower lanes).

Ah, that would be a really good idea. So, it'd look like this slightly better crude kolourpaint diagram then.

Hmm, the shape of the lanes got me thinking. What kind of wheels would 4D vehicles have, and how would turning me achieved? This would probably dictate what sort of shapes are best for lanes (I haven't thought about it this way before).

If you had spherindrical wheels, they would have the unpleasant property of being able to roll along any direction in the plane, which means that they are hard to control. (The axle would only be able to constrain their motion to a plane, but not to a specific direction in the plane.)

If you had cubindrical wheels, they would only roll straight ahead, which is good for vehicles. Cubindrical wheels can be turned on two axes to effect a curving motion along two cardinal directions, so this in effect gives us the possibility of turning left/right or ana/kata. However, these two directions of turning are perpendicular, which means that it is difficult to orient the vehicle on the road: the orientation of the vehicle relative to the cross-section of the road will be more or less fixed, so that to turn into a lane located diagonally from the current lane will require turning the wheel on two axes simultaneously.

It is conceivable, of course, that the axles would be mechanically designed so that the driver simply has to specify the direction to turn in, and the mechanism will automatically apply the right proportion of turning to each axis. However, we still have the problem that the driver can't easily re-orient the vehicle to a more comfortable viewpoint (more on this later). And it may also be desirable to allow the wheels to only turn on one axis at a time (perhaps older vehicles have this limitation), and so one needs to orient the vehicle correctly before one can turn into the next lane with the right orientation.

One conceivable way to allow a vehicle to change orientation is by "twisting" the front wheels from the back wheels. Assume for simplicity that vehicles have 8 wheels, located at the corners of an elongated cuboid, and that the direction of motion (relative to us 3D observers) is upwards. If you rotate the top face of the cuboid slightly, so that the top 4 wheels are skewed, the vehicle can be brought into a twisting motion that will change its orientation but not its direction of travel.

The upshot of this is that 4D vehicles not only can turn in 3D, but they can also "twist". So drivers have to be able to master not just turning, but also "twisting" (or orienting).

Now, as far as orienting is concerned, note that in 4D, you can stand upright and stare in the same direction, and twist your head around at the same time! This extra orientation is analogous to looking straight ahead and turning your head sideways on either shoulder, except that in 4D, you can still keep your head vertical. In terms of projections, this reorientation does not change what you can see, but it does rotate the 3D image formed on the retina in the horizontal plane. I don't think there's necessarily a "preferred" orientation in this sense, but a driver may find it more comfortable to align his vehicle orientation-wise with the lane he's turning into, instead of turning on a corner, so to speak.

This also opens up a new vista of facial expressions: you could, in effect, shake your head without having to look from side to side! This is quite distinct from shaking your head in the other plane, which does change your direction of sight. These two could form the basis of two different facial gestures. (Don't you wish you had a 4D face? Think of the eloquent expressiveness of it! )

Ah, but you're dealing with polynomial powers here. Efficiency in supply/demand ratio really only buys you a constant factor, but a difference in degree between two polynomials (in this case, x^4 vs. x^3) quickly grows out of control (i.e., K*x^3 grows significantly slower than K*x^4, no matter how small K is). So at best, an efficient supply/demand ratio can only hold out temporarily; it doesn't scale.

Well, in any case, the point is that their ratio would grow while ours is pretty much constant.

Yes.

As for the time to walk from car to building, I think 3f0l is actually better: most buildings would probably have only a few entrances (otherwise you have an excessive number of doors, which could be a security concern), so a parking lot confined to a cubical hyper-area (yes, yes, I'm avoiding the term "volume" because it could be misconstrued to mean 4D bulk), say 10x10x10, has a shorter average walk time than, say, a 100x10 parking area (you may end up walking 50m to get to the entrance).

I'm thinking of town center streets which are composed of a lot of small shops in a row. Sure, for enormous supermarkets, there wouldn't be many doors, and there would be a 3f0l car park, but 2f1l parking in high streets would remove or at least reduce the need for multi-storey carparks around town.

True.

[...]
Here's a better idea: we both come up with a 4D conlang, then put them into the same 4D world and have them interact with each other. (Here's to eliminating that boring single-climate, single-race, single-culture, single-facial-appearance, single-language trope in too many alien planet/fantasy scenarios! Heck, even good ole Earth has more variety than that, and we're talking about the infinite variety in space here.)

