Building 4D objects and worlds

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 10:02 pm

Mrrl wrote:
quickfur wrote:
This then becomes the challenging concept. How do you turn your head through two axis of left-right while still having one down.

Um... because in 4D your neck has two degrees of freedom in turning?

In other words, your neck is spherindrical (extrusion of a sphere), and it can rotate in 2 directions while still remaining upright.


Do you mean three degrees?

Yes you're right. I was thinking of the two degrees which changes the direction you look at, but of course there's that additional degree which changes your orientation but not the direction you look at.

Of course, if you include up/down directions then it adds another 3 types of rotation, corresponding to tilting your head sideways (two directions possible, each one with positive/negative rotations) and looking straight up/down. EDIT: And the tilting doesn't change the direction you look at, but looking up/down does.

Actually, my neck has three DOF (I can turn head in front-left, front-up and left-up planes), so 4Der's neck should have complete set of 6 DOF (if we consider them as upright walking beings)

Yes you're right. :)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Mon Nov 21, 2011 10:21 pm

hi quickfur.

I apologise, you are right, I was referring to a hypersphere though and not meaning to refer to a spherinder. It still remains as for our spheres that the mass curves off 3 axis for a hypersphere; as it curves off 2 axis for our sphere.

However, gravitational force is actually calculated combinationally. F = G(m1m2))/r2.
So both the bulk of the hyperplanet and the body contribute to the combined gravitational force.

I think you also have to remember that no matter how big an object is it will still fall at the same speed which is purely dependent upon the object it is falling towards.
It's bulk does not contribute to its own falling speed.
What it's own bulk does contribute to is the speed the planet falls towards it so they will actually meet sooner if one or both are bulkier.

Also, although we can see all sides of a square when it is planewards towards us we can not see the opposite inside or those same sides from the other side. If we turn a square so that it is edge on to us and we close one eye than it becomes hard to see any of the other edges or either square inside. The same goes for a 4Der who's eyes can see all 'faces' of the cubes when it is at the right 'angle' to them. They can 'rotate' the cube so that they are looking at it 'edge' on as well; the edge being a plane.

The 4Der doesn't have eyes that surround the cube; instead they have eyes that exist like hyperspheres. The back of a cube to us is like the back of a tesseract to them. If something has a back in the 4D world then the 4Der will no more see that back then we see the back of anything. What they will see is the front spread through 3 of their 4 dimensions; whereas what we see is the front of things spread through 2 dimensions. So I strongly disagree that they will see cubes as having a top, bottom, back, front, and two sides. They will not see an enclosed cube at all.

I don't think it is simply a case of translating the 3rd dimension into the 4th. The behind of a cube in our world remains part of the behind of a tesseract in their world. To them a cube would be a completely different beast to our cube. Our cube does not share the same 'space' as their cube.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 11:01 pm

gonegahgah wrote:[...]However, gravitational force is actually calculated combinationally. F = G(m1m2))/r2.
So both the bulk of the hyperplanet and the body contribute to the combined gravitational force.

Yes, and in 4D, you have F = G(m1m2)/r3.

But we're talking about gravity on the surface of the planet here, so r and m1 are fixed. So that amounts to F = Km2 where K is (approximately) a constant and m2 is the mass of the object on the planet. In this sense, it's identical to the 3D case. Things only become different when the value of r is significantly different, say when you're flying out into orbit or something.

I think you also have to remember that no matter how big an object is it will still fall at the same speed which is purely dependent upon the object it is falling towards.
It's bulk does not contribute to its own falling speed.
What it's own bulk does contribute to is the speed the planet falls towards it so they will actually meet sooner if one or both are bulkier.

You mean the acceleration is the same. The speed at which an object falls changes over time. But you're right that if two objects, one heavy and the other light, were to be dropped from the same height, they would hit the ground at the same time. That's because they are experiencing exactly the same gravitional force, which gives them identical acceleration. The bulkier object feels more force, but it's directly in proportion to its mass, so the resulting acceleration is equal.

One way to think about it is that the atoms in the object are identical (or, if you like, the subatomic particles or whatever it is at the most basic level), so they all fall down with the same acceleration and reach the ground at the same time. Just because a small number of atoms happen to comprise a light object doesn't change this, and just because a large number of atoms happen to comprise a heavy object doesn't change this either.

Also, although we can see all sides of a square when it is planewards towards us we can not see the opposite inside or those same sides from the other side.

That assumes the square has non-zero thickness. In other words, it's actually a very flat cube. In a truly 2D universe, a square has zero thickness, and 3D light would pass right through it (i.e. it will be invisible). But if we can see it somehow, we'd be able to see all of it at once, because its "other side" is actually its "near side" too.

If we turn a square so that it is edge on to us and we close one eye than it becomes hard to see any of the other edges or either square inside.

Again, that assumes that the square is actually a very flat cube. If it's a true 2D square, it would literally disappear when viewed edge-on because it will have 0 thickness. In this case we won't even be able to make assumptions about magically seeing it regardless, because it will project to a line of 0 width in our eye, which cannot be seen. But if we make the additional assumption that we can somehow see a mathematical line segment, then that's what we will see: just that one edge of the square (or 2 if you look at it from an angle).

