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 gonegahgah » Thu Nov 17, 2011 9:41 am

It's certainly worth looking at further. The main concern I have is the expression of the missing bulk. To give some idea, if a 2Der drew a cube and wanted to demonstrate all the 'bulk' of the cube then they would have to draw multiple lines above each other; but how many? The answer is an infinite number of lines; because you can't represent true area in 2D. This highlights one of the problems of different spatial dimension counts. The 2Der knows the cube is L3 but they can only represent L2.

The same goes for the hypercube. How many cubes would we have to draw to represent the bulk of hypercube? The answer again is infinite. We just cannot represent the bulk of a 4D object in any meaningful way in our diagrams. What we see is, by strong imagination, only 8x the volume instead of the needed infinity by the volume. So we still have the question about where the bulk is hiding in the 3D diagram.

You can't say the hypervolume is L x L x L x L volumes. ie 3cm would give 81cm3. It is L x L x L x infinity volumes. If you did try to put that few volumes together you would end up with a very thin tessarism.

If we appreciate that then we realise that we can't see even close to what a 4D object looks like as far as I can tell. The 2Der would have to imagine infinite squares to be close; and we would likewise have to imagine infinite cubes. How does that sound?
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Re: Building 4D objects and worlds

Postby quickfur » Thu Nov 17, 2011 3:48 pm

But why do we have to represent the bulk in the 2D diagrams? Our own eyes don't even see 3D bulk. It only sees 2D images! How do we represent 3D bulk in 2D images anyway? No matter how you draw it, it's still only a 2D diagram. There's no 3D bulk in it at all. Even if you build a 3D model, your eyes still only see 2D images. The model itself has 3D bulk, but our eyes can't see it. In other words, even we 3Ders can't represent 3D bulk.

But the fact is, our mind understands 3D bulk perfectly fine. How? It is inferring the 3rd dimension from the 2D projections that our eyes capture. So our mind is doing something to get from 2D to 3D. Somehow, it's inventing this concept of 3D bulk based only on 2D images seen by the eyes, and it's doing a darned good job of it. And by analysing how it does this, we can, in principle, explain the process to a 2Der, who can then follow the same steps and arrive at more-or-less the same mental model of a 3D object. There's no need to directly represent the 3D bulk -- if we infer 3D from the 2D images correctly (and we do, every day), our mind is what supplies the "missing bulk". This bulk isn't represented at all in our eyes.

So the key is that the reconstruction from 2D to 3D is done by our mind. The 3D volume isn't represented in what our eyes see -- it cannot be, since our retina is only 2D. In the same way, the 3D to 4D reconstruction is done by the 4Der's mind. They can't see 4D bulk either... but they can infer it from the 3D images they see. That's the point. The process that their mind goes through to reconstruct the 4D model is fundamentally the same process that our mind goes through to construct 3D from 2D. It's just taking the images seen by the eye and building a model of one higher dimension from it. This is the key point. The mind is capable of going one dimension higher. We do this every day, going from 2D to 3D. All I'm saying is that we take this one step further: after our mind goes from 2D to 3D, take the 3D model and feed it back into the same process: now you're going from 3D to 4D. The details are more complex but the principles are exactly the same.
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Re: Building 4D objects and worlds

Postby gonegahgah » Thu Nov 17, 2011 7:49 pm

Hi quickfur. This is where I'm trying to bring in the connection between my 2.5D cube and what I call our 3.5D tesseract. When we draw a tesseract we are really only drawing the hyperfaces of the tesseract. In the diagram we see 8 cubes (some of them superimposed; unlike how they are in a real tesseract; but as you say this happens even when we draw a cube) which is exactly what the faces only of a tesseract are made up of.

We are, in relative terms through this process, fairly closely demonstrating the actual volume of the faces of the tesseract; even though some are superimposed. What we aren't drawing is the bulk of the tesseract. It gets lost in a dimensional upstep.
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Re: Building 4D objects and worlds

Postby quickfur » Thu Nov 17, 2011 8:17 pm

Well actually, we aren't drawing the volumes of the cubes either. We're only drawing the faces of the cube. After all, these are only 2D images. The volume of the cube isn't drawn; it's implied by the faces that we draw. That's my argument. Even the volume of the cube isn't depicted here, at least, not directly. What is really here is just a bunch of pixels on a 2D screen, representing the projection of a cube's faces. Where is the cube's volume? No matter where you point to in the image, I can say it's just a point on the cube's faces. No matter how you try to draw the cube's volume on a 2D surface, you can't. Even if you draw the diagonals of the cube, one could easily argue that they are just painted on the faces of the cube in such a way they happen to line up. Can you draw a diagram of only the cube's insides without drawing its faces?

My point is that the 3D volume doesn't exist in the picture at all. It cannot be depicted. The 3D volume is supplied by your mind. That's what I'm getting at. When you look at the picture of a cube, your mind sees the 3D volume, not because it actually exists in the image, but because your mind has already constructed the 3D model of the cube from the 2D image you see. You're actually seeing the 3D model your mind has created. The cube's volume doesn't exist in the image.

This is what I've been trying to say: it's your mind that created 3D from 2D. Now I'm just going one step further, to train it to create 4D from 3D. The mind can already create 3D images, as you have so aptly shown: you can see the cube's volume by looking at a 2D picture that, strictly speaking, doesn't depict any volume at all! All I'm saying is that now, you take the 3D images your mind has created, which happens to be what a 4Der would see in her eye, and do the same dimensional upstep to it -- that gets you into 4D.
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Re: Building 4D objects and worlds

Postby Mrrl » Thu Nov 17, 2011 9:03 pm

Next version of 4D Viewer is ready. Now it can show objects in projection mode as well as 3D sections. It's my first experience with OpenGL, so transparent projections look ugly enough. Link is here: http://astr73.narod.ru/4DView/View4Dv3.zip . Archive includes program with necessary dlls and three examples.
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Re: Building 4D objects and worlds

Postby quickfur » Thu Nov 17, 2011 9:54 pm

The question of how to render transparent ridges effectively in opengl is something I've been struggling with. A long time ago I wrote a test program to do transparent ridges, but it turned out really hard to see. Somehow, it's just not the same when rendered using povray. So I gave up and have stuck to povray since.

But i'd like to get back into opengl eventually, because the problem with povray is that you can't manipulate the object in real time, which makes it quite limited in terms of true 4D visualization. It's one thing to stare at a static image and try to extrapolate 4D from it, but to actually move it around and play with the object in real time makes a world of a difference. For example you can confirm whether or not your mental 4D model is accurate by checking whether it fits with how the object rotates.

The basic problem is how to simulate transparency in opengl. Suppose i have a bunch of ridges, some have 50% transparency, some have 25% transparency, some have 10%, etc.. How to correctly render this in opengl? Which compositing operation is the correct one? Should the ridges be rendered front-to-back or back-to-front? The lack of native transparency support in GL makes this quite difficult to get right.

Unfortunately currently my video drivers aren't working with 3D acceleration so I haven't been able to work on this for a while now.
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Re: Building 4D objects and worlds

Postby gonegahgah » Fri Nov 18, 2011 10:13 am

Hi quickfur. I agree with you that the cubes we draw are flat; but we can still see where the insides of the cube would go. When we draw a tesseract I can't see where the insides would go. All I can see are the hyperfaces squashed together.

In a cube the square faces only touch 4 of their sibling square faces via their edges. In a square the line edges only touch 2 of their sibling line edges via their corners. It stands from this that in a tesseract that its cube hyperfaces only touch 6 of their sibling cube hyperfaces via their squares. That is kinder shown correctly by the tesseract diagram though some of the squares seem to connect to the inside faces of the outside cube which isn't technically right. It would probably be more technically correct if the rest of the universe were the insides of the outer cube I guess rather than as we tend to view it. I guess that is just one of the problems of superimposition; but its good to understand these things.

Looking at the insides now. For a square its 'bulk' - I like that term; nice one Wendy - lies between its 4 edges. Unlike the 2Der we can certainly see that clearly. For a cube its bulk lies between its 6 faces. This should mean that for a tesseract that is bulk should like between its 8 cubes. Unfortunately all I can see are 8 cubes and I can't see anywhere left for the bulk in the diagram. This is different to a cube where even when we draw on a sheet of paper we can see the 6 faces and this fairly clearly leaves an apparent space for the bulk. It's quite clear their is a square at the top, the bottom, and on 4 sides leaving a very obvious space for the cubes bulk. This isn't the case for the tesseract because the cubes take up all the apparent space and leave no readily apparent location for the bulk. Where is the bulk to be found in the diagram? Do you think we can see the bulk in the diagram; or is it only inferred?
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Re: Building 4D objects and worlds

Postby gonegahgah » Fri Nov 18, 2011 10:45 am

hi mrrl. Thanks for that. That's even better. That certainly allows us now to compare the two styles.
I would love it if you would be able to add a ground in there as well.
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Re: Building 4D objects and worlds

Postby Mrrl » Fri Nov 18, 2011 12:30 pm

Actually there is work with transparent objects (they call them translucent, I guess), and I use it in my viewer: you have to enable blend, set some blend function (for example, set coeff Asrc for source and (1-ASrc) for destination) and if you send objects in back-to-front order it will give you pretty good results. But if you have depth buffer enabled and your order is not exactly correct, it will be not so good. When you add translucent triangle, it will hide all triangles (including solid ones) that you add after it and that are deeper in Z-buffer then it. So you will get hole in your picture. A way to fight it - add all solid triangles first, and continue with translucents sorted by depth more of less accurate (if you have time for it). It is what I do.
Probably it's not enough. Now I see two ways: First, I can split large triangles to smaller ones and sort them by depth of center (it's easy and will not take much CPU time). Or I can improve sorting function: compare depths or triangles not in centers, but in the center of their overlapping area. But this function is not transitive and not well-defined (if two triangles are not overlap, result is unknown). So we have a problem of topological sorting of digraph (probably with loops) :) Not for real-time rendering :(
But there is another thing. What I want do is to show 3D picture that is 4Der sees. For the point (x,y,z,w) screen coordinates will be (x/w,y/w), so I use combination of perspective projection in w direction (it gives the picture of 4Der's view) and orthogonal projection in z direction (what I see). For the proper view first projection should be opaque (with solid cells) and the second is translucent. But I do them in wrong order! First I calculate set of "contour" ridges, project them to z=0 space, and then feed resulting triangles to opengl using w as depth! Looks very wrong.
Probably first thing I'll try is to sort triangles by z (or by z/w) coordinate and use it for the output order. But I afraid that result will be even worse... I don't want to compute 4D occlusion in real time!
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Re: Building 4D objects and worlds

Postby quickfur » Fri Nov 18, 2011 4:08 pm

gonegahgah wrote:Hi quickfur. I agree with you that the cubes we draw are flat; but we can still see where the insides of the cube would go. When we draw a tesseract I can't see where the insides would go. All I can see are the hyperfaces squashed together.

