Could you have more than one building in the same spot by "stacking" them in 4th dimensional space?

In my universe a 4th SPACIAL dimension exists around our 3 dimensions. I'm playing with ideas for a society that has access to a 4th spacial dimension and am currently thinking of what they would be able to construct. My thought process is this:

Say you are a 3D being and have a square 2D hotel in a 2D universe. You are thinking of adding an expansion but all available room near your hotel is occupied. So you build a whole new hotel and push it up one foot on top of your original hotel and install a way for the 2D people to move up and down between hotels. To them it would seem like they hadn't moved at all in 2D space and the hotel just changed around them.

Is it possible for multiple 3D buildings to occupy the same location in 3D space but be "stacked" in the 4th dimension? (Under the assumption a 4th spacial dimension exists around our 3D universe.)

Edit: Added emphasis on spacial because too many people think I am asking about a temporal dimension.

• Comments are not for extended discussion; this conversation (including several recommendations for stories) has been moved to chat. Please continue the discussion there. Jul 21, 2017 at 3:32

Yes

It isn't any different to occupants of a 2D world asking whether - if there were 3 dimensions - could we stack objects on top of each other in a third dimension?

One thing to note, however, is that if the fourth spatial dimension exists then the buildings must have some size in the fourth dimension. As long as you space them accordingly along this spatial dimension - just as you would any of our three - then you can 'stack' the buildings in this way.

• Yeah so you need your buildings arranged along the fourth dimension just as you would if your hotel was comprised of 2D floors but each floor was separated in the third dimension. Jul 17, 2017 at 16:20
• @immibis: You don't say. Jul 18, 2017 at 0:45
• Dont use any non allocated space in any dimension. It will segfault. Jul 18, 2017 at 8:20
• @aloisdg Or throw an exception, if it uses any flavor of managed God.NET. Jul 18, 2017 at 17:37
• The mathematicians who live in a 2D world might be able to imagine "stacking" objects in a 3D space, but if they actually are 2D beings, how could they even exist in a 3D space? Think about this: What contains their guts in the 2D world? It's their skin (i.e., their perimeter), and the fact that there's no place else their guts can go. What happens to them when you suddenly release them from the constraints of their 2D plane and give them freedom to explore three dimensions? Jul 18, 2017 at 19:05

You are the master of your world, so yes. Some practical concerns:

Your guests are only three-dimensional and would like to stay that way. How do you get them from one "hyperfloor" to the next? Well, you put them in a "hyperelevator" and move the whole room across. Peasy, easy.

Are there alternate Earths under the other "hyperfloors"? You can answer that question both yes and no resulting in different problems. Yes can mean problems with the natives of those worlds. No means there is no gravity on those floors. And you will have to make things completely airtight, and either import or recycle oxygen. (Or both)

A related question is what is the view from the windows is like.

What do you do about power failure and fire safety? The "hyperelevators" are not running, the lights are off and the guests are panicking. Your move.

You need "hyperstairs". They will probably not look like ordinary stairs. The most reasonable look is a corridor that somehow starts on one hyperfloor and ends on another. Actually, you probably want something like that for water, waste and electricity too.

Just problematic is that power failure anyway? The "hyperstructure" of the hotel... is it something you build once and then it is stable, or is it something that is more like a force field that need continuous power to maintain? Your choice, but in the force field scenario you will need an UPS (Uninterruptible Power Supply) better than the Pentagon. I am talking Tesla Batteries and Diesel Generators and Cold Fusion Generators and exercise bikes with dynamos and ...

Finally, for marketing reasons you really need to find better words than "hyperwhatever".

• love the Hyperelevator - great stuff. Jul 17, 2017 at 12:40
• I like this answer because it deals more with the practicality of it than with the sciency stuff. Jul 17, 2017 at 16:16
• I wonder if one could build a "stairwell" that as well as shifting them along their normal three dimensions, could slide them along the fourth dimension as well. To them, they would see a lobby with like 3 stairwells, each stairwell takes them to the "above" five floors, but the five floors are different for each stairwell. Very practical and fire-safe. However, I make no claims about what someone halfway up the stairs would see when looking out. Jul 18, 2017 at 0:56
• Also consider that if there's a device able to rotate things through the 4th dimension, it's possible to effectively mirror people and objects. So someone who's right-handed could go through a 4D turntable and come out left-handed. From their POV it would appear as if the rest of the universe got mirrored.
– Kyle
Jul 18, 2017 at 13:53
• @Kyle there are some interesting and subtle problems with that, such as the structure of protein molecules, all explored very nicely in Roger Zelazny's Doorways in the Sand. Jul 18, 2017 at 15:33

Initially consider it as time.

There was a building before the one you're in and a building after it. All you need to be able to do is move between those buildings.

Consider a co-ordinate system x,y,z,t. x and y are the ground level, z vertical and t your 4th.

Just as you move between your building and the one next to it, which shares your x,z,t co-ordinates, by moving in y, you can move in your 4th dimension between the buildings sharing your x,y,z co-ordinates by moving in t.