The same goes for a 4Der who's eyes can see all 'faces' of the cubes when it is at the right 'angle' to them. They can 'rotate' the cube so that they are looking at it 'edge' on as well; the edge being a plane.

Yes, and I see this in projection all the time when looking at polytopes with equatorial facets. Facets like cubes show up as squares or hexagons when seen from that angle.

The 4Der doesn't have eyes that surround the cube; instead they have eyes that exist like hyperspheres. The back of a cube to us is like the back of a tesseract to them. If something has a back in the 4D world then the 4Der will no more see that back then we see the back of anything. What they will see is the front spread through 3 of their 4 dimensions; whereas what we see is the front of things spread through 2 dimensions. So I strongly disagree that they will see cubes as having a top, bottom, back, front, and two sides. They will not see an enclosed cube at all.

OK, I still can't figure out if you're talking about 4D cubes or 3D cubes. A 4Der who looks at a 3D cube sees all 6 faces at once, plus the entire inside of the cube.

When they see a tesseract, they only see at most 4 of its cubical cells at once. (Unless it's transparent, of course.) They have to turn it around to see the other 4 cubes. But nevertheless, in their mind's 4D model of the tesseract, there will be a top cube, a bottom cube, and 6 lateral cubes.

I don't think it is simply a case of translating the 3rd dimension into the 4th. The behind of a cube in our world remains part of the behind of a tesseract in their world. To them a cube would be a completely different beast to our cube. Our cube does not share the same 'space' as their cube.

OK, you're using very confusing terminology. I can't figure out where "cube" means 3D cube and where "cube"' means 4D cube. But they are very different beasts.

The behind of our 3D cube is completely laid bare before the 4Der's eyes. They can see all of it, back faces, front faces, insides, everything. In fact, to them there is no such thing as a front face or a back face; all 6 faces are equivalent. It's only from our 3D-centric view that there's a difference between a cube's front face and its back face.

A 4D cube, on the other hand, can only be seen one side at a time from the 4Der's eyes. If they look at the 4D cube from one of its facets, all they would see is a 3D cube. This 3D cube, of course, has no "front" or "back" faces from their point of view; they can see all of it simultaneously. However, this 3D cube obscures their view of the other 7 cubes in the 4D cube. If they were to turn the 4D cube a little, then they might see a second 3D cube come into view (and the other 6 remain hidden from their view in the 4th direction). Or if they turn it a different way, they may see three 3D cubes (and the remaining 5 remain hidden from view). Or, if they turn it so that they are looking at one of its 16 corners, then they would see 4 3D cubes simultaneously. (And by that I mean that all 4 3D cubes along with all of their faces, edges, and vertices, are all visible at the same time.) The other 4 remain out of sight. And this is the best they can do: they simply cannot see more than 4 3D cubes on the tesseract at one time. They have to rotate the tesseract around to its far side to be able to see the other 4 3D cubes, at which point, these 4 3D cubes will no longer be visible.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Tue Nov 22, 2011 11:34 pm

hi quickfur. Sorry I made some silly mistakes, thanks for correcting those.

In that formula: F = Km the mass is actually that of the planet; not the object.
The gravitational force acting on any object (in the same circumstances) is the same regardless of its bulk.
I think we are describing the same thing but I would describe it differently.
I would say that the gravitational force acts on every micro-bulk of an object equally which causes each micro-bulk to fall with the same rate of change.
This is why we don't have to worry about momentum affecting the rate of fall because the gravitational 'force' acts on all parts of the object equally.
This is unlike a rocket which divides its force over the bulk which means that a greater bulk will result in slower acceleration.
Sorry, reading on you've pretty much said this anyway.

The other thing to remember with Galileo's acceleration experiment is that the two objects and Earth form part of the one system. The smaller object and the larger object are both together pulling the planet towards them. The result is that everything will meet at the same time. If you brought in a moon and dropped it on a planet it would appear to fall faster than if you brought in a pebble and dropped it on the same planet. The reason is that the moon+planet have a greater combined gravitational force than the pebble+planet.

This is where I talk about systems being important. If you instead dropped the moon and the pebble side by side they would both hit the planet at the same time as the pebble and moon's gravity would combine and accelerate the planet towards the both of them equally.

But back to the tesseract and cubes. Sorry for confusing the terminology; I am trying to present them distinctly.
I also apologise, you are right that each face of each cube on the outside of the tesseract will be visible to the 4Der. You are correct.
They will see it in some sort of 4D form. Just like the lines we see describing a cube are in 3D form. ie. as well as the defining property of length, those lines are actually atoms thick and atoms wide as well.

But, I would suggest that straight line projections don't make this obvious to the 3D born observer.
Instead, they make the outside cubes look like the bulk of the tesseract.

If we compare like for like we would compare the 4Ders view of a cube to our view of a square.
We see a square primarily via its area but still importantaly as defined by four constraining equal straight lines.
We see a cube primarily via its squares; the volume is just assumed.
A 4Der however sees a cube primarily via its volume; though importantly as defined by six constraining equal squares.