Ahhhh yes, that is exactly the point that that I addressed on my website.

Basically, you're interpreting the image as 3D. Which is perfectly normal, since our brain is trained to do just that. Just like when a 2Der looks at the projection of the cube, it can see nowhere for the inside of the cube to go, because, from its 2D perspective, that inner square and the 4 trapezoids have already completely filled up the area of the projection. Where is the space for that "inside"??

Try to think of how the 2Der sees the cube projection vs. how we see the same projection. What is it that makes us see 3D depth in that image? Try to think of how the 2Der needs to interpret that projection in order to be able to see the 3D depth in it. Perhaps the first thing to understand is that the bigger square and the smaller square are actually the same size; the smaller square looks smaller only because it's farther away. And those trapezoids are actually also squares, but they look distorted into trapezoids because they are being seen at an angle.

In a cube the square faces only touch 4 of their sibling square faces via their edges. In a square the line edges only touch 2 of their sibling line edges via their corners. It stands from this that in a tesseract that its cube hyperfaces only touch 6 of their sibling cube hyperfaces via their squares.

This is exactly right.

That is kinder shown correctly by the tesseract diagram though some of the squares seem to connect to the inside faces of the outside cube which isn't technically right.

You don't think so? Then what about the projection of the cube into 2D, where you have an outer square, and inner square, and 4 trapezoids? The trapezoids connect to the inside edges of the outside square. Is that right or wrong?

Or perhaps a better question to ask is, why do they appear to connect to the "inside edge" of the outer square? Is that really different from connecting to the "outside edge"? More on this below.

It would probably be more technically correct if the rest of the universe were the insides of the outer cube I guess rather than as we tend to view it. I guess that is just one of the problems of superimposition; but its good to understand these things.

And here you touch on another important point about these projections. The superimposition happens because we're not clipping surfaces that lie on the far side of the (hyper)cube. For this, you really should read this.

If you're confused by the superimposition, you're not alone. Basically, in our experience with 3D solid objects, most of the time they are opaque, so we actually only see the faces that lie on the near side, because the far side is hidden from our view. That's why we don't see any superimposition. For example, when you look at a cube face-on, you don't see the square-inside-a-square projection at all. You only see a square, and that's all. Where did the other 5 faces go? They lie on the far side of the cube. They have been obscured by the near face!

Now if the cube was made of glass, then it's a different story. Now we can see all 6 faces. And now we get superimposition: the front face superimposes over the 5 faces behind it -- you get the square-in-a-square projection. The 4 squares around the far face appear to connect to the "inside" of the front square, but we know that's actually just an illusion. If you turn the cube a little, it's obvious that they are on the surface of the cube, just like the front face. In fact, there's no such thing as an "inside edge" or an "outside edge"; they are all on the surface of the cube. The inside/outside distinction is really just an artifact of projection.

This is why I try to use HSR (hidden surface removal) whenever possible. It gets rid of the superimposition (at least for convex objects), and is closer to how we see solid, opaque objects.

Looking at the insides now. For a square its 'bulk' - I like that term; nice one Wendy - lies between its 4 edges. Unlike the 2Der we can certainly see that clearly. For a cube its bulk lies between its 6 faces. This should mean that for a tesseract that is bulk should like between its 8 cubes. Unfortunately all I can see are 8 cubes and I can't see anywhere left for the bulk in the diagram. This is different to a cube where even when we draw on a sheet of paper we can see the 6 faces and this fairly clearly leaves an apparent space for the bulk. It's quite clear their is a square at the top, the bottom, and on 4 sides leaving a very obvious space for the cubes bulk. This isn't the case for the tesseract because the cubes take up all the apparent space and leave no readily apparent location for the bulk. Where is the bulk to be found in the diagram? Do you think we can see the bulk in the diagram; or is it only inferred?

Again, we can see the 3D bulk of the cube only because our brain is interpreting the image as a 3D projection. There is nothing in the image itself that contains the 3D bulk! If a 2Der looks at the same image, it wouldn't see any 3D depth in it at all.

Or, to use a different illustration, think about this one: suppose we paint that cube projection on a piece of paper. Now hold up the piece of paper and look at it from the edge of the paper. What do you see? The projection image is now squished into a very flat image. Can you still see the 3D bulk in the cube? Maybe, maybe not. If you use your imagination, perhaps you can still imagine the thing expanding outwards from the paper to fill up a cubic volume. But the point is, this interpretation is done by your brain; the image itself is nothing but a flat 2D thing.

So the 3D bulk of the cube is actually only inferred. It's not in the image at all! But because our brain is so good at reconstructing 3D from 2D, most of the time we think in 3D without ever realizing that actually we only see in 2D; the 3D is all in our head!

So yes, the bulk is inferred... just like the cube's 3D bulk is inferred by the 2D diagram. But inferred doesn't mean that it's something nebulous -- our brain is inferring 3D all the time, but we don't feel like 3D is some nebulous abstract concept; we see it in our mind like it's real. I've spent a lot of time practising to infer 4D from 3D projections, so when I look at those tesseract projections, I can see how the thing "bulges outwards", not in the 3D sense, but in the 4D sense. It does take some effort to see this, and I don't know if i'll ever be able to see it without consciously wanting to do it (unlike seeing 3D, which we all do unconsciously), but it's certainly real, at least for me. I'd like to think that i'm relatively normal, not some insane freak who sees things others can't see. :lol:
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Re: Building 4D objects and worlds

Postby quickfur » Fri Nov 18, 2011 4:21 pm

Mrrl wrote:Actually there is work with transparent objects (they call them translucent, I guess), and I use it in my viewer: you have to enable blend, set some blend function (for example, set coeff Asrc for source and (1-ASrc) for destination) and if you send objects in back-to-front order it will give you pretty good results. But if you have depth buffer enabled and your order is not exactly correct, it will be not so good. When you add translucent triangle, it will hide all triangles (including solid ones) that you add after it and that are deeper in Z-buffer then it. So you will get hole in your picture. A way to fight it - add all solid triangles first, and continue with translucents sorted by depth more of less accurate (if you have time for it). It is what I do.

Hmm. Perhaps the solution is to disable depth buffering, and make sure rendering is back-to-front?

Probably it's not enough. Now I see two ways: First, I can split large triangles to smaller ones and sort them by depth of center (it's easy and will not take much CPU time). Or I can improve sorting function: compare depths or triangles not in centers, but in the center of their overlapping area. But this function is not transitive and not well-defined (if two triangles are not overlap, result is unknown). So we have a problem of topological sorting of digraph (probably with loops) :) Not for real-time rendering :(

Yeah this sounds quite complicated. :( Although it might be good enough for convex objects, I think. I'll have to experiment with it to find out. But first I have to install 3D drivers for my video card. :D

But there is another thing. What I want do is to show 3D picture that is 4Der sees. For the point (x,y,z,w) screen coordinates will be (x/w,y/w), so I use combination of perspective projection in w direction (it gives the picture of 4Der's view) and orthogonal projection in z direction (what I see). For the proper view first projection should be opaque (with solid cells) and the second is translucent. But I do them in wrong order! First I calculate set of "contour" ridges, project them to z=0 space, and then feed resulting triangles to opengl using w as depth! Looks very wrong.
Probably first thing I'll try is to sort triangles by z (or by z/w) coordinate and use it for the output order. But I afraid that result will be even worse... I don't want to compute 4D occlusion in real time!

The way I do it, I actually use two different projections: first 4D to 3D which gives 3D coordinates, then 3D to 2D using a different, not necessarily related projection. (Of course, in code this can be combined into a single matrix, but the idea is that I treat the 3D camera as an independent thing from the 4D camera.) This way, I can keep the 4D camera fixed, and then rotate the 3D camera to give the user a good view of the 3D structure seen by the 4D camera without changing the 4D viewpoint.
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Re: Building 4D objects and worlds

Postby Mrrl » Fri Nov 18, 2011 5:09 pm

quickfur, answer is in the new topic: viewtopic.php?p=16404#p16404
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Re: Building 4D objects and worlds

Postby gonegahgah » Fri Nov 18, 2011 9:02 pm

I wish I knew how to use a 3D modelling program because I want to offer what I believe is going to be about as close to showing us what a tesseract would approximately look like as I think we are going to get; at least as I envisage it in my head. Instead I'll just have to describe it and maybe someone here can produce an image for me if they kindly would.

