3D 'slit' view into 4D world

[gonegahgah]

I think there can be misconceptions about a fourth spatial dimension and hopefully this helps.

To help clarify 4D it is safest for us to first think down to 2D in a 3D world.
This might be done by locking ourselves in a 2D plane but this is not the best way to do it.
The best way actually to do it is to put yourself in a 3D world but give yourself a 2D slit to look through.
In this way your 2D viewing slit can be moved within the 3D world.

You can see then that if you watched the moon pass your slit it would follow the standard process.
That is it would not be there then it would start from a dot and grow rapidly then slowly to the full line then shrink back in reverse.
Of course you could rotate the plane of the slit to keep the moon in view for longer too.

This also answers a question that I had about gravity.
Naturally if you are in a 3D world with a 2D viewing slit you are going to stay upright to the 3D world.
So as you move around the world your slit view will stay upright to the world; not to its original orientation.
So you won't see the world grow smaller as you move right or left until your standing on a dot.
Instead the 2D view of the world will remain the same size and the distance to the horizon will stay the same.
This will remain so even if you 'tilt' your head (and slit) sideways.

The trick then becomes to translate this to the 4D world with our 3D 'slit' view into it.
A 4D world has a centre of gravity just like a 2D and a 3D world. So you will tend to stay upright to that centre of gravity.
So the 4D world would not shrink or grow bigger as you move around it (excepting hills and valleys of course).

What does become different is enclosures.
To enclose a house in a 2D world you need a line for the floor, a line for the back and a line for the front, and a line for the roof.
To enclose a house in a 3D world you need a square for the floor, a square for the front, back and sides and a square for the roof.
So we've gone from 4 continuous lines to needing 6 filled squares.
To enclose a house in a 4D world you need a cube floor, a cube front and back, 4 'side' cubes and a roof cube.
The cubes also have to be 'solid' as otherwise any empty space anywhere inside the cube will be a door or window.

The same goes to contain water.
In a 2D world you need a front line, a back line and a bottom line.
In a 3D world you need a front side, back side, two sides and a bottom side.
In a 4D world you need a front solid, back solid, four side solids and a bottom solid all joined internally.
If you don't have those 4 sides the water will simply spill out just like if our 3D glass only had a front and back but no sides.

[gonegahgah]

Let's look at another standard structure: the roof.
Roofs are usual angled to allow rain to fall off them.

In a 2D world a roof could be 2 lines; one from font rising to centre, the second line decending again to back.
In a 3D world a roof could be 4 triangles; each rising to a centre point and joined at the edges.
In a 4D world a roof could be 6 square based pyramids; each rising to a common centre point and joined innerly to each other.

I assume that this corresponds to 2D line walls not joined, 3D square walls joined by edges, and 4D cube walls joined by their innerness.

I used inner to hopefully differentiate from inside.
Although 4D walls are joined innerly they still manage to leave an inside that is empty and separate from them.

[gonegahgah]

Oops, error alert!
The cube walls are connected by their faces; not by their insides.
Same for the 6 square pyramid roof segments.
The square base of each is connected to a 'side' cube face.
Their triangle faces are connected to 4 other faces of different pyramids each face.

[gonegahgah]

The other thing is that the cube walls are not cube thin.
Just like our 3D walls are not 2D square thin but are made up of an infinite series of stacked squares.
Our 3D walls connect at their square edges and so do the series of stacked squares that complete each side wall.

The same goes for 4D walls. The 'face' of each wall is a cube and its front insides.
Just like a square in our world has a front inside and a back inside so do cubes in a 4D world have a front inside and a back inside.
So similar to our 3D walls, the 4D thickness of the 4D wall is also an infinite series of cubes.
Each of the four side faces of each cube in the 4D thickness series connects to an adjacent corresponding face in a side wall.
There are four adjacent walls in a 4D world (as well as one roof and one floor).
The bottom face of a wall cube connects along with its series of cubes in its 4D thickness to the ground's up faces.
The top face of each outer wall cube connects to the roof along with their series of solids in the 4D thickness via their down faces.

Each wall only has so much 4D thickness (like our walls have only so much 3D thickness) which allows for the 4D explosion of cubes inside, being empty space, to reside.

[gonegahgah]

The other interesting thing this provides is the amount of space.
To accommodate 64 (same size) items a 2D room has to have a floor 64 units long.
To accommodate 64 (same size) items a 3D room has to have a floor 8 units by 8 units.
To accommodate 64 (same size) items a 4D room has to have a floor 4 units by 4 units by 4 units.

From this it would appear that in a 4D world things need only be a lot smaller than they need to be in our 3D world.
And, as with all things related to raising to different powers, this scales up a lot faster in a 4D world.

[quickfur]

[...]There are four adjacent walls in a 4D world (as well as one roof and one floor).[...]

Yes, and a hypercubic room in 4D is bounded by 6 walls.

But these 6 walls are not in a hexagonal arrangement; they are in a cubical arrangement. Which means that unlike 3D, 4D rooms have two kinds of corners, one where two walls meet at a face, and one where three walls meet in a line. (In 3D, there's only one kind of corner: where two walls meet in a line.) A 3D room has 4 corners; whereas a 4D room has 12 corners of the first kind and 8 corners of the second kind. That's a lot of corners for clutter to accumulate. I surmise that 4D housekeeping is a much more tedious task than it is in 3D. (Esp. given the vastly increased floor area per unit measurement. Think of how tedious it must be to sweep the floor.)

[quickfur]

The other interesting thing this provides is the amount of space.
To accommodate 64 (same size) items a 2D room has to have a floor 64 units long.
To accommodate 64 (same size) items a 3D room has to have a floor 8 units by 8 units.
To accommodate 64 (same size) items a 4D room has to have a floor 4 units by 4 units by 4 units.