It is as much a difference of perception as how we and a 2Der view a square. They see a line and we see an area.
And like us for the square; the 4Der has two views of a cubes volume; a 4D front view and a 4D back view.
So whereas we see only one volume. They see two possible perspectives on the volume.
Again this is like a 2Der sees one area where we see two possible perspectives of the area of a square.
How does that sound?
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Wed Nov 23, 2011 2:20 am

gonegahgah wrote:[...] This is where I talk about systems being important. If you instead dropped the moon and the pebble side by side they would both hit the planet at the same time as the pebble and moon's gravity would combine and accelerate the planet towards the both of them equally.

OK, so we're actually saying the same thing after all, just in different terms. Different representations. :P

[...] But, I would suggest that straight line projections don't make this obvious to the 3D born observer.
Instead, they make the outside cubes look like the bulk of the tesseract.

If you look at the projections from a 3D-centric point of view, yes, they look like oddly-shaped volumes embedded inside a outer 3D volume. And if you see a projection 4D rotation from a 3D-centric point of view, it will look like impossible shape-shifting and turning inside-out.

I don't think this is something that can be helped, in the sense that a 3D born observer, being accustomed to interpreting things in 3D, will of course carry over this 3D-centric interpretation when looking at these projections. After all, they can be interpreted in the 3D sense, just like a 2D projection of a 3D object forms a 2D image, and a 2Der who looks at it can also interpret it as just a 2D object.

That's the key issue. In order to "see 4D" using these projections, you have to be actively looking to interpret it differently from what it seems. It seems like a funny-looking 3D object, but you have to "turn off" that 3D interpretation in your brain and use a 4D-centric interpretation instead. So this is the hard part. And it doesn't just apply to projections, any method you use for visualizing 4D, eventually you get to a point where you just have to "switch off" your 3D worldview and adopt a new kind of view, and new kind of understanding, or metaphorically speaking a "new pair of eyes" to see things in a different realm, a strange new world where there are 4 mutually perpendicular directions (that cannot exist in 3D).

If we compare like for like we would compare the 4Ders view of a cube to our view of a square.
We see a square primarily via its area but still importantaly as defined by four constraining equal straight lines.
We see a cube primarily via its squares; the volume is just assumed.
A 4Der however sees a cube primarily via its volume; though importantly as defined by six constraining equal squares.

Yes, while we can only infer the cube's volume, the 4Der can see it.

This reminds me of that old novel "Flatland", which talks about a sentient square living on a 2D plane who one day meets a 3D sphere, who tries to explain 3D to him. In 2D, angles are not visible; all you can see are lines, and those lines are always parallel (to your eyes, anyway). The existence of angles between lines in the 2D world is something you infer, but it's impossible to see it with your 2D eyes (which only has a 1D retina: it's impossible to represent an angle). So when the 3D sphere says that he can see an angle, the square thinks that he is talking nonsense. Nobody can see an angle! Well. Unless you're in 3D. :) Then angles are as obvious as day and night.

However, the 2Ders can measure angles. They can't see it, but based on measurements taken between two line segments, they can infer that there's such a thing as an angle, and they can take measurements and compare it, and do useful stuff with it (like making sure two walls meet at 90° instead of 89°, for example). Based on this, they can construct a model of what an angle is in their mind, even though they will never be able to see such a thing.

So in the same way, some aspects of 4D cannot be seen by us no matter how you try to represent it. But you can learn how to handle it by using various tools like measurements, inference, etc., and acquire the ability to do stuff with it. You then form a model of 4D in your mind that no physical image can ever represent directly.

It is as much a difference of perception as how we and a 2Der view a square. They see a line and we see an area.
And like us for the square; the 4Der has two views of a cubes volume; a 4D front view and a 4D back view.
So whereas we see only one volume. They see two possible perspectives on the volume.
Again this is like a 2Der sees one area where we see two possible perspectives of the area of a square.
How does that sound?

Yes, that's correct.

So here's the thing: we can't see a 3D volume directly, but our mind surely does a good job of making a model of it. A 4Der can see the volume directly, so what is only indirectly known to us is known directly by them. But whether you know it directly or indirectly, eventually both the 3Der and the 4Der "knows" what a 3D volume is. The 4Der does this just by looking; the 3Der does this by learning how to interpret indirect 2D images of it.

My thesis is simply that a 4Der can't see 4D bulk directly, but they can infer it from 3D images. So here's the connection: they start from 3D and arrive at 4D; we start from 2D and arrive at 3D. But if we've arrived at 3D, then we can repeat what they do to get to 4D, and so we now have a way to see 4D (albeit indirectly, in fact, double-indirectly, but the end result is the same, in principle).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Tue Dec 13, 2011 3:24 pm

The question arises how to transfer a co-ordinate system to my spherical representation of a 4D space.
I think that we need to go through how a co-ordinate system is currently done using the bulk missing representations.
Before I do that I just want to examine some things about the various dimensions further.

This is a look at the side transcribed edges.
1D: if you move an edge (a dot) through a square equivalent (a line) up and down it will have 0 transcribed edges as there is nowhere to transcribe them to.
2D: if you move an edge (a line) through a square equivalent (a square) up and down it will leave 2 transcribed line edges.
3D: if you move an edge (a square) through a square equivalent (a cube) up and down it will leave 4 transcribed square 'edges'.
4D: if you move an edge (a cube) through a square equivalent (a tesseract) up and down it will leave 6 transcribe cube 'edges'.