Well, I wouldn't attempt this on my own - there are a lot of practical effects ( ) that we have to take into account and it's easy for one person to ignore the differences sometimes.

The 4D part of the conlang is pretty much determined by the spatial properties of 4-space, so this part we can work out together. One may even say that as far as spatial relationships are concerned, any 4D conlang will express them in more or less the same way. The differences are in the parts that aren't directly related to spatial relationships.

[Keiji]

Stuff about wheels

Let me just check if I'm making sense of this properly, first.

The simplest analog to 3D is spherindrical wheels. There would be four axles, each with a pair of spherindrical wheels attached. The four axles would be positioned on four parallel edges of the cube as our two axles are positioned on two parallel edges of a rectangle.

For cubindrical wheels, we'd need to attach each wheel to TWO axles. Therefore there would be two sets of eight axles, each axle in a set being on an edge of a square, the two squares being two opposite faces of a cube. Each wheel would be at a vertex of the cube, attached to the axles corresponding to the adjacent edges.

As you mentioned, a spherindrically-wheeled car (what an awesome constructed adjective ) would have steering difficulties, and a cubindrically-wheeled car could not be reoriented without lifting it up from the road. So, I present two possibilities:

1. What about if the vehicle had two independently-turnable axles, each with a pair of spherindrical wheels? Each would constrain the car to the ground-plane perpendicular to it, so the movement of the car itself would be constrained to the two planes' linear intersection, i.e. the ground-line perpendicular to the plane containing both axles' axes.

2. A car could be built in the shape of a cubinder, with cubindrical wheels. The intersection of the car with a horizontal realm would be a cylinder. The cubindrical wheels would be attached as described above, but there would be a mechanism to rotate the inside of the car independently of the outer body. Thus a driver can orient his vision how he sees fit, and he can turn the car in two different dimensions at once. Alternatively, a mechanical or computer-based system could be used to automatically choose the turning direction based on the angle of the inner body.

The 4D part of the conlang is pretty much determined by the spatial properties of 4-space, so this part we can work out together. One may even say that as far as spatial relationships are concerned, any 4D conlang will express them in more or less the same way. The differences are in the parts that aren't directly related to spatial relationships.

Wouldn't everything be somewhat related, though? Even, say, colours or sounds would be affected by dimensionality.

[quickfur]

Stuff about wheels

Let me just check if I'm making sense of this properly, first.

The simplest analog to 3D is spherindrical wheels. There would be four axles, each with a pair of spherindrical wheels attached. The four axles would be positioned on four parallel edges of the cube as our two axles are positioned on two parallel edges of a rectangle.

Correct.

For cubindrical wheels, we'd need to attach each wheel to TWO axles.[...]

Not really. You still only need one axle per wheel. For steering, you just have a joint (or joints) in the axle that allows it to turn in two dimensions. (Think of how steering is achieved in a 3D car.)

[...]As you mentioned, a spherindrically-wheeled car (what an awesome constructed adjective ) would have steering difficulties, and a cubindrically-wheeled car could not be reoriented without lifting it up from the road.

Actually, I don't think you need to lift it off the road; you just need the right twisting motion by slightly changing the skew between the front and back wheels. Imagine if you will a 3D cylinder with 4 wheels on each end, enclosed inside a hollow tube, and moving upwards. If the top 4 wheels are slightly rotated in the same sense (clockwise/counterclockwise), the resulting motion will be spiralling. Now extrude this setup in the 4th direction, and you have something that can reorient itself on a 4D road (the wheels extrude into cubinders, and so does the body, but of course the body can be a base on which the 4D vehicular body is mounted).

So, I present two possibilities:

1. What about if the vehicle had two independently-turnable axles, each with a pair of spherindrical wheels? Each would constrain the car to the ground-plane perpendicular to it, so the movement of the car itself would be constrained to the two planes' linear intersection, i.e. the ground-line perpendicular to the plane containing both axles' axes.

This is an interesting way to make spherindrical wheels to move only in one dimension. Another way might be to constrain the axle itself so that it only turns one way. Think of a 3D car with spherical wheels, if you will: the fact that the wheels are spherical doesn't necessarily mean they will move in all 2 dimensions; it depends on whether the axle (or whatever they are attached to) allows them to roll like that. If you constrain the sphere to only roll in 1 dimension, then the car will still only move in 1 dimension, even though the wheels in themselves can roll in 2 dimensions.

