Topology & modelling 4D objects in 3D

[quickfur]

Recently, I've been thinking about the possible ways of representing 4D polytopes by mapping to the surface of a 3D sphere, as some have suggested. I've discovered that this is impossible without compromising the connectivity structure of the 4D polytope in some way. Here's why:

Topologically speaking, convex polytopes like the tesseract and the other convex symmetrical polytopes that we are most interested in are essentially tilings of the 3-sphere (4D sphere). Which means they share the same topology as the 3-sphere.

For those who are unfamiliar with topology, here's a quick-n-dirty layman's introduction: topology is concerned with how points on a surface are connected to each other, without regard for the actual shape of an object. That is to say, you can deform an object in any way you want, and it has the same topology as long as you do not introduce self-intersections in the object. So a donut (doughnut for you brits) has the same topology as a coffee cup, because you can deform the coffee cup by flattening it so that its cavity is eliminated, then squeeze the flattened cavity so that it acquires the same shape as its handle, and massage it into a donut shape. A sphere has the same topology as an ellipsoid, which has the same topology as a cylinder, and so on. But you can never deform a donut shape into a sphere, because there's no way of eliminating the donut hole without introducing self-intersections, or breaking the surface apart to introduce new holes.

Since an object retains the same topology regardless of what deformations are introduced, as long as the deformations don't break connectivity of the surface (create new holes) or add connectivity that wasn't there before (eliminate existing holes), they represent a fundamental property of a particular class of shapes, and they allow us to reason about shapes in a general way that applies regardless of what deformations are introduced. So this is a useful way of reasoning about the possibility of mapping the connectivity structure of a 4D polytope on the surface of a 2-sphere (i.e. 3D sphere), because no matter what kind of deformations are introduced, the topology remains the same. We don't want to break existing connectivity or introduce connectivity that wasn't there before, since otherwise it would not be a faithful representation of the original 4D polytope. So topology is perfectly suited for our task.

Now, since (convex) 4D polytopes are tilings of the 3-sphere, they have the same topology as the 3-sphere (you can "inflate" any 4D polytope so that its facets become "puffed up", so that it becomes a 3-sphere). So the question reduces to whether it's possible to reduce the 3-sphere into the 2-sphere (i.e., map the cells of the 4D polytope onto the surface of a 3D sphere while retaining the connectivity structure of the cells -- note that because we're using topology here, we're taking into account any deformations that you may dream of, or even deformations that no one has dreamt of yet). Note that this essentially equates to whether the 2-sphere and the 3-sphere have the same topology -- since by definition, having the same topology means you can deform one into the other without compromising the connectivity structure of the surface.

So let's look at some properties of the 2-sphere and the 3-sphere to decide whether this is possible. Obviously, if they have the same topology, then their respective properties must match, since otherwise they can't possibly be the same topology.

One interesting feature of the 2-sphere is the so-called hairy ball theorem. In layman's terms, the hairy ball theorem states that if you have a hairy ball, then it's impossible to comb the hair on the ball without creating a cowlick. That is, any continuous vector field on the 3D sphere must have at least one vanishing point.

What about the 3-sphere? The 3-sphere has a remarkable property that it does admit a continous vector field without any vanishing points. One such vector field is related to the Hopf fibration. The structure of this fibration is interesting, because it manifests itself in the m,n-duoprisms, which are 4D polytopes consisting of two mutually perpendicular rings of cells: one ring of m-gonal prisms, and another ring of n-gonal prisms. These two rings are interlocked, such that the ring of m-gonal prisms have exactly n members, and the ring of n-gonal prisms have exactly m members. The two rings lie on two great circles on the 3-sphere that do not intersect each other. They induce a structure of concentric tori that are nested one inside another, none of which intersect with each other. By assigning a direction to the tori such that the directions on adjacent tori are consistent, we successfully create a non-vanishing continuous vector field on the 3-sphere: that is, we've managed to successfully comb a "hairy 4D sphere" such that no cowlicks are created.

By the hairy ball theorem, such a structure is impossible on the 2-sphere. So this conclusively proves that the 2-sphere and the 3-sphere have distinct topologies, and so a faithful mapping from a 4D polytope to a 3D sphere is impossible. Furthermore, this particular property of the 3-sphere gives us a concrete example of a substructure of the tesseract that cannot possibly be faithfully represented on the surface of a 2-sphere: the 8 cells of the tesseract can be partitioned into two rings of 4 cells each. These two rings are (1) interlocking each other, (2) do not intersect each other, and (3) are equidistant from each other. Property (3) poses no problem for us, since we allow arbitrary deformations as long as they do not alter the connectivity structure of the tesseract. But properties (1) and (2) are impossible on the surface of a 2-sphere: there is no way you can have two rings on a 3D sphere that are both interlocking and non-intersecting. (In fact, (1) alone is already impossible: two rings that lie on the surface of a 3D sphere can't possibly be interlocking: it's only possible to have intersecting rings. To have interlocking rings you need to leave the surface of the 3D sphere.)