Looking at this differently they also:
1D: nothing to transcribe.
2D: the dot ends of moving line transcribe 2 line edges.
3D: the line ends of moving square transcribe 4 square 'edges'.
4D: the square ends of moving cube transcribe 6 cube 'edges'.

This is just another way of looking at how each of the 6 faces of the cube each individually transcribe a cube through their movement when the cube is extended into the 4th dimension to describe a tesseract. This also means that when producing a co-ordinate system for any of the models - that we may chose to use - we need to be able to demonstrate how the start cube 'edge' transcribes these 'side' cube 'edges'.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby Secret » Wed Dec 14, 2011 12:38 pm

gonegahgah wrote:The question arises how to transfer a co-ordinate system to my spherical representation of a 4D space.


If I have not misinterpret what you mean here
You should ask quickfur to send you the "article of the puffy flat misconception of the bulks"

What I want to say is the 4D bulk is something not anywhere within the projection envelope, it is "behind" (in the sense that it is not in that 3D solid model) the projection

No amount of bending and squishing can force the 4D bulk to show up in the projection image
I never realize this before I receive that article from quickfur.

The point is, flatness is relative. A 3D cell or a 3D rind is not paper thin as we know it, it is still as puffy as we used to know them, even when seen from 4D
Secret
Trionian
 
Posts: 162
Joined: Tue Jul 06, 2010 12:03 pm

Re: Building 4D objects and worlds

Postby Mrrl » Wed Dec 14, 2011 12:57 pm

Secret wrote:What I want to say is the 4D bulk is something not anywhere within the projection envelope, it is "behind" (in the sense that it is not in that 3D solid model) the projection

No amount of bending and squishing can force the 4D bulk to show up in the projection image


It is a question of visualization. You can see 4D bulk, for example, if you add some fog to 3D projection: close (in 4D) areas will be brighter and far areas more foggy. You'll see, say, 3D sphere as a ball with foggy surface and bright center, and for concave hemisphere you'll get a ball with foggy center and bright surface. Or you may use colour coding for the depth function. The problem is to teach our brain to work with volume 3D scenes (not with 2D images as we use to do).
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby Secret » Wed Dec 14, 2011 1:17 pm

It's not that simple, according to quickfur, say the hemisphere projection (a series of concentric 2spheres with decreasing radius difference between two successive 2spheres to form the curvature in the three sideways directions. Innermost sphere is solid while it gets more foggier as it approaches the outermost spherical envelope) no matter which point you point to (even the insides of the individuals 2spheres) they are just part of the rind. The 4D bulk is not anywhere in the projection. It is "behind" it. What we need is to train our brains to infer it's existence.

In the past, I used to point to somewhere inside the 2sphere thinking it is part of the bulk
Secret
Trionian
 
Posts: 162
Joined: Tue Jul 06, 2010 12:03 pm

Re: Building 4D objects and worlds

Postby Mrrl » Wed Dec 14, 2011 1:37 pm

There is no "behind" in 3D, so bulk should be in this projection. And it is. We can see it as particles of the fog that are covered by solid sphere surface close to the center of the ball. There is a segment of bulk in every point of 3D and we see where it's closest point is (and we don't see its far end because it's 4D-behind the sphere).
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby quickfur » Wed Dec 14, 2011 4:59 pm

Mrrl wrote:There is no "behind" in 3D, so bulk should be in this projection. And it is. We can see it as particles of the fog that are covered by solid sphere surface close to the center of the ball. There is a segment of bulk in every point of 3D and we see where it's closest point is (and we don't see its far end because it's 4D-behind the sphere).

The thing about perceiving bulks is that it is a matter of interpretation.

To restate what I wrote to Secret: no matter how you try to draw a 3D object on a 2D surface, you can only ever succeed in representing a 2D area. It is impossible to draw 3D volume on a piece of 2D paper. Take the image of a cube, for example. Can you point to the 3D bulk of the cube in the image? No matter where you point, it's only a point on the surface of the cube. We can see the 3D bulk only because our brain interprets it that way, but the 3D bulk actually isn't really represented on the paper. If you ask a 2Der, it will say that there's no 3D bulk at all, just a 2D area. No matter how we try to color it, make it transparent, use fog effects, etc., it is still merely a 2D projection. Point anywhere on the image. The 2Der will only see you pointing to the 2D surface of the cube. Sure, it looks like some parts are colored weirdly, but it's still just a point on the 2D surface. The 2Der still can't "see" the 3D bulk, because it isn't represented by the image. It can't be represented by the image. How do you represent 3D volume with 2D area? Even if you use "transparency" or "fog", how do I know that you didn't just paint it on the faces of the cube?

The only way we see it is because our brain interprets the 2D image in a 3D way. Instead of seeing trapezoids in the cube projection, our brain interprets it as square faces slanted into 3D. Instead of seeing an area with funny color shading, our brain interprets it as lighting and shade caused by 3D slope and lighting. Instead of seeing faces of different sizes, our brain interprets it as the faces of the same size at different distances. In fact, these distortions are interpreted as cues to the 3D-ness of the represented object. When the 2D image is interpreted in this strange way, then suddenly our brain realizes that those trapezoids are actually squares enclosing space in 3D -- that's when we "see" the 3D bulk of the cube.