Basically you need a translucent sphere so that we can see that it is a sphere.
On the surface you first need eight points at equal distances apart (like the eight points of a cube; though we aren't depicting a cube with this).
Each of the adjacent points are connected to each other via direct (not straight) lines that follow the surface of the sphere.
It would look a little like a cube with curved edges with its corners and edges on the sphere's surface.

Now you repeat this but displace the second cube at a short distance at an angle from the first though with the edges and corner points still touching the sphere's surface.
Then you connnect all the corresponding points between the two 'cubes' with lines that follow the surface of the sphere.

This should hopefully turn the surface of the sphere into a canvas where we have drawn eight cubes as we would draw them on paper; but because the surface is curved we can show the 8 cubes connected together via their side faces; leaving one hyperface (window into the cube) on the surface and one on the inside towards the bulk for each cube.

Effectively, you end up with 8 flat cubes equally dispersed across the face of the sphere. The final thing to draw is some short arrows from the centre of each of these flat cubes towards the centre of the sphere to show that the entire inside of the 'surface' cubes are connected to the surface of the bulk inside of the tesseract.

To my mind this would help more people more clearly understand what a tesseract is even though it is still falls short of the real thing.
It depicts that the cubes are merely 'surface' features of a tesseract and in no way represent any of the bulk inside.
It tries to depict that the insides of each 'surface' cube connect to the bulk.
It shows that the bulk is so much more than the 'surface' cubes.

We know that cubes don't have curved edges. By representing them as curved edges we are merely converying the idea that they are flat as far as the tesseract is concerned.
Throwing in a bit of non-Euclidean geometry I guess.

Is it possible to do a diagram up like this for us to discuss?
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Re: Building 4D objects and worlds

Postby quickfur » Fri Nov 18, 2011 9:50 pm

If I understand you correctly, you're projecting the 8 cubes onto the surface of a sphere, so that all 8 cubes are flat?

Sure, sounds possible. You'll have lots of intersecting lines, though, because the edges of the tesseract do not form a planar graph. Another thing is I don't know if you can arrange it such that each cube is equally deformed. I think they will end up unequally deformed, so it won't be easy to see their equivalence. Also, you won't be able to see the 3D volume of each cube, since they would all be flattened.

Under this representation, the 4D bulk would be represented by the 3D volume inside the sphere, right? I guess that works... except that I'm not sure if I can see how exactly this 3D volume would correspond with the 4D interior of the tesseract. For example, if I were to pick some random 4D coordinates inside the tesseract, I don't think i'd know how to map that to this diagram correctly. (And vice versa.)

As for actually drawing this diagram... i haven't done anything like this before, so i'll have to try and see if I can work out a spherical mapping that would work, but no guarantees. ;) Maybe you could take an image of a transparent sphere and try to sketch out the diagram on it with a paint program or something?
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Re: Building 4D objects and worlds

Postby gonegahgah » Fri Nov 18, 2011 10:34 pm

hi quickfur. That's a good idea although I wouldn't have the artistic ability to draw it in a paint program. I might be able to buy a rubber ball, mark out 8 points that are each equal distance from 3 of their neighbours, connect these. Mark another 8 points a little away from the 1st 8 points that also are each equal distance from 3 of their neighbours and connect them. Finally, connect the corresponding corners from one to the other. Then take a picture. That should work hopefully.

If you can find a way to draw it that would be even more fantastic too.

Hopefully it is worthwhile to explore this approach. I'm not sure it if can be turned into something more generally useful but I'm hoping.
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Re: Building 4D objects and worlds

Postby quickfur » Sat Nov 19, 2011 12:26 am

OK, if you draw it that way, then 6 of the cubes would appear like small distorted hexagons, whereas the first two will fill up essentially the entire surface of the sphere. And there will be lots of superimpositions between all the cubes.
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Re: Building 4D objects and worlds

Postby gonegahgah » Sun Nov 20, 2011 7:59 pm

hi quickfur. I was able to initially draw up a diagram which showed me that I was approaching it slightly wrong. I still think that what I am seeking is do-able but I am struggling with trying to draw in 3 dimensional space using 2D tools. I guess I'll have to initially try to describe it further.

One of the first steps is to realise that a sphere can be divided into 8 equal segments which corresponds to the number of cube hyperfaces on a tesseract. Now again we only want to use the surface of the sphere for drawing our cube hyperfaces as basically this gives us a better - not perfect - understanding of how the 4Der sees a tesseract. To them a volumetric figure, such as a cube or sphere, is pretty much near to flat relative to the hypervolume of their 4D objects; in relative size comparison.

The next thing to realise is that you can get a cube to face all of its 8 faces in a general sideways direction by standing it on one of its corners. Normally we think of a cube with 4 faces facing sidewards and one upward facing face and one downward facing face. By standing it on its point we end up with 3 downwards/sidewards facing faces and with a 60 degree twist above this 3 upwards/sidewards facing faces. This will be handy.

The other thing to realise is that each tesseract hypersurface cube connects all 6 of its square surfaces each to a different hypersurface cube; and the only one that they are not directly connected to is the hypersurface cube on the opposite hyperside. Going back to our sphere canvas this means that each of the 8 segments needs to be connected to 6 of its hyperneighbours.

Now, normally an eighth segment in a sphere is only connect by 'face' to 3 of its neighbours and this is where we also need to connect it to its diagonal neighbours as well to achieve the desired 6 connections; which also means that each segment is still not connected to the segment on the opposite side of the canvas sphere; exactly as we desire.

So to achieve this we need to distort 8 cubes onto the surface of the sphere as though they are standing on one of their points with the bottom 3 'squares' facing their immediate neigbours and the top 3 'squares' facing their diagonal neighbours.

Now normally, if you stick 8 cubes in the surface of a sphere so that the corners touch, only 6 corners of each cube will touch or aim at another. Of the remaining two corners on each cube one will point up and the other will point down towards the opposite cube. However, instead of corners touching, we want the square faces to be flush touching.

**** Is it possible, quickfur for demonstration purposes, for me to get you to draw 8 regular cubes with one corner point of each cube facing a common centre and the opposite point pointing away from the common centre and each of the other 6 points aiming at 6 other neighbours. (So wish I knew how to use 3D apps). Do you know what I mean? ****

To get the square faces to flush with each other we need to distort our cubes even further. We do this by taking the topmost corner point and bottom most corner point and open them up. This actually creates an analogy that we want. By opening just these two points up we are creating our 2 hyperfaces into the cubes. Just as 4D extrapolation tells us, this allows us to see inside the cubes; just as we understand a 4Der is able to do. We do have to realise that these are 3D views into the inside of the cube so; and not 2D, so I would put an arrow into these with the head of the arrow having 3 points with little arrows on each to represent 3D co-ordinates; just as a clue to the viewer.

The other analogy that these apparent hyperfaces provide is that they help us to understand that we can only see into the outer hyperface while the inner hyperface is connect to the inside bulk of the tesseract.

So that's a general description of what I am trying to draw. There will be 8 cubes drawn on the surface of the canvas sphere. Each cube will be standing on a corner with a corner pointing out from the centre of the sphere. Each of those outside corners will be shown as a gaping hole into the cube and each innermost corner will be shown as a gaping hole connecting the cube insides to inside of the sphere; or the bulk of the tesseract. Each of the faces of the cube will connect with its corresponding neighbour via this process. One last thing, the faces of the cube will be shown as fairly narrow to give a better representation of how the cubes form a very small part of the tesseract.

Hopefully we can get to representing what I mean. As I mentioned quickfur, is it possible to draw 8 cubes where all corners, ignoring the top and bottom corners, point to one of 6 neighbour cubes? What does a cube standing on its point look like?
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 1:21 am

gonegahgah wrote:[...]
One of the first steps is to realise that a sphere can be divided into 8 equal segments which corresponds to the number of cube hyperfaces on a tesseract.

Unfortunately, what you described is impossible to draw on the surface of a sphere. The only possible 8-fold subdivision of a sphere's surface that is perfectly symmetrical is the octahedral division. But in such a division, each subdivision can only touch 3 neighbours. There is no subdivision of a sphere into 8 equal regions where each region touches 6 other regions. It's a topological impossibility.

Besides, the symmetry of the octahedron, even though it is 8-fold, is different from the 8-fold symmetry of the tesseract. They coincide only in number, unfortunately.

[...]
To get the square faces to flush with each other we need to distort our cubes even further. We do this by taking the topmost corner point and bottom most corner point and open them up. This actually creates an analogy that we want. By opening just these two points up we are creating our 2 hyperfaces into the cubes. Just as 4D extrapolation tells us, this allows us to see inside the cubes; just as we understand a 4Der is able to do. We do have to realise that these are 3D views into the inside of the cube so; and not 2D, so I would put an arrow into these with the head of the arrow having 3 points with little arrows on each to represent 3D co-ordinates; just as a clue to the viewer.

Y'know, what you just described is just a highly-distorted form of the vertex-first projection of a tesseract into 3D:

Image

This particular image shows only 4 of the cells (because the other 4 are on the "far side"), but basically the cubes project to 4 parallelopipeds lying in a tetrahedral configuration. The other 4 cells, if you were to include them, project to another 4 parallelopipeds lying in a dual tetrahedral configuration. And it so happens that the convex hull of 2 dual tetrahedra is just an octahedron.

So basically what you're describing is to take this projection and disconnect the square faces that meet at the yellow vertex and squish them to the projection envelope, then inflate the result into a sphere.