IOW, no matter what convoluted mapping you may invent to map the cells of a tesseract onto a 3D sphere (no matter how you stretch, squeeze, shear, or otherwise deform the cells), it will lose the interlocking rings structure that is present between the cells of the tesseract in 4D. That is to say, any mapping from the tesseract's cells to a 3D sphere's surface will lose some of the connectivity between cells that is present in 4D. So it will not be a faithful representation of the topology of the tesseract.

So the final answer to this subject is that, unfortunately, there's no way to represent the surface of a 4D polytope on the surface of a 3D sphere, without fundamentally altering its topology.

[wendy]

Since four dimensional polytopes are indeed tilings in S3, it follows a mapping exists.

One should note that the rings of cubes in the tesseract are indeed interlocking. One can not reduce the circles to vanishing without crossing the other circle, so they must be interlocked. The rings don't interlock in 4d, but then this is only through the additional dimension. You can uncouple any knotted structure of lines in 4d, though.

[gonegahgah]

Hi quickfur. Thanks for having a think about it. I must admit I haven't done much further thinking about it presently. Been too busy.

One of the things intriguing me at the moment is this idea of having an in²side as well as insides.
It's actually nice that we call it the in-'side'. That's probably an accident but its probably what a 4Der would call it as to them it is a side; of which they can see two whereas we conceive it as one.

The same goes for the poor 2Der who can only conceive of their objects having a single inside whereas we perceive their objects as have two sides.
It is a little difficult for the 2Der to understand how we can see their objects from two sides; which they think of as inside.

But, I guess we don't think of a line as having a side so perhaps 4Ders won't consider our planes as sides at all. Wonder what they would call them.

So, just as a 2Der struggles to depict their objects as having 2 insides; I wonder how we go about depicting our 3D objects as having two insides?

[quickfur]

[...]One of the things intriguing me at the moment is this idea of having an in²side as well as insides.
It's actually nice that we call it the in-'side'. That's probably an accident but its probably what a 4Der would call it as to them it is a side; of which they can see two whereas we conceive it as one.

The same goes for the poor 2Der who can only conceive of their objects having a single inside whereas we perceive their objects as have two sides.
It is a little difficult for the 2Der to understand how we can see their objects from two sides; which they think of as inside.

But, I guess we don't think of a line as having a side so perhaps 4Ders won't consider our planes as sides at all. Wonder what they would call them.

So, just as a 2Der struggles to depict their objects as having 2 insides; I wonder how we go about depicting our 3D objects as having two insides?

The way I think about it is that what we perceive as purely 3D objects are actually very thin prisms of said 3D objects. For example, what we perceive as a cube is actually a tesseract with an imperceptibly narrow 4D thickness, such that its 6 faces are actually very thin cubes (from a 4D point of view) that we see as squares, and the "inside" is actually the inside of the tesseract compacted so that only 3 dimensions are measurable. This "inside" has two boundaries, from a 4D point of view, which are the two cubical facets that a 4Der sees when she looks at the object.

Of course, in a mathematical idealization, a 3D cube's 4D thickness would be actually zero, not just some imperceptibly small number. So you just start with the narrow prism and take the limit (ala calculus) as the thickness approaches 0, and you'll have a pretty good idea of what happens in the ideal case.

[4Dspace]

Interesting discussion. A bit over my head though. I have a question:

As I am looking at 3d object from 4D, I see how the 3 faces of a cube seen vertex first, merge into a plane. Yes, a perfect 2D plane, delivered to my eyes by parallel rays of light (and parallel rays are interpreted as 2D surfaces).

I wonder how difficult it would be to make a visualization of a cube with its faces colored differently: say, green, yellow, red, orange, purple and blue. Then, seen from 4D, you see how 3 faces merge into a plane divided into 3 segments. So, essentially, seeing a 3D object from 4D is actually very easy, because 3 faces become a plane, 3D->2D, putting us back to our familiar, homy, 3D (4D - 1D = 3D easy!)

From 4D POV you can see this divided into 3 colored segments plane sort of wobble into a segment of a cube as you approach it or cross to another direction. You can see 3D faces proper only from 3D. From 4D again they merge into a plane, making sort of 2d->3d->2d->3d..

I wonder how it would look, how the colors on the wobbling plane would change as you go around the cube looking at it from all 8 directions -?



Another question I have: to me it looks as if Abbott made a mistake in Flatland when said that one can see "insides" of a 3D object from 4D. I don't see how it is possible. Here is what I see when I look at a 3D object from 4D:

Imagine a cube made up of 27 stacked balls (3 per edge). Imagine that the balls are projected onto each face (so the ball in the very center of this cube is obscured by its neighbors on all sides). Now, looking at the vertex of the cube from 4D, the 3 faces seen, orthogonal in 3D, collapse into a 2d plane divided into 3 segments.