In the same way, even a 4Der can't see 4D bulk, in the literal sense. A 4Der's eyes only sees 3D projections; no matter where you point in the 3D projection, it's still only a point on the surface of the 4D bulk. The inside of the 4D bulk is not represented at all, and is not representable. From our 3D-centric point of view, there are only funny-looking 3D volumes in the projection image; there's no 4D at all. But the 4Der would say that she can see 4D bulk -- because her brain interprets it that way. What we see as strange squished parallelopipeds, she interprets as cubes bending into 4D. What we see as a smaller cube embedded inside a larger cube, she interprets as two cubes of the same size, at different distances. The point is that she is looking at a 3D image, but interpreting it in a "funny way" that causes her to "see 4D".

As long as the 2Der looking at the 2D projection of the cube tries to interpret the image in a 2D sense, it will only see the area of the cube, not its bulk. The only way it can see 3D bulk is to interpret the image in a "funny way" - to interpret the distortions as indications of 3D-ness rather than distortions. That is to say, it needs to interpret 2D images in the same weird way that our brains interpret them, to "invent" 3D bulk out of a flat 2D image. The 3D bulk isn't in the image at all, but our brains invented the concept. Which, oddly enough, is close to reality. :)

In the same way, we cannot possibly see 4D bulk. But if we interpret those 3D projections in a "funny way", that is, interpret the distortions as indications of 4D-ness rather than distortions, then we will be able to "see" 4D bulk. In other words, the 4D bulk isn't represented at all, because it cannot be represented in 3D, but the 4Der's brain invented the 4D bulk out of a "flat" 3D image. But it's somehow pretty close to reality. :) So for us to see 4D, we have to follow the same weird interpretation that the 4Der's brain is doing, to "invent 4D" out of a 3D image, just like we instinctively invent 3D from 2D.

It's all a matter of interpretation.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby Mrrl » Wed Dec 14, 2011 5:26 pm

quickfur wrote:In the same way, we cannot possibly see 4D bulk. But if we interpret those 3D projections in a "funny way", that is, interpret the distortions as indications of 4D-ness rather than distortions, then we will be able to "see" 4D bulk. In other words, the 4D bulk isn't represented at all, because it cannot be represented in 3D, but the 4Der's brain invented the 4D bulk out of a "flat" 3D image. But it's somehow pretty close to reality. :) So for us to see 4D, we have to follow the same weird interpretation that the 4Der's brain is doing, to "invent 4D" out of a 3D image, just like we instinctively invent 3D from 2D.

It's all a matter of interpretation.


Everything is a matter of interpretation. Including interpretation of neural impulses from our eyes as 2D images. But I don't agree that "4D bulk cannot be represented in 3D". When you look at 2D topography map of surface of 3D planet, you can see isolines that represent elevations of different points. And by these lines complete geometric shape of the surface is represented in 2D (with some level of accuracy). You can do the same with 3D projection of 4D object and get representation of shape of its visible side in 3D. You don't need to guess 4D shape of the object by such image - just read it from isolines :)
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby quickfur » Wed Dec 14, 2011 6:03 pm

Mrrl wrote:[...] But I don't agree that "4D bulk cannot be represented in 3D". When you look at 2D topography map of surface of 3D planet, you can see isolines that represent elevations of different points. And by these lines complete geometric shape of the surface is represented in 2D (with some level of accuracy).

But the topography map only shows the 2D surface of the planet. It doesn't show the planet's bulk. Sure, it shows how the surface curves in 3D, but it is still only a 2D surface. The bulk itself is not represented, but is inferred by the curvature of the surface.

You can do the same with 3D projection of 4D object and get representation of shape of its visible side in 3D. You don't need to guess 4D shape of the object by such image - just read it from isolines :)

Well, topography is certainly a good idea -- one that I haven't considered before. It certainly deserves further exploration. But it still doesn't represent 4D bulk. :) It represents the curvature of the 3-manifold that forms the surface of the object, but the 4D bulk itself can only be inferred.

And that's really what I was getting at. 3D bulk is not representable in a 2D image, although you can certainly adopt a certain interpretation of a 2D image (e.g. topography lines, shading, etc.) that lets you infer the curvature of the surface of the bulk, which in turn lets you infer the shape of the bulk. Similarly, 4D bulk is not representable in a 3D image, but if you adopt certain interpretations (e.g., interpret "squishiness" as perspective foreshortening, topography ridges, etc.), then you can infer the curvature of the bounding 3-manifold. Which then lets you infer the 4D bulk that lies beneath the surface.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Thu Dec 15, 2011 12:45 am

My aim is to have something that you can look at and see what is happening in a sense.
It will unavoidably introduce its own distortions of course.
I'm starting to teach myself how to use Blender to hopefully work it out.

The 'projection' model is fine for 4D games like worms where the worm is a 3D line.
When you introduce 4D objects and/or rotate them into the 4th dimension it appears to morph the object even though it is the same object.
The 'slice' model is a bit like a blind man figuring out an elephant without having seen any animal before.