Honestly, I see very little value in this kind of contrived diagram; you're not actually seeing the tesseract as it should be; you're just seeing a distortion of it where basically you have ripped open each of the 8 cubes and forced them into the limb of the projection. In doing this, you have actually lost the very thing that would have given you insight into the 4D bulk of the tesseract, because the two vertices that were 'exploded' in this way are actually the vertices lying along the line-of-sight of the 4D viewer: they are the corners of the tesseract that, when a 4Der looks at the tesseract, "sticks out" into the 4th direction. This "sticking out" is what gives the 4Der the impression of the tesseract's bulk protruding into the 4th direction; it's what makes it a full-bodied 4D bulk as opposed to a 'flat' 3D region.

Besides, in such an 'exploded' diagram, the 6 faces of the cube are no longer equivalent, so it fails to convey the equivalence of the tesseract's 24 square faces. Not to mention two of its 16 vertices have been "exploded" and missing from the diagram, along with 8 of its 32 edges. I would hardly call such a diagram a faithful depiction of the tesseract, sorry to say.

Moreover, while you have certainly managed to clear out the inside of the spherical volume by pushing everything to its boundary, it is by no means an adequate depiction of the 4D bulk of the tesseract. While it may in some sense convey that the tesseract's facets lie on its boundary and not in its interior, making the association between the 3D interior of the sphere and the 4D bulk of the tesseract is misleading at best. It gives a false sense of confidence that you have perceived the 4D bulk, when in reality it is not only merely a 3D volume, but a distorted one that has no connection with the real shape of the tesseract's 4D bulk.

And pushing everything to the boundary of the sphere also fails to convey the 3 degrees of freedom that you have on a real tesseract's surface, and gives you the false sense of confidence that you perceive the boundary of the tesseract, but actually it is only a "torn" version of it, where two holes (corresponding to the two vertices removed) have been torn in it, and the rest squished into a distorted 2D area. You are thus left with only 2 degrees of freedom, and it only covers an incomplete portion of the real tesseract's surface.

In other words, you're trying so hard to make a diagram that is inadequate in conveying the tesseract's true shape and also very hard to understand because of the distortions that have been applied to it -- why not use the projection in the first place? All it takes is to learn how to interpret the projection as a 4Der would interpret it, and you get the benefit of an accurate understanding of the topology of the tesseract's surface in addition to being able to perceive the real shape of the tesseract's 4D bulk. Why put so much effort into an inadequate shoehorning of the tesseract into a sphere, when a much more accurate tool is already available?
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Re: Building 4D objects and worlds

Postby gonegahgah » Mon Nov 21, 2011 3:14 am

hi quickfur. I'll try to build a model towards what I'm doing. I'll need a small cube and I'll place 8 cubes apex first onto each of the corners with some wire somehow and twirl them around so that each face is approximately facing the face of another outer cube. The small cube is simply to give me 8 points equidistant from each other. This should be possible because there is one cube to 0deg, another cube to 60deg, another cube to 120deg, another cube to 180def, another cube to 240deg, and a final cube to 300deg of each cube. And, the faces of a cube on its apex point off into these directions. I'll try to do this so I can visualise it better.

The problem I have with the projections are that they show 0% of the tesseract bulk. They give the impression of representing something when they really are the showing even less than the shell of something; because you have outside faces connecting to inside faces which just doesn't happen in a tesseract. They are not much better at representing a tesseract than a single line is at representing a cube.

I'm seeking to convey to even a novice that a tesseract is so much more than we can perceive. You by no means are a novice but you are locked in your thinking that a 3D projection can convey any of the sense of a 4D object. They don't even show how the faces connect outwardly to each other.

I'm looking at a way to represent this outwards connectedness and to convey some sense of the bulk at the same time. Unfortunately we can know how something works in principle without being able to properly have a sense of its being. It has concerned me a little that you have not been able to accept the 2.5D model of a cube as another representation of a cube; so I am growing doubtful that you will be able to accept anything but projections as 'true' representations of 4D objects. Non-euclidean geometry should make us realise there are other ways of representing things. A cube doesn't always have to have straight sides. Deformation is a natural part of the order; and your projections are every bit as much a deformation from the truth as any else. I'm afraid that straight lines have to be abandoned simply because they limit the concepts that can be projected. By keeping the lines straight; you have to deform everything else too much.
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 6:01 am

gonegahgah wrote:[...]The problem I have with the projections are that they show 0% of the tesseract bulk.

The problem I have with diagrams of cubes is that they show 0% of the cube's volume.

They give the impression of representing something when they really are the showing even less than the shell of something; because you have outside faces connecting to inside faces which just doesn't happen in a tesseract. They are not much better at representing a tesseract than a single line is at representing a cube.

I still don't understand what you mean by "inside faces".

I'm seeking to convey to even a novice that a tesseract is so much more than we can perceive. You by no means are a novice but you are locked in your thinking that a 3D projection can convey any of the sense of a 4D object. They don't even show how the faces connect outwardly to each other.

And neither does a photograph of a cube convey how the faces connect outwardly to each other.

[...] I am growing doubtful that you will be able to accept anything but projections as 'true' representations of 4D objects. [...]

I never said projections are the only "true" representations of 4D objects. What is a "true" representation anyway? The fact that something needs to be represented means that the representation is not the thing itself, so what is the meaning of "true representation"? But since we're back in this topic of representations:

If you really want to represent a tesseract in its full, unadulterated, undeformed state, then here's a representation that provides all of that: draw the Hasse diagram of its face lattice. At the top is a single node, representing the tesseract itself (if you like, its 4d bulk). Below this node are 8 nodes, each representing its 8 faces. Below these 8 nodes are 24 nodes, representing its ridges, and each of the 24 nodes have 2 edges to the layer of 8 nodes: each face node connects to 6 face nodes. Then below the layer of 24 nodes are 32 nodes representing edges. Each of the 24 nodes are connected to 4 of the 32 nodes, and each of the 32 nodes are connected to 3 of the 24 nodes. Then below the 32 nodes are 16 nodes representing vertices. Each of the 32 nodes are connected to 2 of vertex nodes, and each vertex node is connected to 4 edge nodes. Finally, below the vertex nodes is the null node, where everything connects to. Now feed this into a graph layout program, and you will get a diagram that represents the true tesseract. Here you have everything: all its vertices, all its edges, all its ridges, all its facets, and even its 4D bulk.

All other representations are, strictly speaking, a subset of this representation. Including what a native 4Der sees in her eye. Even what a 4Der sees is not the real tesseract; it is only a projection of the tesseract. Does this representation help you visualize the tesseract? Maybe. It doesn't help me, though. But your mileage may vary.

But while we're at it, why not look at other representations as well? Since you like curved lines, here's a diagram with curved lines:
Image

And if you like non-uniform deformations, what about this:
Image

But these diagrams only show edges, so they don't really show the "real thing", right? OK, what about this:

Image

This one shows the 8 faces of the tesseract folded out. All you have to do is to imagine that the faces at right angles are actually the same face once folded up, and the faces of the bottom cube are the same as the outer faces of the top 4 cubes.

But you may argue, this doesn't show the connectivity of those 8 faces. OK, then we have this:

Image

Each red dot represents a cube, and as you can see, each cube is connected to 6 other cubes. If you like, substitute a drawing of a cube for each red dot, and you'll have a pretty good representation of the tesseract's faces and how they connect to each other. This diagram is even better than trying to draw the thing on a sphere, because on a sphere, our poor 2D eyesight cannot fully capture all the information, so we're bound to make mistakes. With this diagram there's no ambiguity, and it's even perfectly symmetrical. If you bend the edges so that they fit on the outside of the octagonal boundary, then you can fit the representation on a circle. What better way to describe a tesseract to a 2Der?

I can go on. There are many other representations of the tesseract. How about this one:
Code: Select all
#
# 4D hypercube definition
#

polytope tetracube {
   dimension 4
   vertices {
      # Format: just the vertex coordinates
      # The order vertices appear here is significant, since it
      # defines the index of each vertex referenced by the face
      # lattice. Numbering begins from 0.
      <1,1,1,1>
      <1,1,1,-1>
      <1,1,-1,1>
      <1,1,-1,-1>
      <1,-1,1,1>
      <1,-1,1,-1>
      <1,-1,-1,1>
      <1,-1,-1,-1>
      <-1,1,1,1>
      <-1,1,1,-1>
      <-1,1,-1,1>
      <-1,1,-1,-1>
      <-1,-1,1,1>
      <-1,-1,1,-1>
      <-1,-1,-1,1>
      <-1,-1,-1,-1>
   }
   lattice {
      # Format: dimension + set of vertex indices defining the face
      # For facets (dimension = polytope dimension - 1), an optional
      # halfspace definition vector (normal vector + constant, such
      # that N.v - C <= 0 for all v in polytope).