Notice that the ball at the vertex is represented 3 times on this divided into 3 segments plane. The balls at the edges, are seen twice (on each side of the edge). The balls that lie in the center of the face are seen once each.

The central ball is not seen no matter how you look at this cube from 4D (just like you'd not see it from 3D either).

And this is clear from a primitive example of "matter" that is made up of 27 atoms. Only 26 of them can be seen from 4D, and only after looking at the cube from all 8 directions. Only a cube made up of 8 atoms shows all of them (not at once, just like in 3D).

And so, from 4D we are also limited by seeing surfaces only. And because of that, we cannot see inside a 3D object from 4D no more than we can see inside it from 3D. In fact, from either number of dimensions we can see the only the faces and not insides.

To me looks like Abbott made a wrong analogy here. Only because from 3D you can see the whole of 2D lined up does not mean that you can do the same for a 3D object in 4D. You need to look at 3D from 5D for that trick to work.-?

is that right?

[wendy]

It's not. It is quite possible to project a movie onto the surface of a sphere, or to suppose the decorations on a ball somehow live.

In the case of the 4d sphere, one can have perfectly equidistant lines, but these 'rotate' around each other. One can demonstrate by noting that one can have two lines, each equidistant from a third, but none the less cross.

In the case of two lines that are perfectly as far from each other, one can not remove either circle without crossing the other circle, in the sense that one can not shrink to vanishing without crossing the other. In this sense, two orthogonal planes cross a glome as a pair of knotted lines, even though these are the very essence of un-knottedness.

[quickfur]

Interesting discussion. A bit over my head though. I have a question:

As I am looking at 3d object from 4D, I see how the 3 faces of a cube seen vertex first, merge into a plane. Yes, a perfect 2D plane, delivered to my eyes by parallel rays of light (and parallel rays are interpreted as 2D surfaces).

If, by "merge into a plane", you mean that the projection images of the squares fill up a 2D area (in this case a hexagon) with no superimpositions, then that's exactly what you'd get in a 4D->3D projection of a convex polytope with hidden surfaces culled. That's what I do all the time in my polytope visualizations. For example, my favorite projection of the 120-cell is:



All the dodecahedra seen here occupy their own volume in the image, without overlapping with its neighbours, because cells on the far side of the 120-cell have been culled, so you could think of this as what a 4Der would see in her native space. I don't have all the cells colored differently, because there aren't enough easily-distinguishible colors available, but I think the separation into 3 layers helps you discern where each cell is without too much difficulty. There are 45 cells visible here: the nearest one in blue (center), the 12 green ones surrounding it, and 20+12 more red cells. This is actually about all the cells you can see simultaneously if you were looking at a 120-cell in 4D, because the remaining cells lie on the far side (30 of them are on the "equator" - 90° from the 4D viewpoint, and the remaining 45 are on the far side, having exactly the same structure as seen here on the near side).

I wonder how difficult it would be to make a visualization of a cube with its faces colored differently: say, green, yellow, red, orange, purple and blue. Then, seen from 4D, you see how 3 faces merge into a plane divided into 3 segments. So, essentially, seeing a 3D object from 4D is actually very easy, because 3 faces become a plane, 3D->2D, putting us back to our familiar, homy, 3D (4D - 1D = 3D easy!)

From 4D POV you can see this divided into 3 colored segments plane sort of wobble into a segment of a cube as you approach it or cross to another direction. You can see 3D faces proper only from 3D. From 4D again they merge into a plane, making sort of 2d->3d->2d->3d..

I wonder how it would look, how the colors on the wobbling plane would change as you go around the cube looking at it from all 8 directions -?

OK, you lost me here. I don't understand what you mean by "3 faces merge into a plane divided into 3 segments", or what "wobbling" means. Could you explain what you mean?

Another question I have: to me it looks as if Abbott made a mistake in Flatland when said that one can see "insides" of a 3D object from 4D. I don't see how it is possible.

There is no mistake. We 3D beings can see the inside of a polygon just as plainly as day. But a 2D being can only see it edge-on: they can only perceive the edges that surround the polygon, not the body of the polygon. Our ability to see the inside of a polygon would seem utterly magical to a 2Der, but it's actually just plain old 3D geometry: we're looking at a 2D object from a point outside the 2D plane in which it lies, so there's an unobstructed path for light to travel to our eyes from every point in the polygon. A 2Der can't see the inside of the polygon simply because any light rays from an internal point would be obstructed by the points in front of it up until the polygon's edge, so only the edge is visible to the 2Der.