As I say distortions would still occur but hopefully it might be possible to see the sense of it with a bit more ease.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Thu Dec 15, 2011 5:58 pm

gonegahgah wrote:[...] The 'projection' model is fine for 4D games like worms where the worm is a 3D line.
When you introduce 4D objects and/or rotate them into the 4th dimension it appears to morph the object even though it is the same object. [...]

This "morphing" is unavoidable. You do realize that a tesseract can rotate in six different planes, right? No matter how you try to force its surface to fit on a sphere, I just don't see how you can possibly do it in such a way that all 6 planes of rotation will not cause a "morphing" effect. A 2-sphere only has 3 planes of rotation... If you can find a way to do it, I'd love to hear it.

Not to mention that I have my doubts whether it's even topologically possible to fit a tesseract's surface onto a 2-sphere. But hey, what do I know? If you discover something, I'd love to see it.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Fri Dec 16, 2011 1:34 am

Hi quickfur. Hopefully but the proof will be in the pudding at that time. I'll love to see it too ;-) The main effect I'm hoping to get around is the appearance of a 4D object seemingly turning in upon itself as occurs with the projection model. Other sorts of distortions will occur but I hope they are more easily understood.

Regarding something else. I saw following image when going through the intro notes on here:
Image

I personally suspect that this diagram is not correct and that it is not possible to 'walk around' a 4D river.
I think there will still be a hyperdown and that we will be as much bound to the hyperdown as the river is.
What do you think?
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Fri Dec 16, 2011 3:04 am

gonegahgah wrote:[...]Image

I personally suspect that this diagram is not correct and that it is not possible to 'walk around' a 4D river.
I think there will still be a hyperdown and that we will be as much bound to the hyperdown as the river is.
What do you think?

Um... you do realize that this is a "floor plan" diagram, right? That gravity is in the 4th direction relative to this diagram? That is, all 3 axes in this diagram lie flat on the floor in 4D?

It seems that you are failing to understand that in 4D, surfaces are 3-manifolds. Or should I say, hypersurfaces? Lest you imagine that said surfaces are 2D, which they are not. In particular, floors occupy a 3D volume. That means anything with a linear track, like rivers, roads, etc., do not divide the floor, because of the plain and simple fact that a line does not divide 3D space.

In fact, 3D is the only dimension where rivers divide land and have two banks. In 2D, rivers completely fill up the floor so you can't even cross a river, you have to walk in it. In 3D, rivers don't fill up the floor, but they do divide it, so if you want to get to the other side, you have to cross the river. In 4D and above, rivers don't divide land and you can just walk around them.

In the same way, roads in 2D have to stop once it reaches a building; there's no way at all for it to go around the building. In 3D, roads can go around buildings, but they also divide land, so if you want to get to the other side, you have to cross the road (unless you make use of the 3rd dimension by building a bridge or a tunnel). Roads also divide the city into blocks, and you have to cross roads when moving from one block to another. In 4D, roads not only go around buildings, but they don't divide land. To cross to "the other side" (which is actually meaningless in 4D because land surrounds the road in 360°) you simply don't walk straight into the road, just go around it. Roads do not divide the city into blocks at all; in fact, there's no need of the concept of a "block" in the first place: as long as you leave enough space between buildings to put in a road, you can put buildings wherever you like. You can even build a single, undivided building that completely surrounds a road without any overpasses or tunnels.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Fri Dec 16, 2011 2:49 pm

I have to agree that I disagree with you on this one. I think this comes back I believe to our different perceptions of gravity in a 4D world.

As I have already allured to, it is my opinion that if you have a hypersphere then in my view the gravitational pull is a sum of the gravitational pull of all the so called 'spheres' that go to make the hypersphere. It also means that the direction of pull - or down - is towards the centre of the hypersphere and not towards the centre of any of the decreasing spheres that are represented in the slice model. So down always occurs in the one direction from any point which is towards the centre of the hypersphere.

This does mean that there is a lowest hyper-region where water will gather; even with it being 4D water.

Even as a 4D creature we are also bound to this same gravity constraint. So down will be towards the centre of the hypersphere. To approach a river we have to go downhill to it even if that hill and down exists in 4D. The water will still be at the lowest point and will prevent us moving any further beyond it without us going down into the river itself.

Water will only fall towards this single down hyperpoint. It will not fall towards multiple down hyperpoints that exist who knows where.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Fri Dec 16, 2011 4:10 pm

gonegahgah wrote:I have to agree that I disagree with you on this one. I think this comes back I believe to our different perceptions of gravity in a 4D world.

As I have already allured to, it is my opinion that if you have a hypersphere then in my view the gravitational pull is a sum of the gravitational pull of all the so called 'spheres' that go to make the hypersphere. It also means that the direction of pull - or down - is towards the centre of the hypersphere and not towards the centre of any of the decreasing spheres that are represented in the slice model. So down always occurs in the one direction from any point which is towards the centre of the hypersphere.

OK, clearly we have communication problems here. I don't know where you got the idea from that "my" idea of gravity is that it's pulling towards the center of the spherical slices of the hypersphere. I never said that, I never meant that, and I never thought that. Gravity is always towards the center of the hypersphere -- that is the only way it makes any sense at all. So I don't know what you're disagreeing with. We both have the same understanding here.