      # Cells
      3, { 0, 1, 2, 3, 4, 5, 6, 7 }, < 1, 0, 0, 0, -1>
      3, { 8, 9,10,11,12,13,14,15 }, <-1, 0, 0, 0, -1>
      3, { 0, 1, 2, 3, 8, 9,10,11 }, < 0, 1, 0, 0, -1>
      3, { 4, 5, 6, 7,12,13,14,15 }, < 0,-1, 0, 0, -1>
      3, { 0, 1, 4, 5, 8, 9,12,13 }, < 0, 0, 1, 0, -1>
      3, { 2, 3, 6, 7,10,11,14,15 }, < 0, 0,-1, 0, -1>
      3, { 0, 2, 4, 6, 8,10,12,14 }, < 0, 0, 0, 1, -1>
      3, { 1, 3, 5, 7, 9,11,13,15 }, < 0, 0, 0,-1, -1>

      # Ridges
      2, { 0, 1, 2, 3 }
      2, { 4, 5, 6, 7 }
      2, { 0, 1, 4, 5 }
      2, { 2, 3, 6, 7 }
      2, { 0, 2, 4, 6 }
      2, { 1, 3, 5, 7 }

      2, { 8, 9,10,11 }
      2, {12,13,14,15 }
      2, { 8, 9,12,13 }
      2, {10,11,14,15 }
      2, { 8,10,12,14 }
      2, { 9,11,13,15 }

      2, { 0, 1, 8, 9 }
      2, { 2, 3,10,11 }
      2, { 0, 2, 8,10 }
      2, { 1, 3, 9,11 }

      2, { 4, 5,12,13 }
      2, { 6, 7,14,15 }
      2, { 4, 6,12,14 }
      2, { 5, 7,13,15 }

      2, { 0, 4, 8,12 }
      2, { 1, 5, 9,13 }
      2, { 2, 6,10,14 }
      2, { 3, 7,11,15 }

      # Edges
      1, { 0, 1 }
      1, { 2, 3 }
      1, { 4, 5 }
      1, { 6, 7 }
      1, { 8, 9 }
      1, {10,11 }
      1, {12,13 }
      1, {14,15 }

      1, { 0, 2 }
      1, { 1, 3 }
      1, { 4, 6 }
      1, { 5, 7 }
      1, { 8, 10 }
      1, { 9, 11 }
      1, { 12, 14 }
      1, { 13, 15 }

      1, { 0, 4 }
      1, { 1, 5 }
      1, { 2, 6 }
      1, { 3, 7 }
      1, { 8, 12 }
      1, { 9, 13 }
      1, { 10, 14 }
      1, { 11, 15 }

      1, { 0, 8 }
      1, { 1, 9 }
      1, { 2, 10 }
      1, { 3, 11 }
      1, { 4, 12 }
      1, { 5, 13 }
      1, { 6, 14 }
      1, { 7, 15 }

      # We omit vertices 'cos they're simply all possible singletons
   }
}

This one encapsulates the entire Hasse diagram of the tesseract's faces (i.e., surtopes). Like the Hasse diagram itself, this one is a pretty darned good representation of the tesseract, since it leaves nothing to question, everything is laid bare before your eyes, every vertex's coordinates, every interconnectivity, the dimension of every aspect of it. If you don't like the omitted parts, you could even list all the vertices (just replace the comment with the set of all singletons in the form "0, { i }" for 0<=i<8). In fact, this is the representation my polytope projection program uses internally. I chose to use this format because it is (one of) the truest representations you can get.

I can go on... there are many other ways to represent the tesseract. For example, here's yet another way: the tesseract is the set of all points satisfying max(w,x,y,z)<=1 for all real numbers w,x,y,z. This is also a pretty darned accurate representation of it. It has absolutely no distortion, no loss of information, and every part of the thing is represented, and it's pretty darned concise you can write it on a single line. You don't even need to draw a diagram, and you don't need to deal with long strings of numbers like the representation my polytope viewer uses.

But let's talk about representations for a moment.

That last representation I talked about is a good starting point. If you like, it is the "real" tesseract, not in the sense that that particular sequence of characters is equal to the tesseract, but in the sense that the set it describes is precisely the tesseract itself. Everything you can ever imagine about the tesseract is encapsulates in this set. Of course, it is also an uncountable set, being of equal cardinality to the set of real numbers, so there are an uncountable (infinite) number of members in this set. But that's beside the point. The point is that this representation is complete.

The Hasse diagram I talked about is also complete. It's actually "better" than the max(w,x,y,z)<=1 definition because instead of drowning you out with an uncountable number of points in the set, the Hasse diagram only contains the features of interest: the 4D bulk of the tesseract itself, the 8 facets, the 24 ridges, the 32 edges, and the 16 vertices. It even has the null element, which is a nothing associated with every polytope. The Hasse diagram doesn't tell us anything about, say, the point (.5, .123, .321, .92981), which happens to be inside the tesseract. Nor the point (1,0,1,0.399192931). Which is on one of the tesseract's ridges. So in this sense, it's "incomplete". It's missing information about points that are in the tesseract, in a sense. But in another sense, this information is already included in the Hasse diagram, because its nodes represent the features of interest of the tesseract, and these features already include these points. So in that sense, the Hasse diagram is also an accurate representation of the tesseract.

The only problem is, if you were to actually draw the Hasse diagram of the tesseract (or have a program draw it for you), it's pretty big, and there are lots of lines everywhere. Not the easiest thing to sort through. Sure beats trying to sort through an uncountable number of points, though!

Now what about this representation:

Image

This is actually a subset of the Hasse diagram. It only contains the 8 nodes representing the tesseract's facets, and has an edge where in the Hasse diagram there is a ridge node shared by two facets. This is pretty convenient, because it reduces the rather large Hasse diagram to something more manageable. Something that can fit in a nice little octagon with not too many edges to confuse things. And it shows you how the 8 facets are joined to each other. This diagram isn't complete, however. Although you can infer that each facet has 6 faces, it doesn't tell you what shape these faces have. It doesn't tell you where the edges are. It doesn't tell you how many vertices there are. And, perhaps more importantly for us, it doesn't tell us anything about the bulk of the tesseract.

But no worries, there's always the good ole cross-section method to help us on that front:

Image

If you stack these shapes together in 4D, you will get the 4D bulk of the tesseract. So this is a not-bad representation of the tesseract. It's kinda hard to see where the vertices, edges, ridges, and facets are, but the 4D bulk is the important thing, right?

But we're talking about representations. Here's a type of representation that we haven't really highlighted yet: T-E-S-S-E-R-A-C-T. These 9 letters represent the tesseract too. It's kinda hard to deduce very much from it, but it's certainly a valid representation. After all, we've been using it all along! And here's another kind of representation: point, tetrahedron, truncated tetrahedron, octahedron, truncated dual tetrahedron, dual tetrahedron, point. This is the same as the cross sections image above (well, it adds two points that are missing from the cross-sections... but that's just a nitpick). It's actually better than the image, if you think about it. After all, it's not all that clear what those objects are in the image, unless you've already seen them before. If you show it to someone who hasn't seen a truncated tetrahedron before, he may have the wrong 3D model of the shape in his mind, which would lead to a wrong understanding of the tesseract's 4D bulk when he puts the sections together. By using the precise name of each cross section, we ensure that there will be no misunderstanding.

What's my point, you ask?

I purposely brought out as many representations of the tesseract as I can find in the time it took to write the reply, because I wanted to highlight the fundamental idea of representations, and that is, you're using something to represent something else. Strictly speaking, we can't show you a real cube, because there's no way to transmit that cube directly into your mind. It has to go through that distorting process of projection into your eyes (and even worse, this process depends on circumstantial parameters like lighting, view angle, whether the air is foggy, the presence of other objects that clutter the projection image, etc.), and then your mind has to second-guess from the inadequate 2D image how to construct a 3D model of the cube. In all of these stages, the cube is being represented, often imperfectly, by other things, until eventually it arrives in your brain. And actually, what your brain perceives isn't a cube at all -- it's just a bunch of neural signals flying around in your brain cells. Those signals aren't a cube. For sure, they aren't anywhere near being in the shape of a cube -- that much we know for certain. They represent a cube. But they aren't the thing itself.

When you represent something, you often lose information. There are certainly ways of preserving information, such as the max(w,x,y,z)<=1 representation. But honestly, who besides mathematicians can really make much use of that representation? It's not easy to do very much with it, unless you have the necessary mathematical tools to work with it. The Hasse diagram representation is better, even though in the strict sense it doesn't preserve all the information about the tesseract. But it does point out the most salient features of the tesseract -- so it's useful in bringing your attention to features of interest, instead of drowning you out in non-descript points like (0.2, 0.341, 0.3921) which, although it is a part of the tesseract, isn't very interesting at all. So you lose some information, you gain the ability to easily single out features of interest.

But what is interesting? Everyone is interested in different things. For a set theorist, the max(w,x,y,z)<=1 representation is perhaps most interesting, because it is important from his point of view to be able to tell which points are in the set (like (0.1, 0.321, 0.994, 0.231)), and which points aren't (like (0.321, 0.323, 3.412, 0.321)). For the abstract polytopist, the Hasse diagram is most interesting, because it captures all the structure of the tesseract that he might want to work with: things like whether it's transitive on its flags, the in-degree of its j-faces, and so forth, all of which are readily available from the Hasse diagram.

So the whole argument about "true" representations is perhaps an unprofitable discussion, because ultimately, what representation is most useful depends on what you're interested in.

I happen to be interested to understand what goes on in a 4Der's eyes when it looks at 4D objects. How does a 4Der see 4D objects? It's by perspective projection. But perspective projection is a poor representation of 4D objects: after all, it's perspective, which means that it suffers from foreshortening and other such crippling deformations. And secondly, it's a projection, which means that you're actually losing the 4D bulk; you only see 3D facets. And even worse, half of opaque objects is not even visible, so actually you're only seeing half a shell of the 4D object itself! And to top it off, you run the risk of obscuring factors like the presence of other objects in the scene, which may be cluttering the background of the projection image, making it hard to understand, or worse yet, which may be obscuring part of the object. So all in all, the poor 4Der's eyesight is actually a pretty darned lousy way of understanding 4D. It would be much better off reading Hasse diagrams, for example.