From this it would appear that in a 4D world things need only be a lot smaller than they need to be in our 3D world.
And, as with all things related to raising to different powers, this scales up a lot faster in a 4D world.

Yes, and buildings would generally tend to be lower than in 3D, because the floor area increases so much faster, so it can accomodate more per storey. (E.g., the number of units per floor in an apartment building would be much larger, so you'll need fewer floors to fit everyone in.) Plus, due to the fact that a lot more building material is needed (walls are 3D, that means to cover an cubic area of N units you need N^3 units of wall area, so the cost of adding more walls and floors increase dramatically), and much stronger support is required to bear the weight of 4D storeys, it would seem that tall buildings would be much more expensive to build. So the tendency would be to stick with lower buildings: the increased floor area compensates for that already.

[gonegahgah]

Yes, and buildings would generally tend to be lower than in 3D, because the floor area increases so much faster, so it can accomodate more per storey. (E.g., the number of units per floor in an apartment building would be much larger, so you'll need fewer floors to fit everyone in.) Plus, due to the fact that a lot more building material is needed (walls are 3D, that means to cover an cubic area of N units you need N^3 units of wall area, so the cost of adding more walls and floors increase dramatically), and much stronger support is required to bear the weight of 4D storeys, it would seem that tall buildings would be much more expensive to build. So the tendency would be to stick with lower buildings: the increased floor area compensates for that already.

It would probably be a bit of a process to work out the overall math for this.
I wonder what the ideal relative sizes would be?

Material wise they might have an excess over us for one reason but less for another reason? I'm not sure...

Firstly, the house doesn't have to be as big on its measured lengths when it comes to holding the same number of contents as discussed.
Also, mining materials still produces a comparable amount of material when it comes to building houses whether in 2D, 3D or 4D.
Though I guess it might take longer to mine the material in the higher dimensions.

Also, the planet would probably have a smaller diameter and offer up less comparable material for that reason?

I'm not sure?

[gonegahgah]

But these 6 walls are not in a hexagonal arrangement; they are in a cubical arrangement. Which means that unlike 3D, 4D rooms have two kinds of corners, one where two walls meet at a face, and one where three walls meet in a line. (In 3D, there's only one kind of corner: where two walls meet in a line.) A 3D room has 4 corners; whereas a 4D room has 12 corners of the first kind and 8 corners of the second kind. That's a lot of corners for clutter to accumulate. I surmise that 4D housekeeping is a much more tedious task than it is in 3D. (Esp. given the vastly increased floor area per unit measurement. Think of how tedious it must be to sweep the floor.)

Cool, I never thought of that.

[gonegahgah]

As I mentioned the best way for us to contemplate what a 2Der in a 3D world sees is to build a slit to look through.
Even better may be to build special goggles that have two periscopes to rotate our vision from left/right to top/bottom with only small slits of view.
I'm guessing a 2D person would need vertically placed eyes to gauge distance.
Would be interesting for our brains to interpret this different orientation and thin information.
I wonder if our brains could learn to deal with the world that way?

Again, the 4Der has to do something similar when they try to see the world from our perspective.
They are used to looking at the complete volume of one of the two inner sides of solid faces whereas we can only see the outside of a solid.
So they need special goggles as well that reorient their 4D eyes to see in our 3D way.

[gonegahgah]

It might be interesting to have a programmed world that is 3D but we can only see a bi-2D slit view of it.
Maybe those 3D glasses could show the bottom image only to the left eye and the top image only to the right eye.
And with just this information we would have to navigate around the world.

Our vision point movements would have to be: forward/back, left/right, up/down, plus: tilt forward/back, rotate left/right, tilt left/right.
Probably the most important of those extra movements would rotate left/right so that we can build up a picture in our mind.

The slit views picture would need to be a few pixels wide or more but only one colour per line to simulate 0 thickness.

I wonder if there is anything like this already? It might almost be like being one step from blind for us...

[gonegahgah]

Firstly, the house doesn't have to be as big on its measured lengths when it comes to holding the same number of contents as discussed.
Also, mining materials still produces a comparable amount of material when it comes to building houses whether in 2D, 3D or 4D.
Though I guess it might take longer to mine the material in the higher dimensions.

Though I wonder about this too. In order to build two units of space a 2D excavator might have to mine 1 unit twice, a 3D excavator might be able to mine 2 units at once side by side, and a 3D excavator may be able to mine more again at once. So maybe construction would even be faster overall in higher dimensions?

[gonegahgah]

It's also interesting to think about a computer keyboard in all different dimensions.

In 2D a computer keyboard might be a reverse C shape to allow easier access to all those keys.
In 3D we have our rectangular keyboard with keys on top.
In 4D you would have a 'solid' with all the keys spread evenly around inside it.

Using our present visualisation techniques we would first think it was just a block.
A 4Der would see it as it truly is and that is as keyboard buttons on a 'flat' 3D 'inside~surface' that are all easily accessible from the 4th direction.

[gonegahgah]

You can imagine writing in a 4D world as well.

The simplest notion of writing is that it will be made of lines. Even in our 3D world this is so.
But 4D characters will be lines that travel on the 'inside~surface' of the 4D page.
So a character line will swirl about within the 3 directions creating all sorts of interesting characters that we can't create.

eg. a line goes one direction then u-turns back around then turns to circle partially the first line without touching it.

Their range of possible character shapes would make ours look puny.
They probably wouldn't even think to repeat patterns like we do for 'd' and 'b' or 'q' and 'p'.

How cool would that be!