This does mean that there is a lowest hyper-region where water will gather; even with it being 4D water.

Yes, and that would be the sea.

Even as a 4D creature we are also bound to this same gravity constraint. So down will be towards the centre of the hypersphere. To approach a river we have to go downhill to it even if that hill and down exists in 4D. The water will still be at the lowest point and will prevent us moving any further beyond it without us going down into the river itself.

No, the lowest point is just the sea, which fills up a 3D region. You cannot walk around this kind of sea, because it covers a 3D region. That's not what we're talking about here.

Water will only fall towards this single down hyperpoint. It will not fall towards multiple down hyperpoints that exist who knows where.

Nobody ever said anything like this. Where did I say that it's falling towards multiple hyperpoints?

The water begins at some point at an altitude, and it will always flow towards the direction of lower elevation. We agree on this, correct?

Therefore, starting from point X on the surface of the planet, the water will always flow in some direction, let's call it D, such that X+D is at a lower elevation than X. Then from X+D, it will flow in some direction, let's call it D' (which may or may not be the same direction), where X+D+D' is at an even lower elevation. And so on. At each point, the water will keep flowing in the direction that leads to the lowest elevation. This is obvious, right?

OK. So now you have a trail that begins at point X and travels in some direction, D, D', and whatever other direction follows after that. So this traces out a 1D curve starting from point X and eventually reaching to the sea, where the water accumulates (because there is no lower elevation it can flow to).

But since the surface of the planet is a 3-manifold, this 1D curve, which we call a river, traces out some kind of 3D curve through the surface of the planet. But since the surface is 3D, that means there's plenty of room all around the curve that you can walk on. You may think of it as a circular river bank that wraps around the river. And by "wrap around" I do NOT mean in the sense of surrounding it like a bridge and tunnel at different elevations, but that the ground around the river is at the SAME ELEVATION. (Well, slightly higher, because by definition the banks of a river is at a slightly higher elevation.) This is possible because the ground is 3D. It is obviously impossible in 3D because the ground here is 2D, and a 1D curve cuts the ground into the left side and the right side, and the two banks are disconnected. But in 4D, there is only one river bank, because a 1D curve does not divide a 3D ground. There's plenty of room around the river that you can walk.

Another way to think about this is to consider that a 4D being walking on the surface of a 4D planet is essentially confined by gravity to the surface, so she only has 3 degrees of freedom, not 4, as far as walking is concerned. So the path she walks is equivalent to some path that a rocket in our 3D space flies through space. Similarly, the shape of the 4D river traces out a 3D curve, which you can think of as some curved path traced out by a 3D rocket flying through space. Now let's say the "river rocket" traces out some curve through space, and we're in the "walking rocket" trying to get from point A to point B past the region where the river curve is. It's obvious that there is absolutely no need to intersect the river's curve at all. We just fly past it. It's really just that simple.

Now, things change when the river reaches the 4D sea, which again, is confined by gravity, so that its surface would occupy a 3D region. So we can compare this to a nebula in 3D space that the "river rocket" reaches at some point. Now if we were flying towards this "sea nebula" in our "walking rocket", we can no longer just "fly past it", because the nebula occupies a 3D region. To get to the other side, we would need to either fly through it (equivalent to sailing across the sea in 4D) or take the long way by changing our course and making a big detour to get around the nebula without flying through it.

Furthermore, in 4D there's a third kind of body of water, that is neither a river (it doesn't occupy a 1D curve) nor a sea (it doesn't occupy a 3D region either). Instead, it occupies a 2D region. Or to be precise, it's approximately the shape of an ellipsoid with two long axes and one narrow axis. This kind of body of water does divide the ground, so to cross it, you will need a bridge. You will only need a bridge to cross its narrow axis, however, because if you're approaching it from a long axis, you could just walk beside it -- it doesn't divide the ground in that direction.

Anyway, none of this is a new idea. This stuff has been known and proven for almost (if not already) a decade now. It's a natural consequence of the very definition of gravity being towards the center of the hypersphere, and the plain and simple fact that the surface of the hypersphere is a 3-manifold.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby gonegahgah » Fri Dec 16, 2011 10:01 pm

Hi quickfur

The following picture:
Image

Depicts 3 slices of a planet in slice model. We can see a river on the surface of the centre slice. The river also spreads into some of the adjacent slices that aren't shown.

Just like water will spread across 2 dimensions in our world it will spread across 3 dimensions, which all consider themselves to be perpendicular to downwards, in a 4D world.

I've shown the river turning into a line (even though that wouldn't occur as it spreads across 3 dimensions always) to show that some of it spreads into the hidden dimension onto the immediate adjacent slices.

We can see part of a creature on the leftmost sphere. Most of the creature is off in the adjacent slices towards the centre slice.

Our creature can walk all over the left sphere that we see with no trouble - with the rest of its body existing in the immediate adjacent slices between.

The trouble comes when the creature wants to walk to a place that places some of themselves onto the sphere that we see on the other side of the central slice.
To get to the other side they have to approach the river.