In the same vein, our own eyesight is a pretty darned lousy way of understanding 3D. I mean, why do you think people need maps to get to places without getting lost? Because their eyes can't see 3D, for crying out loud, the 2D perspective projection images they do see simply do not convey 3D adequately! Their view gets obscured by such things as buildings and mountains, so they can't see where they're going! And I mean, when you look at a tree, do you even have any idea what's inside the tree? No? What do you mean you have to cut it up to see what's inside? What kind of lousy representation of the tree does your eyes give you anyway?!

I'm not too concerned about all this, though. Perhaps I'm just deluded, but I think I've a pretty good grasp of 3D. Over the years of living in 3D space, I've somehow managed to get a good grip of 3D even though the only thing I have is this lousy pair of eyes that can only see distortions. I've learned how to figure out 3D based on these distortions, and in fact, I'm ashamed to admit, I've become pretty comfortable with figuring out 3D using only these lousy distorted images.

And when I think of the poor 4Der who suffers from the same problem of lousy perspective projection eyesight, my sympathy for her is perhaps a bit tempered by the fact that she seems to be quite comfortable navigating the complexities of 4D space in spite of only being able see in 3D. Perhaps my compassion for her plight made me want to empathize with her by going into her eyes, figuratively speaking, to find out how she sees her 4D world. What I found there struck a chord in me -- it's pretty similar to my own experience trying to puzzle out 3D using only lousy distorted 2D images of it. In fact, I observe how she also uses the same type of techniques that I use to deduce 3D from 2D, to deduce 4D from 3D. And that gives me a similar kind of comfort that I have with the 3D world. Even though I can't see 3D, I've learned how to make use of the lousy 2D images my eyes give me to figure things out pretty well. And I'm learning to now follow the same principles that the 4Der uses to figure out the 4D world from those lousy distorted perspective projections. And I think i'm starting to get a bit more comfortable in 4D space.

I can't speak for you, though. Perhaps this is all just a silly exercise for you. I mean, why torture yourself by using imperfect information to figure out something, when there are much better ways of doing things? Like going back to the Hasse diagrams or the max(w,x,y,z)<=1 representation, which don't suffer from visual artifacts, perspective distortions, illusions, and all that nonsense. And even if you decide to take the hard way, why follow in somebody else's stupidity, right? After all, perspective projections aren't the only way to distort things. :P There are plenty of other ways. And perhaps you find those other ways better, because they preserve what's important for you, and only distort things that aren't so important. My priorities are not the same as yours, after all.
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Re: Building 4D objects and worlds

Postby Mrrl » Mon Nov 21, 2011 6:48 am

One more way to see a tesseract is like this:
Image
Yes, it's our old familiar view of "cube inside cube", but here we have free navigation over the surface...
Here you see only seven cells, like you look inside the tesseract box without the lid. You see seven distorted cubic cells and may imagine the bulk inside it, if you want.
Last year and half I've used to work and play with such images of polychora. The latest example was the omnitrucates simplex:
Image
But when I play with such models, I feel myself not like 4Der, but like habitant of 3D sphere. I have free navigation in it, I can operate with objects, I can investigate their relative positions - but when I play with puzzle based on 3D honeycomb (there are no such things yet :) ) or on hyperbolic honeycomb (and I really wrote it and solved couple of twisting puzzles in it), it's the same thing for me as image of 4D polychoron. Something is missing here. And I know what is it - it's the bulk of the polytope. For the puzzles we need to give to user as much information as possible - and in the most understandable form. And inner view of 3D sphere is much more convenient than any other form of projection of 4D body. Yes, mathematically the puzzle that we solve is 4D Rubic's cube - without any flaws. But we solve it as 3Ders, we don't need to use 4D imagination for it.
Now there are about 150 humans who solved 4D cube :) Is it many or few? Probably, there are much more people who could develop some 4D (and more) imagination... and we are only in a sandbox from their 4D point of view :) So let's play!
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Re: Building 4D objects and worlds

Postby gonegahgah » Mon Nov 21, 2011 12:33 pm

hi mrrl. It is a sandbox :-) Fun place to be even when the castles look different ;-)

hi quickfur. What will be interesting is if we can devise some way to steer through these different worlds in ways that make any sense.
What I hope can be done is to produce a way to steer through a 4D world.

It is interesting that in a 2D game that your main directional controls are generally left-right and for some free movement games forward-back.
These correspond to the 4 edges of a square.
In 3D games usually you have left-right + forward-back (& maybe jump) or in free movement games you will also have up-down or have this instead of forward-back.
These correspond to the 6 faces of a cube.
In a 4D game you may want to move optional up-down but the other important directions are the 3 'sideways' directions. I call them sideways because they all have equal value to a 4D worlder. In some respects this takes away the notion of left-right movement. Instead we may have to think of movement towards our muffa paw or its opposite umpta paw; movement towards our ulta paw or its opposite butay paw; and movement towards are contal centre or to its opposite.
These correspond to the 6 cubes of a tesseract.

But movement direction is always through the centre of the lower dimensional object. ie square directions are through the centre of the line sides, cube directions are through the centre of the square faces, and hypercube directions are through the centre of the cube hyperfaces.
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 3:49 pm

gonegahgah wrote:[...]
hi quickfur. What will be interesting is if we can devise some way to steer through these different worlds in ways that make any sense.
What I hope can be done is to produce a way to steer through a 4D world.

It's strange, I've managed to teach myself to navigate through 4D using only perspective projections. I can see elementary 4D scenes, in fact -- the 2D horizon, mountains in the distance, roadways that go around rivers, buildings, etc..

[...]
In a 4D game you may want to move optional up-down but the other important directions are the 3 'sideways' directions. I call them sideways because they all have equal value to a 4D worlder. In some respects this takes away the notion of left-right movement. Instead we may have to think of movement towards our muffa paw or its opposite umpta paw; movement towards our ulta paw or its opposite butay paw; and movement towards are contal centre or to its opposite.

Well, left-right movement has been expanded into a full 3D freedom. You can, while standing upright, rotate in 3 different ways, 2 of which change the direction you're looking at, and one of which only changes the orientation of what you see but doesn't change the direction you're facing.

I've always considered the idea of using the mouse to control horizontal viewing direction. Let's say you're looking into +W, and up is +Z. Then you can use the mouse's 2 axes to point in any direction in the XY plane (the corresponding rotations are in the WX and WY planes). If you rotate in the XY plane itself, then you're only changing the orientation of what you see, but not the direction you're facing. This is a less important movement for navigation, so I'd relegate it to auxiliary controls, perhaps two keys for spinning the view orientation left or right.

One caveat, though, is that rotating your facing direction this way changes your orientation depending on the order in which you do things, so if you move the mouse from point A to point B, you will always end up facing the same direction no matter how the mouse moves from A to B, but your final orientation changes depending on the path your mouse follows. For example, if you go horizontal first then vertical, you end up with one orientation, but if you go vertical first then horizontal, you end up in another orientation at 90° from the first. And if you trace out other kinds of curves, you end up in various other orientations. But you always end up looking at the same thing, so in the end it doesn't really matter that much (you can always use XY rotation to adjust your orientation afterwards).

What this means is that a 4D creature has an additional plane in which it can rotate without changing what it's looking at, and still keep itself upright. Now, from the perspective of what a 4Der looks like when she rotates in this way, it's exactly the same as if you, the viewer, change orientation (except that if she's the one doing the turning the background doesn't rotate).

Because of this additional freedom of movement, as well as the change in orientation depending on the path you turn through to look from point A to point B, this means that a 4Der's brain must be hardwired to automatically compensate for rotation in the XY plane, so it would recognize that you're looking at the same thing even though the image is arbitrarily rotated in the XY plane. They would be excellent at solving orientation puzzles, for example. Reading backwards, forwards, sideways, mirror-imaged, etc., would all be no problem at all.

(And I got all that just from perspective projections, can you believe it? :evil:)

[...]
But movement direction is always through the centre of the lower dimensional object. ie square directions are through the centre of the line sides, cube directions are through the centre of the square faces, and hypercube directions are through the centre of the cube hyperfaces.

And that is why when you bump into a tesseract, what hits your nose first is the center of a cube.
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Re: Building 4D objects and worlds

Postby Mrrl » Mon Nov 21, 2011 5:27 pm

quickfur wrote: If you rotate in the XY plane itself, then you're only changing the orientation of what you see, but not the direction you're facing. This is a less important movement for navigation, so I'd relegate it to auxiliary controls, perhaps two keys for spinning the view orientation left or right.


Actually it's one of the most important movement for us. It is equivalent of rotating yor 3D camera around the 4D view - you look at the projection from different XY directions and try to recover actual 3D scene from what you see. Or if you work with sections instead of projections, this movement gives you complete set of slices of what 4Der sees ahead.

[...]
(And I got all that just from perspective projections, can you believe it? :evil:)

I don't beleive. Why do you need projections for it? Just think about 4D and then translate what you see to projections (if you wish).
But yes, for investigation of additional rotation projections are useful. Namely, parallel projection in Z direction (top view). It will translate 4Ders to more familiar 3D habitants, for example, sentinent space rockets. They have eyes to see (in W+ directions), arms to work, legs (probably with rocket engines) to move across space... And when they spin around their W axis, they see that image of the space rotates around this axis too... Easy to imagine, easy to translate to 4D and then to perspective view projections... For example, when I make half-turn in that direction, I'll see the same scene is section viewer, but with swapped left and right sides :)

And that is why when you bump into a tesseract, what hits your nose first is the center of a cube.