In 4D, there are two kinds of axles: cubindrical and spherindrical. A cubindrical axle connected to a spherindrical wheel constrains it to only roll in 1 dimension. A spherindrical axle can have ridges or other such features that can be connected to locking mechanisms that constrain it to roll in 1 dimension or permit it to roll in 2 dimensions. By designing the appropriate mechanism, it may be possible to control this at the driver's will.

By comparison, though, a cubindrical wheel is much easier to design: you don't need complicated mechanisms to keep things moving in the right direction.

We could even have a combination of cubindrical and spherindrical wheels: cubindrical for the back wheels so that they always follow the path set by the front wheels, and spherindrical front wheels with the complex steering mechanism to control where they go.

2. A car could be built in the shape of a cubinder, with cubindrical wheels. The intersection of the car with a horizontal realm would be a cylinder. The cubindrical wheels would be attached as described above, but there would be a mechanism to rotate the inside of the car independently of the outer body. Thus a driver can orient his vision how he sees fit, and he can turn the car in two different dimensions at once. Alternatively, a mechanical or computer-based system could be used to automatically choose the turning direction based on the angle of the inner body.

This is certainly possible, but I would imagine it would be more efficient to achieve re-orientation by twisting the wheels, than to build the equivalent of a platform mounted on a ring (like a tank turret in 3D). The ease of re-orientation may also not be that desirable, since the driver may get confused by continual changes in orientation (of course, this can be designed to only happen in controlled cases).

The 4D part of the conlang is pretty much determined by the spatial properties of 4-space, so this part we can work out together. One may even say that as far as spatial relationships are concerned, any 4D conlang will express them in more or less the same way. The differences are in the parts that aren't directly related to spatial relationships.

Wouldn't everything be somewhat related, though? Even, say, colours or sounds would be affected by dimensionality.

That depends on whether you assume Tetronians see more colors or not, but that really is an independent choice (it can go both ways). Colors and sounds are just waves, and while it is true that 4D waves can be much more complex, the same can be said for 3D waves w.r.t. 1D waves; yet our ears mainly hear in 1D only. (There is a little bit of 2D in spatial location, in that the brain uses discrepancies in the 1D waves to deduce the origin of the sound.) Color is really just the resonance of our retinal receptors to certain frequencies; incoming light carries a particular frequency, and different frequencies causes different levels of resonance in the R, G, B receptors. Frequency is independent of dimension! So we really aren't hearing in more than 1 dimension, and the number of perceived colors depends on the number of different types of receptors, not the dimensionality of space. (Some animals see in ultraviolet, for example; if they also see in RGB, then they could be said to see 4D color, but this 4D is not the same as 4D space.)

[Keiji]

For cubindrical wheels, we'd need to attach each wheel to TWO axles.[...]

Not really. You still only need one axle per wheel. For steering, you just have a joint (or joints) in the axle that allows it to turn in two dimensions. (Think of how steering is achieved in a 3D car.)

Oh, right.

Would it be possible to attach two axles to a wheel and spin it?

Actually, I don't think you need to lift it off the road; you just need the right twisting motion by slightly changing the skew between the front and back wheels.

Oooh, that's a brilliant idea!

Then, the best solution is probably to take my second original suggestion, but use this wheel-skewing to re-orient the car instead of having an inner body.

Wouldn't everything be somewhat related, though? Even, say, colours or sounds would be affected by dimensionality.

That depends on whether you assume Tetronians see more colors or not, but that really is an independent choice (it can go both ways). Colors and sounds are just waves, and while it is true that 4D waves can be much more complex, the same can be said for 3D waves w.r.t. 1D waves; yet our ears mainly hear in 1D only. (There is a little bit of 2D in spatial location, in that the brain uses discrepancies in the 1D waves to deduce the origin of the sound.) Color is really just the resonance of our retinal receptors to certain frequencies; incoming light carries a particular frequency, and different frequencies causes different levels of resonance in the R, G, B receptors. Frequency is independent of dimension! So we really aren't hearing in more than 1 dimension, and the number of perceived colors depends on the number of different types of receptors, not the dimensionality of space. (Some animals see in ultraviolet, for example; if they also see in RGB, then they could be said to see 4D color, but this 4D is not the same as 4D space.)