The only problem is moving in that 4th dimension when you only exist in 3.

There can be an indefinite number of stacked Flatland planes of existence, none of them are aware of the others, none of them can interact with the others or pass between them. They exist only in x and y, they have no z and no understanding of z.

But

There's always a but. Your society has a 4th spacial dimension, so they'll use it as a matter of course. Flatland has no access to the 3rd dimension so you can stack Flatland planes without a second thought and with no issues. Your society is going to have buildings that already exist and have length in this 4th dimension, you can't just casually stack them like 2D worlds in 3D, you'll have to put them alongside each other. You'll also have to work out terminology and decide whether like left and right, you can just go that way or whether like up and down, it has a force and you need to have something to support you in "flight".

Going back to the example of time as a 4th dimension, while we can't control our movement a building still has a length in time. A building may be 10metres x 5metres x 20metres x 100years, but they still exist and have a fixed length in that dimension.

• Time has a "force", it always draws us forward (like gravity pulls us down)
• We can't travel against the force without assistance (which we haven't developed yet)
• The question does specifically say a spatial dimension. Using time adds the problem of the buildings moving in time as they age - they don't necessarily move in the spatial dimension. Jul 17, 2017 at 8:19
• @FreeElk, usually someone turns up with a bulldozer ;) There are a few meters between buildings in x and y and a couple of months in time. The gaps are measured in whatever units you use to measure that dimension. Jul 17, 2017 at 8:35
• Using time as an example just makes it more confusing, because when you move through time, you still exist in the other times - i.e. if you travel back in time, you would run into yourself. This is not a problem with a spatial dimension, as no matter how you move through the 4th dimension you won't meet yourself. Jul 17, 2017 at 9:37
• The trouble is that as 3D beings it's very difficult for the average person to comprehend a direction which is at right angles to all three of the spatial dimensions we know. Hence using time, it's imperfect but at least you can wrap your brain around it. Jul 17, 2017 at 9:46
• @MooingDuck, can you clap one hand? Jul 18, 2017 at 3:07

Really it depends. Let's consider the fourth dimension to be a spatial dimension perpendicular to all other three spatial dimensions. The main problem will be this: does gravitation extend into or along that fourth spatial dimension?

If gravitation is fourth dimension, and unless the stack of buildings isn't secured, the stacked buildings will fall down, in the direction of the fourth dimension, and crash into the buildings stacked in the other three spatial dimensions.

This might look like the buildings in three dimensional space inexplicably undergoing massive damage and deformation, apparently without a cause (as the stacks of buildings falling down in four dimensions won't be observable in three dimensions).

This suggests you should four dimensional gravitation out of it. In which case, stacking extra buildings in four dimensions may be fine.

• Gravity only applies on the axis inline with the force, it doesn't apply on the other perpendicular axes. The 4th dimension may have its own force, but it won't be gravity. Jul 17, 2017 at 14:05
• Stacked in the "other three" spatial dimensions? That's not how it works, you don't get to choose a dimension. Jul 18, 2017 at 0:45
• @Separatrix what does "the axis inline with the force" mean? Jul 18, 2017 at 0:45
• @a4android, gravity acts along the z (vertical) axis, x,y and '4th' are all perpendicular to this. There's no reason why gravity should act any differently in 4th than it does relative to x and y i.e. it acts only in z. Flatland has no gravity because it has no z. Jul 18, 2017 at 9:32
• @Separatrix That's a reasonable way of looking at things in three dimensional space. The difference is I am prepared to entertain the idea that gravitational field effectively acts along the 4th dimension. Your approach is equally reasonable. It's the deplorable lack of 4d structures that prevents us from testing our ideas. It's all the fault of the universe! Jul 18, 2017 at 9:59

A good use for 4D would be to house a 3D prison. In 4D space you can achieve a vantage point where you can see all 3D points simultaneously, with no occulusion from walls. Similar to how if you stand in 3D above a 2D sheet of paper you can see the whole thing, irregardless of how many lines are on the paper that would block the view of 2D beings on that paper. While a 3D being might not be able to achieve such a 4D vantage point, they might be able to create a camera of sorts that can.

As for hotels, as hotel rooms are independent of one another (ie you don't need to see the contents of one from another) all you need in 3D space is reception, an elevator and a single hotel room. The button in the elevator room transfers the elevator to a different 4D plane to face a different room. In any 3D instance this makes your hotel very small, no need for separate floors, but it can support infinite rooms. I'm trying to figure out if the elevator room could simply rotate in 4D space to face different rooms instead of translating, might be safer or more energy efficient.