Especially when the tesseract is 100 meters high building and you approach it from 1D edge side :) Exactly.
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 6:14 pm

Mrrl wrote:
quickfur wrote: If you rotate in the XY plane itself, then you're only changing the orientation of what you see, but not the direction you're facing. This is a less important movement for navigation, so I'd relegate it to auxiliary controls, perhaps two keys for spinning the view orientation left or right.


Actually it's one of the most important movement for us. It is equivalent of rotating yor 3D camera around the 4D view - you look at the projection from different XY directions and try to recover actual 3D scene from what you see. Or if you work with sections instead of projections, this movement gives you complete set of slices of what 4Der sees ahead.

Thing is, I'm thinking of this from a 4Der's point of view. As far as the 4Der is concerned, there's no additional information gained from this rotation, so it's not really that important. (Although one can imagine such a rotation as being some kind of body language when engaged in conversation, for example, without ever looking away from the other party.)

As for rotating 3D camera, I always consider that as a secondary issue, because it is completely absent from the 4Der's consciousness. In fact, in my mind I always treat the 3D view completely independently of the 4D view. The 3D view is only for the benefit of us poor 3Ders who can't see the entire projection at once; from the point of view of the 4Der (who, if this is a maze game, for example, is the player character) there is only the 4D->3D projection and nothing else. When we examine the projected 3D image, we may rotate it however we like, but the 4Der (the player character, using the maze example) is not even moving at all.

[...]
(And I got all that just from perspective projections, can you believe it? :evil:)

I don't beleive. Why do you need projections for it? Just think about 4D and then translate what you see to projections (if you wish).

You don't understand. I don't use equations or vectors or any of that stuff at all. I visualize 4D in terms of projections. I only do the math when I wish to confirm that my visualization is accurate. Usually I work entirely in terms of projections, manipulating 4D objects, moving around in 4D space, etc.. The fact that my visualizations are accurate in retrospect, when I check the math, gives me a lot of confidence that my method works. Maybe you don't think so, but that's OK, everyone has his own methods. :)

(Well, to be accurate, I'm not dealing with the projection images per se, because those are only 3D; what I'm really doing is applying 4D intuition that I developed from working with projections. Just like when we visualize 3D objects, we think in terms of what our eyes see -- i.e., in terms of 3D->2D projections, so when I play with 4D, I think in terms of 4D->3D projections.)

But yes, for investigation of additional rotation projections are useful. Namely, parallel projection in Z direction (top view). It will translate 4Ders to more familiar 3D habitants, for example, sentinent space rockets. They have eyes to see (in W+ directions), arms to work, legs (probably with rocket engines) to move across space... And when they spin around their W axis, they see that image of the space rotates around this axis too... Easy to imagine, easy to translate to 4D and then to perspective view projections... For example, when I make half-turn in that direction, I'll see the same scene is section viewer, but with swapped left and right sides :)

Hmm. If i understand you correctly, you're thinking in terms of orientation in 3D space?

That's not what I meant by working with projections. I think in terms of 3D projected images that the 4Der sees in her eye. The 4th direction is not completely lost, because in general, the perspective foreshortening still gives you diagonal lines parallel to the viewing direction. For example, if you imagine you're looking into a cubical tunnel (i.e., the inside of an elongated tesseract), then the floor is a frustum and you work in terms of (w,x,y) coordinates from the floor plus height z, which is vertically upward from the floor. Completely analogous to looking down a square tunnel (i.e., the inside of a long cuboid): you can see the 3D space you're in by tracing (x,y) coordinates on the frustum image that is the floor, then tracing the z coordinate vertically upwards.

In the cubical tunnel, when you rotate to +x, the projection image shifts from right to left in "inside out" rotation (for lack of a better word) and the floor spins around y axis. Then what was left frustum in the original view is now cuboidal in the center, and left frustum is now what was behind you. So this changes your viewpoint. The XY rotation I was talking about is rotation only of the projection image, nothing new comes into view.

And that is why when you bump into a tesseract, what hits your nose first is the center of a cube.

Especially when the tesseract is 100 meters high building and you approach it from 1D edge side :) Exactly.

Really? If you approach it from the 1D edge side, you hit the edge first. That's like walking into the corner of a building. Except that in 4D there are two kinds of corners, so you can also walk into a ridge. Both will give you a bloody nose, because ridges in 4D are sharp. I was talking about walking perpendicular to the wall. You only get a bruise. :)
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Re: Building 4D objects and worlds

Postby Mrrl » Mon Nov 21, 2011 7:11 pm

quickfur wrote: Thing is, I'm thinking of this from a 4Der's point of view. As far as the 4Der is concerned, there's no additional information gained from this rotation, so it's not really that important. (Although one can imagine such a rotation as being some kind of body language when engaged in conversation, for example, without ever looking away from the other party.)

As for rotating 3D camera, I always consider that as a secondary issue, because it is completely absent from the 4Der's consciousness. In fact, in my mind I always treat the 3D view completely independently of the 4D view. The 3D view is only for the benefit of us poor 3Ders who can't see the entire projection at once; from the point of view of the 4Der (who, if this is a maze game, for example, is the player character) there is only the 4D->3D projection and nothing else. When we examine the projected 3D image, we may rotate it however we like, but the 4Der (the player character, using the maze example) is not even moving at all.

I see. Single 4D camera may not give me all kinds of views. For example, if 3D->2D projection is made always in Y direction, it will be impossible to look at 3D view from the top: that would require from character to lay on the ground and game controls may not enable this kind of orientation: when you go by the ground, you should be in straight position. But I'll get all sideviews by spinning characted in XY plane

You don't understand. I don't use equations or vectors or any of that stuff at all. I visualize 4D in terms of projections. I only do the math when I wish to confirm that my visualization is accurate. Usually I work entirely in terms of projections, manipulating 4D objects, moving around in 4D space, etc.. The fact that my visualizations are accurate in retrospect, when I check the math, gives me a lot of confidence that my method works. Maybe you don't think so, but that's OK, everyone has his own methods. :)

(Well, to be accurate, I'm not dealing with the projection images per se, because those are only 3D; what I'm really doing is applying 4D intuition that I developed from working with projections. Just like when we visualize 3D objects, we think in terms of what our eyes see -- i.e., in terms of 3D->2D projections, so when I play with 4D, I think in terms of 4D->3D projections.)
[...]
Hmm. If i understand you correctly, you're thinking in terms of orientation in 3D space?


In 3D you may think in terms of perspective images as well as in terms of technical drawing (parallel projections) or maps (may be with depth layer) - all kinds of projections are good for us. Same is in 4D: you can imagine for 4Der sees, you can look at the map of 4D city or lanscape and traces of its habitants (if it is what you investigate now), sometimes it's useful to works with 3D sections or even with 3D nets (when we think about 4D clothes)... And sometimes math goes to the front level and all other things are just interpretations of mathematical results.

Orientation in 3D is useful method when I imagine behaviour of 4Ders in the global scale: not how they move legs and arms, but where they go and where they look. "Space rockets" model is good enough, at least for me. Then I can add depth to it, if I like, and switch to perspective views only when I'll work with vertical shapes (buildings, trees, road curbs and so on).

Really? If you approach it from the 1D edge side, you hit the edge first. That's like walking into the corner of a building. Except that in 4D there are two kinds of corners, so you can also walk into a ridge. Both will give you a bloody nose, because ridges in 4D are sharp. I was talking about walking perpendicular to the wall. You only get a bruise. :)


So you were serious about the center of the cube? Sorry, I didn't get it... Of course, you can hit any point at the cubic wall if it is on the proper height above the ground. I'm not sure what is worse: to bump in the edge or in the ridge of the building. Edge looks more sharp, but it'll damage you only along a vertical line, while ridge can cut you in halves :)
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 7:40 pm

Mrrl wrote:[...]
I see. Single 4D camera may not give me all kinds of views. For example, if 3D->2D projection is made always in Y direction, it will be impossible to look at 3D view from the top: that would require from character to lay on the ground and game controls may not enable this kind of orientation: when you go by the ground, you should be in straight position. But I'll get all sideviews by spinning characted in XY plane

This is why I treat the 3D camera as completely separate from the 4D camera. Like you said, if the 3D->2D projection is fixed, then sometimes you cannot see certain things in the projection. So the player needs to be able to rotate the 3D camera independently of moving the 4D character.

[...]
In 3D you may think in terms of perspective images as well as in terms of technical drawing (parallel projections) or maps (may be with depth layer) - all kinds of projections are good for us.

Yeah, I use perspective projections when I want to know how a 4Der sees something, but for more precise tasks like measuring lengths or estimating angles, parallel projection is better. Especially when visualizing complicated polytopes, you need parallel projection so that when you get to the limb of the projection the facets at 90° to the camera project to 2D areas in the image; otherwise they fall to the far side of the projection and is clipped by visibility, and also you can't say for sure if they are at 90° from the camera or 91° or 89°.

Same is in 4D: you can imagine for 4Der sees, you can look at the map of 4D city or lanscape and traces of its habitants (if it is what you investigate now), sometimes it's useful to works with 3D sections or even with 3D nets (when we think about 4D clothes)... And sometimes math goes to the front level and all other things are just interpretations of mathematical results.

I usually don't have a need to use 3D sections, although they are somewhat useful when decomposing a complicated shape into simpler building blocks (e.g., cut a 24-cell into a tesseract and 8 cubical pyramids).

But in the end, everything is just an interpretation of mathematical results, right? :) They are just a way to relate complicated math results like objects with 4 variables to something we're more familiar with -- visuals.