Oh, forgive me, I actually forgot how sound waves were longitudinal there.

As for light, though - light (electromagnetic) waves in 3D take up all three dimensions in a 1f2l structure, with the "electric" part on one lateral and the "magnetic" part on the other lateral. What would take up the third lateral in 4D? Similarly, electromagnetism has the three perpendicular lines of force, field and current in a 3f0l structure; what would take up the fourth frontal?

However, I note that sound is 3f0l; as a one-dimensional longitudinal wave is immersed in three dimensions it spreads out and follows an inverse cube law. Perhaps this would mean that EM waves would spread themselves over a plane (by becoming 2f2l instead of 1f3l) when immersed in 4D? This would unfortunately make light a lot weaker. However, it doesn't answer the question of how electromagnetism would work.

[quickfur]

For cubindrical wheels, we'd need to attach each wheel to TWO axles.[...]

Not really. You still only need one axle per wheel. For steering, you just have a joint (or joints) in the axle that allows it to turn in two dimensions. (Think of how steering is achieved in a 3D car.)

Oh, right.

Would it be possible to attach two axles to a wheel and spin it?

Yes, think of it this way: a cubinder is bounded by 4 cylinders, in two pairs of cylinders on parallel planes. Where adjacent pairs meet, you have 2 cylinders along perpendicular axes. So you could attach two axles to your wheel in the same manner, by aligning them perpendicular to each other. The wheel will then roll in the same direction the cubinder can (remember that the cubinder rolls on its torus side, not on its cylindrical surfaces! The cylindrical surfaces are flat in 4D.) Keep in mind that this constrains the rotational motion to only lie in the plane orthogonal to the plane spanned by these two axes. (This might be one way of constraining a spherinder to only roll in one direction, for example.)

Actually, I don't think you need to lift it off the road; you just need the right twisting motion by slightly changing the skew between the front and back wheels.

Oooh, that's a brilliant idea!

Then, the best solution is probably to take my second original suggestion, but use this wheel-skewing to re-orient the car instead of having an inner body.

Right, this is what I had in mind before.

Having said that, this does mean that the simplest road layout might actually be the one with the ole boring square-tiling cross-section (so that drivers can switch lanes without needing to reorient).

[...]Oh, forgive me, I actually forgot how sound waves were longitudinal there.

As for light, though - light (electromagnetic) waves in 3D take up all three dimensions in a 1f2l structure, with the "electric" part on one lateral and the "magnetic" part on the other lateral. What would take up the third lateral in 4D? Similarly, electromagnetism has the three perpendicular lines of force, field and current in a 3f0l structure; what would take up the fourth frontal?

Heh, so we're getting into 4D electromagnetism here, aren't we? I'm not sure I want to worry about this quite yet; for macroscopic purposes, we could simply subscribe to the particle model of light. Well, or the "wave packet" model, so that light carries frequency. Electromagnetism, or whatever the 4D analog might be, is another line of research that I'm not sure we want to get into yet, if all we want is to construct a working society!

However, I note that sound is 3f0l; as a one-dimensional longitudinal wave is immersed in three dimensions it spreads out and follows an inverse cube law. Perhaps this would mean that EM waves would spread themselves over a plane (by becoming 2f2l instead of 1f3l) when immersed in 4D? This would unfortunately make light a lot weaker. However, it doesn't answer the question of how electromagnetism would work.

The inverse-square law for light really only applies when we're talking about light in bulk, and then only where individual photons are travelling in all possible directions initially. A photon individually travels in a straight line, and does not diminish over distance (otherwise galaxies would be invisible!); what does diminish is the density of photons per unit volume as you move farther away from the original point source. If the source was a laser, the inverse-square law doesn't apply (or at least, its effect is much less noticeable), since most of the photons in a laser travel in (almost) the same direction.

I guess this does mean that 4D lights disperse much faster, so more lights would be necessary to light up any given area. But if we use spotlights or other more concentrated means of illumination, then we can get away with needing less light (which means more efficient power consumption, necessary for advanced society to work!).