• This is assuming that light casually leaves 3D space so you can see it across all four dimensions, meaning that from our perspective, all objects rapidly lose heat and energy, and even in a vacuum light is lost from a source. I think it would be a bit of a plot hole for that to be the case. Jul 17, 2017 at 21:24
• This also assumes that there is relatively no energy cost in moving between planes. It might actually be cheaper to build many rooms than the energy requirements to move between planes. Granted, using something like an elevator paradigm there isn't much difference. But using something like a portal paradigm, the person moves from the lobby (filled with portals) and steps into a different universe. Jul 18, 2017 at 0:19
• Even better for the prison - you don't need to have any paths out of the rooms in 3D until you actually need to interact with the prisoner. Then just do a bit of four-rotation, and get access to that particular room temporarily. Good luck trying to dig your way out of that with a spoon! :P Jul 18, 2017 at 15:47

No

Unless you just handwave everything, there is no physical sense to viewing any spatial dimension "different" from any other, or to be able to travel "through" an additional dimension.

Flatland is a great example. If you take a Flatland style piece of paper (2D) and a second piece of paper, and put them on top of each other, you do not get 3D space. Topological, you are still locally 2D with some weird effects like being able to "tunnel through" between the sheets of paper by making holes and glueing the edges together, or forming a donut out of a single sheet by glueing opposite edges, or being able to flip beings by glueing a band of paper in the Moebius configuration.

You can do a little experiment: Take a sheet of paper (15x15cm in size) and draw a 10x10cm square on each with a pen. Make it one of those pens which "bleed through" the paper, and thin paper. So it seems like the drawing is really "inside" the paper, just like a Flatlander would be. Now stack many of them, until they build a 10cm high tower. You now have a 10cm high tower of 2D sheets, but what you do not, in no fashion whatsoever, have, is a 10x10x10cm 3D cube.

Thinking the other way round, no 2D Flatlander could ever "lift off" its universe to travel through 3D space, in the same sense that you could never lift off a pencil drawing from its sheet of paper. The operation would just not be defined in any useful mathematical or physical sense.

Our (real) 3D is fundamentally different from a hypothetical 2D stacked upon each other. Even if you take our greatest microscope, at no time does anything fundamentally behave in a 2D way.

A more mathematical way to see how this does not make sense is to see that all dimensions (to be called dimensions in any way making physical sense) would have to be continuous. The stacking 3rd dimension would be discrete and thus people would probably not even call it "space".

All of this scales up seamlessly to 4D.

Bending

What you absolutely could do would be to bend 3D space in creative ways. This is 100% compatible with our real world, as far as we know. This is what happens in the Flatland example, of course - stacking two sheets of 2D paper on top and glueing them is just an easier to imagine way of having a single 2D space in the first place and bending that one into a stacked configuration.

But you cannot have a (physically working) 4D space somehow "embedding" a 3D space and having meaningful travel for the 3D objects.

Multiverse

Your stacking version could also just be a multiverse as we are thinking about today. Those are perfectly happy (and basically required to be) stacking/discrete. But don't call them 4D please, either.

But...

...you can also just forget everything above, you have artistic freedom. It would just grate hugely on everybody who has a passing intuition about these things, spoiling suspension of disbelief absolutely. The broad masses won't probably notice anything, I guess. ;)

As you said it can be visualized as going from a 2D world to a 3D world : everything needs a depth in the new dimension.
To stay with the 2D/3D analogy, your squares have to become cuboids. As you can stack many slim cuboids in a cube, you can stack many 3D hotels in a 4D hotel as long as they have a very limited depth in the 4th dimension.

Note that one tricky thing is to help peoples turn along the new dimension in order to get from a 3D hotel to another... There is a game demonstrating the concept, Miegakure, I believe their videos makes the possibilities quitte clear.

• If you could flesh this out a bit more it would look more like an answer. As it is now is more of a comment and might be deleted as such.
– L.Dutch
Jul 18, 2017 at 9:41
• Wekcome to Worldbuilding SE, NestorDemeure, I agree with L.Dutch your answer needs more substance and should address the question. Answers should ahve facts, information, and reasons to support explanations. Have fun here! Jul 18, 2017 at 9:54
• I agree with you, given more reputation I would have just posted the link in a comment. I added matter to my answer in order to improve it. Jul 18, 2017 at 10:23

Imagine putting multiple flat squares on top of each other. You end up with multiple squares in the same 2D space by stacking them in the third dimension. What you are proposing is identical but going from the third to fourth dimension. The limitation you face is that perception or travel in a fourth spatial dimension is not currently possible.

So, in answer to your question: is it geometrically possible? Yes. Is it physically useful? No.

I have a problem with the answer(s) in this question, even though being an answer I thought of aswell

I'm no expert but quickly I think "yea! you can have a building in one spot, then another in the same spot but in a different time period, and have a 'transportation' device to move between them"

Buut

If we place a building in a spot, then by accessing the 4th dimension we put another building in the same spot but 5s later, what happens once the first object meets the other? Will they crash? An object will suddenly appear where the first was.

You could have buildings be separate more time, like 50 years from one another, but still. I think eventually the buildings ahead of time would have to be destroyed before hand. Placing buildings in the past would cause the same issues. It wouldn't be handy to have whole buildings I guess but rather solo rooms, easily destroyed before the 'crash'

• Hi, yeah that is a problem but this looks like a comment on other answers rather than an answer to the question. The question does specify an extra spatial dimension. Jul 19, 2017 at 19:02