To us it will look as though they are moving uphill to ever larger spheres if we follow their progress. This isn't the case. They are just moving sideways into the hidden dimension.
None-the less they will eventually reach a point where they are looking down into the river who's surface spreads across 3 surface dimension before them.

It is the surface that is important and it is the surface that we have to bypass. We still walk on the surface even if the surface spreads through 3 dimensions instead of our 2.
There just isn't anyway to cross the river - unless you go round the lake at the lowest point - to get from one slice side to the other slice side.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby quickfur » Fri Dec 16, 2011 11:09 pm

gonegahgah wrote:Hi quickfur

The following picture:
Image

Depicts 3 slices of a planet in slice model. [...]

And this is where your trouble lies. You're failing to account for the fact that the center slice is, well, a slice. The 4D being can simply walk kata-wards (from the left slice to the right slice) from where she's standing, without ever needing to pass over the river.

Obviously, if she walks in the direction indicated by your black arrow, then she'll fall into the river. But there's no reason to do that at all. Just stand left of the river and walk kata-wards and you get to the right slice. Then while standing on the right slice, walk west, and then walk ana-wards through the middle slice back to the left slice. Now you've just walked around the river in a full circle without ever needing to cross over it.

To make this explicit: start with the black dot on the left slice. Walking kata-wards, you'll end up on the center slice slightly east of the river, with the same latitude/longitude as the black dot on the left slice. Keep walking and you end up on the right slice at the same latitude/longitude. Now while staying on the right slice, walk westwards. Now your longitude is slightly west of the longitude of the river (but you don't have to cross it 'cos you're standing on the right slice). Now walk ana-wards again. When you get to the middle slice, your longitude is west of the river, so you just missed it. Keep walking ana-wards, and you end up back on the left slice, with the same latitude as before but with longitude slightly west of where you started. Now walk east again and you're back where you started. So you've managed to walk a full circle around the river without ever crossing over it.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby Mrrl » Sat Dec 17, 2011 8:27 am

Map of small part of 4D world - with towns, roads, lakes and channels:

Image

You see that terrain around these features is connected, so you may easily walk around (unless some part is completely surrounded by towns and lakes - then it will be disconnected from the rest of the world)
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby gonegahgah » Sat Dec 17, 2011 9:01 am

This is cool; but which way is down?
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Building 4D objects and worlds

Postby Mrrl » Sat Dec 17, 2011 9:33 am

Double-perpendicular to the screen :) (i.e. perpendicular to 3D scene that is shown on the picture). All objects on the picture are on the same level, planet surface is strictly horizontal.
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby quickfur » Sat Dec 17, 2011 3:25 pm

Mrrl wrote:Map of small part of 4D world - with towns, roads, lakes and channels: [...]

This is very cool. Did you make this model by hand, or was it generated by a program? IIf it was generated by a program, I'd like to adapt it for a 4D exploration game I have. :) (Although currently it only has ASCII output. :()
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby Keiji » Sat Dec 17, 2011 3:49 pm

What does each color represent there?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Building 4D objects and worlds

Postby Mrrl » Sat Dec 17, 2011 4:24 pm

Keiji wrote:What does each color represent there?

Red and orange - for cities and villages, white for roads, light blue for lakes and seas, blue for channels.

quickfur wrote: Did you make this model by hand, or was it generated by a program?

It's complicated. Cells are generated by program, in random model, and I manually build map from them (program checks that cells fit to their neighbours).
Actually it is a game - and it's almost ready :)
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby quickfur » Sat Dec 17, 2011 5:48 pm

Mrrl wrote:[...]Actually it is a game - and it's almost ready :)

Cool!!! Will it feature elevation, or just horizontal movement for now?
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Building 4D objects and worlds

Postby Mrrl » Sat Dec 17, 2011 5:50 pm

It's Carcassonne. So no elevation and no movement - just development :)
Mrrl
Trionian
 
Posts: 165
Joined: Sun May 29, 2011 7:37 am

Re: Building 4D objects and worlds

Postby quickfur » Sat Dec 17, 2011 6:42 pm

Ahh, I see. Nice idea!

Although, I have dreams that one day, once I finally get enough time (har har) to learn to program opengl properly, I will write a 4D FPS where you get to explore a truly 4D world (with gravity), and the auto-map will be a 3D model like the one you have. Of course, there are practical limitations to what you can do within the confines of opengl... for example texturing and detail will be very limited because otherwise you couldn't see a thing in the projection. :P (Although i might be able to get away with more detail if i implement an automatic "line-of-sight" culling in the 3D projection image, by making things that lie in the line of sight more transparent so you can see into the center of the projection. But not sure how usable the result would be.) Also, to achieve real-time rendering, i may have to settle with a block-based world instead of allowing arbitrary slopes.

Well, OK, maybe not an fps -- or that can come later -- but first, a maze exploration program featuring windows, opening doors, trapdoors, rolling boulders in the shape of 600-cells or 120-cells, and chests that open/close with collectible shapes inside. Basically stuff to help with visualization of a 4D scene. I consider 4D visualization as incomplete unless you can visualize a multi-object scene, not just isolated polytopes.

Ah... so many ideas, so little time. :(
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

PreviousNext

Return to Higher Spatial Dimensions

Who is online

Users browsing this forum: No registered users and 15 guests