But if you're talking about field strength and the inverse-square law, then that's a different kettle o' fish. There is currently no known reason for the inverse-square law other than the hypothesis that field charge is carried by virtual particles, and so the density of these particles degrade according to an inverse-square law over the distance from the point of field interaction (one could conceivably imagine that an interaction in the field involves the emission of virtual particles in all directions, some of which reach the other party in the interaction; in this case, the inverse square law can be thought of as the scattering of virtual particles over the surface of a sphere as the distance increases). If we assume this hypothesis, then field strength in 4D would degrade according to an inverse cube law. But if the inverse square law is really caused by something else, then 4D may not be subject to an inverse cube law after all... in which case planets and stable orbits are possible!! (Buahahaha... )

[Keiji]

Whoops, I really messed up there...

I meant to say that sound follows an inverse square law in 3D as it is 3f0l, and light follows a constant law in 3D (considering laser light only) as it is 1f2l.

If sound and light became 4f0l and 2f2l respectively in 4D, then sound would follow an inverse cube law and laser light would follow a inverse linear law.

I actually did not even mention field strength; field strength would probably take an inverse cube law.

But on the subject of electron orbits, do you remember the discussions about coiled orbits?

[quickfur]

Whoops, I really messed up there...

I meant to say that sound follows an inverse square law in 3D as it is 3f0l, and light follows a constant law in 3D (considering laser light only) as it is 1f2l.

If sound and light became 4f0l and 2f2l respectively in 4D, then sound would follow an inverse cube law and laser light would follow a inverse linear law.

I doubt light would become 2f2l; it just seems too arbitrary. Assuming the particle/wave packet model, I'd imagine individual photons will travel in a straight line, so no inverse laws for laser light there, unless, of course, you emit a 2D laser (light constrained to spread along a plane).

I actually did not even mention field strength; field strength would probably take an inverse cube law.

I did a little more readup after I wrote about field strength, and I think an inverse cube law seems more likely now. According to field theory, the inverse square law arises as a consequence of a divergence of 0 over the vector field. So in 4D, the same would give rise to an inverse cube law.

But on the subject of electron orbits, do you remember the discussions about coiled orbits?

Vaguely, yes... but I am more inclined to consider electron orbitals as standing waves than jellybeans flying in circles. True, we've proven in the past that the Schröedinger equation in 4D has no non-trivial solutions for a 4D hydrogen atom, but that depends on whether you want to import quantum mechanics with all of its strangeness into the 4D world. As for me, I'm perfectly happy with treating electron orbitals as standing waves without needing to explain how it could be so. I don't know if I'm ready to reinvent a con-universe just so we can make 4D atoms.

(Aside from the 4D-ness of the 4D world, there really isn't any reason to assume that physics as we know it should carry over without changes. We could play 4D physicists, I suppose, but that's something different from modelling how daily life in 4D would work!)

P.S. Is there anything wrong with simply adopting 4D Newtonian mechanics?

[Keiji]

I doubt light would become 2f2l; it just seems too arbitrary. Assuming the particle/wave packet model, I'd imagine individual photons will travel in a straight line, so no inverse laws for laser light there, unless, of course, you emit a 2D laser (light constrained to spread along a plane).

Well, there are only two possibilities: 2f2l, or 1f3l. 1f3l requires light to have something else on the third lateral, and 2f2l requires it to spread over a plane (regardless of whether you try to make it be a line or not).

I don't know if I'm ready to reinvent a con-universe just so we can make 4D atoms.

I am

[wendy]

I spent a good deal of time in 4d countryside, mainly to watch the mathematics.

WHEELS

If you think of the wheel in the B.C. Cartoon sense, (where the characters stand on the axle), then you of course want rotation in the up/forward direction. Consider what happens if the characters started to rotate on the axle. The thing would be horrible to steer because what's in front is rotating! Wheels, in the main, follow 'great arrow' rotation, with N-2 axis fixed.

SEASONS

If you ever watched the sky, you should understand that all stars go straight across the sky. With a little understanding and following all this, you will see that the seasons are always there (like the time zones), and that there are "season zones" as well. A typical summer's day (like 25 Dec), here on earth, might fall in Winter in the northern hemisphere (which are typically six months behind us), but in 4d, it can fall at any time, for example, the beginning of spring, or in the middle of autumn.

At the polar and equator circles, the sun rises to a certian height every day )eg 23.5 deg and 66.5 degrees respectively. Its only in the tropics that the sun gets to the zenith. In the tropic of Leo, the sun does it in july/august. There are tropics for all months: the tropic of aquarius. In the artic, there are artics for all seasons too: the artic of leo is perpendicular to the tropic of leo, the sun hugs the horizon all day here during July.