On an unrelated note, some math objects are easier to visualize than others... for example, trying to visualize the pieces of the Banach-Tarski paradox gives me a gigantic headache. :) Unless, of course, you make simplifying assumptions like each piece is an amorphous "cloud" of points inside a sphere with infinite density but also infinite holes. But some things cannot even be simplified this way... like trying to visualize aleph_1, for example. At one point, I thought I had managed to see it, but it turns out that what I had was only a picture of the ordinal epsilon_0. Aleph_1 is a lot bigger than epsilon_0. :)

Orientation in 3D is useful method when I imagine behaviour of 4Ders in the global scale: not how they move legs and arms, but where they go and where they look. "Space rockets" model is good enough, at least for me. Then I can add depth to it, if I like, and switch to perspective views only when I'll work with vertical shapes (buildings, trees, road curbs and so on).

I usually think of orientation is 3D as being in the bottom frustum of a tesseractic room. It's the same as 3D, just a bit flattened with a bit of perspective distortion, but that way I can still see the vertical direction outside of the hyperplane the 3D orientation is taking place in. :)

[...]So you were serious about the center of the cube? Sorry, I didn't get it... Of course, you can hit any point at the cubic wall if it is on the proper height above the ground. I'm not sure what is worse: to bump in the edge or in the ridge of the building. Edge looks more sharp, but it'll damage you only along a vertical line, while ridge can cut you in halves :)

True! It's interesting that in 4D, you have two different ways of piercing something, one is using a spike with 0D tip, which will pierce you through a point, and the other is using a "spike-blade" with 1D edge, which cuts you in a line. But this kind of cutting doesn't split you, unlike in 3D; to split you, you need to use a 2D ridge. So cutting with a 1D edge is in between piercing and splitting.

So a butcher will need a knife with a sharp ridge for splitting meat, but for fighting, this kind of knife is too cumbersome (too bulky to have good maneuvering properties). Using a weapon with a 1D edge is better for maneuvering, but you will be unable to block most blows with the weapon. So swordfighting in 4D is not practical. :(
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Re: Building 4D objects and worlds

Postby gonegahgah » Mon Nov 21, 2011 8:43 pm

What I am hoping for is a way to move through a 4D scene without the level of visual morphing that presently occurs. A 4D worlder doesn't see this morphing at all just as when we hold a cube and spin it round it maintains its shape in our minds; even though a picture of different orientations traced produces different line patterns.

Along with hoping to give a better, though still inadequate, visualisation of the step up to 4D I am hoping there is a way to present the movement that gives us a better understanding of how a 4Der understands movement. Although we 3Ders understand tesseracts as infinite 3D cubes inside each other, the 4Der does not see their 'cubes' as having a top, bottom, and 4 sides; as we do. Instead the 3 sidewards axis all have equal direction and can be moved superfluously through as simply as it is for us to turn our head. They don't have to turn and tilt their head as we do to achieve viewing through 3 dimensions. They simply turn their head through two axis of left-rightedness. This then becomes the challenging concept. How do you turn your head through two axis of left-right while still having one down.

We have to remember that for the 4Der down is not towards the centre of a sphere. For them it is towards the centre of a hypersphere. For us, we think the hypersphere is just infinite spheres superimposed; but for the 4Der it is not so. Instead it is a downwards that is defined by 3 perpendicular axis of sideways movement. This is just like our downwards which is defined by being perpendicular to our left-right/forward-back axis and a 2Ders is defined by their forward-back axis.

I'll briefly talk about gravity in this respect again as we did disagree on this quickfur. You felt that gravity would be less in 4D. You said that gravity will drop off at a much faster rate which you are absolutely correct about. Hopefully, this explanation can help you understand why gravity would actually be much greater though on the surface.

We have our current weight at the surface of the Earth. What if we we able to fit two Earths together. The second one would double our weight. If we added a third we would triple our weight. When we step up to 4D we are not double or tripling. We are actually multiplying by infinity because there are multiply superimposed spheres from our perspective. Fortunately we can change the calculation from L3 to L4. This gives us a more rational answer but it still results in a lot more bulk in the same place immediately below us. Gravity does decrease at squared rate from the hypersurface to the hypercentre of a hypersphere; whereas it decreases at a constant rate from the surface to the centre of a sphere. That is correct; but the amount of bulk at each hyperpoint of 4D mass greatly exceeds the amount of bulk at each point of 3D mass. If we had a hypersphere below us now we would be squashed flatter than flat.
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Re: Building 4D objects and worlds

Postby quickfur » Mon Nov 21, 2011 9:24 pm

gonegahgah wrote:What I am hoping for is a way to move through a 4D scene without the level of visual morphing that presently occurs. A 4D worlder doesn't see this morphing at all just as when we hold a cube and spin it round it maintains its shape in our minds; even though a picture of different orientations traced produces different line patterns.

Actually, we do see visual morphing when we spin a cube. It's our mind that interprets it as an unchanging object. In the same way, what appears to us to be crazy 3D morphings of objects is also seen by the 4Der, but her mind interprets it as an unchanging rotating object. Like I kept saying, it all depends on how your mind interprets the thing.

Along with hoping to give a better, though still inadequate, visualisation of the step up to 4D I am hoping there is a way to present the movement that gives us a better understanding of how a 4Der understands movement. Although we 3Ders understand tesseracts as infinite 3D cubes inside each other,

And that would be a not very useful understanding of a tesseract. Do we think of cubes as infinite squares inside each other? I mean, mathematically it's true, but it's not a very useful way to think about it. We think of it as a 3D volume bounded by 6 squares.

In the same way, a more useful way to understand a tesseract is a 4D bulk bounded by 8 cubes. It's straightforward dimensional analogy.

the 4Der does not see their 'cubes' as having a top, bottom, and 4 sides; as we do.

I can't tell whether you're talking about 3D cubes or 4D cubes. If you're talking about 4D cubes, they see it as having a top, a bottom, a 6 sides.

Instead the 3 sidewards axis all have equal direction and can be moved superfluously through as simply as it is for us to turn our head. They don't have to turn and tilt their head as we do to achieve viewing through 3 dimensions. They simply turn their head through two axis of left-rightedness.

They don't need to turn their head to see a 3D cube in full.

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.

We have to remember that for the 4Der down is not towards the centre of a sphere. For them it is towards the centre of a hypersphere. For us, we think the hypersphere is just infinite spheres superimposed; but for the 4Der it is not so.

Infinite spheres superimposed (or stack, which is what I think you mean) is not a hypersphere, but a spherinder. Totally different.

Instead it is a downwards that is defined by 3 perpendicular axis of sideways movement. This is just like our downwards which is defined by being perpendicular to our left-right/forward-back axis and a 2Ders is defined by their forward-back axis.

Correct. And this is actually quite easy to visualize using projections. But of course, you don't like using projections, so I can't help you here.

I'll briefly talk about gravity in this respect again as we did disagree on this quickfur. You felt that gravity would be less in 4D. You said that gravity will drop off at a much faster rate which you are absolutely correct about. Hopefully, this explanation can help you understand why gravity would actually be much greater though on the surface.

We have our current weight at the surface of the Earth. What if we we able to fit two Earths together. The second one would double our weight. If we added a third we would triple our weight. When we step up to 4D we are not double or tripling. We are actually multiplying by infinity because there are multiply superimposed spheres from our perspective. Fortunately we can change the calculation from L3 to L4. This gives us a more rational answer but it still results in a lot more bulk in the same place immediately below us. Gravity does decrease at squared rate from the hypersurface to the hypercentre of a hypersphere; whereas it decreases at a constant rate from the surface to the centre of a sphere. That is correct; but the amount of bulk at each hyperpoint of 4D mass greatly exceeds the amount of bulk at each point of 3D mass. If we had a hypersphere below us now we would be squashed flatter than flat.

You can't transplant 3D measurements into 4D and expect that it would make any sense. Nobody is saying that we have a 4D planet with exactly the same radius as the earth with 4D beings exactly the same height as we are. That simply doesn't make any sense.

Obviously a 4D planet has a lot more mass than a 3D planet of the same radius, but that's comparing apples and oranges. (Or rather, apples and plates.) No matter how much mass the planet has, on its surface gravity would have a constant value, right? And obviously whatever lives on its surface would be appropriately sized so that they don't collapse under their own weight, otherwise they would hardly be able to live on the planet in the first place, right? I mean, it's the same thing in our 3D world. There is a reason whale-sized creatures don't exist on land: they can't, because they would collapse under their own weight and won't be able to breath or move! Whatever exists on the surface of the 4D planet obviously would be appropriately-sized relative to the planet's gravity. Assuming that they live and walk around in a similar way to us, the force of gravity they feel would be approximately the same as what we feel, because otherwise, they would either be leaping across the skies or dying under their own weight!

Now, the difference is that in 3D, if you increase the radius of a ball, its mass increases proportionally to r^3, but in 4D, when you increase the radius of a ball its mass increases proportionally to r^4. So in 3D you have a wider range of sizes that are in that comfortable middle between dying from your own weight and leaping across the skies. In 4D, you have a narrower range, because mass changes faster with changing radius. Does this mean gravity is somehow "stronger"? Not really, in 4D mass simply increases faster with increasing radius, that's all. So increasing the radius by 1 adds so much more mass to a 4D creature than it does to a 3D creature, so the 4D creature would feel its weight increase much more than in 3D. The force of gravity is the same on the 4D planet's surface; what is different is that the 4Der has so much more bulk for gravity to act on.
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Re: Building 4D objects and worlds

Postby Mrrl » Mon Nov 21, 2011 9:34 pm

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?
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)
Mrrl
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