Similarly, in 4D one can look at a 3D object from a point outside of the 3D hyperplane that the object lies in, and so the inside of the object is plainly visible.

Here is what I see when I look at a 3D object from 4D:

Imagine a cube made up of 27 stacked balls (3 per edge). Imagine that the balls are projected onto each face (so the ball in the very center of this cube is obscured by its neighbors on all sides). Now, looking at the vertex of the cube from 4D, the 3 faces seen, orthogonal in 3D, collapse into a 2d plane divided into 3 segments.

Notice that the ball at the vertex is represented 3 times on this divided into 3 segments plane. The balls at the edges, are seen twice (on each side of the edge). The balls that lie in the center of the face are seen once each.

The central ball is not seen no matter how you look at this cube from 4D (just like you'd not see it from 3D either).

That is only true if your viewpoint lies in the same 3D hyperplane as the cube. Once you place your viewpoint outside this hyperplane, every ball is plainly visible.

And this is clear from a primitive example of "matter" that is made up of 27 atoms. Only 26 of them can be seen from 4D, and only after looking at the cube from all 8 directions. Only a cube made up of 8 atoms shows all of them (not at once, just like in 3D).

This is only true if you're talking about the 4D cube. A 4Der that looks at a 3D cube can see all 27 balls. However, a 4Der looking at a 4D cube made of 81 balls can only see a subset of them: at the most, those that lie on 4 of the 4D cube's surface.

And so, from 4D we are also limited by seeing surfaces only. And because of that, we cannot see inside a 3D object from 4D no more than we can see inside it from 3D. In fact, from either number of dimensions we can see the only the faces and not insides.

This is correct, except that you're confusing yourself by using ambigous terminology. In 4D, a "surface" is a section of a 3D hyperplane. Thus, in 4D, you see things not in terms of 2D regions (as we do in 3D): 4D objects have surfaces that have 3 degrees of freedom. In other words, the surface of a tesseract is not, as one might think, a bunch of connected squares, rather, it consists of 8 cubes. A 4Der can no more see all 8 cubes on the surface of a tesseract simultaneously, than we can see all 6 squares on the surface of a 3D cube simultaneously (unless the objects in question are transparent). But a 4Der can see the entire 3D cube, corners, edges, faces, volume, all of it, as plainly as we can see every point of a square, corners, edges, and area.

In the same vein, while we 3Ders can see the inside of a square as plainly as day, a 2Der looking at the same square can only see its edges, and only at most 2 edges at a time. But the area of the square, to the 2Der, constitutes the "inside" of the square, because, from the 2D perspective, there's no way ot reaching into the inside of a square without passing through its edges. But the same area from our 3D point of view isn't an "inside" at all, it's just the area on a surface. We can touch any point on the surface of the square no problem, because we have access to the 3rd dimension.

Analogously, a 4Der looking at a 3D cube does not regard the inside of the cube as an "inside" at all -- to her, it's just a "hyperarea" on a surface. She can touch any point inside the cube without passing through the cube's faces, because she has access to the 4th dimension. But give her a tesseract (4D cube), and she would suffer from the analogous limitation: she cannot touch the inside of the tesseract without passing through one of its 8 cubical cells, and she sure can't see the inside unless the whole thing were transparent. Only (at most) four of the tesseract's 8 cells can be visible at a time, so to see every surface, she would have to look at the tesseract from behind as well.

But a 5Der looking at the tesseract would laugh and declare that he can see every point in the tesseract, no problem, because he looks from a point in 5D outside of the 4D hyperplane that the tesseract lies in, and so the entire inside of the tesseract is as plain as day to him. (Just don't give him a 5D cube. )

To me looks like Abbott made a wrong analogy here. Only because from 3D you can see the whole of 2D lined up does not mean that you can do the same for a 3D object in 4D. You need to look at 3D from 5D for that trick to work.-?

is that right?

Wrong. You can easily see the entire 3D object, insides included, from 4D. And there is no lining up involved here. The 4D viewpoint lies above the entirety of 3D space, just as the 3D viewpoint lies above the entirety of 2D space, so the inside of a 3D object is as plainly visible to the 4D viewer as the inside of a 2D object is plainly visible to us 3D viewers.

Again, you should not confuse yourself with ambiguous terminology. What is a surface to us 3D beings is but a mere ridge to a 4D being (just as what is a surface to a 2Der is just an edge to us), so to correctly understand 4D, we must be careful not to wrongly assume that a 2D surface divides space in 4D. It doesn't; it takes an entire 3D volume to divide space in 4D. So to speak of a "surface" in 4D, one must keep in mind that this surface is not 2D, but 3D.

[4Dspace]

Thank you for your replies.

I feel like such an oaf.. messed up a perfectly good thread discussing 4d objects in 3D with my apparently wrong visualizations of a 3d object in 4D. It was late for me, sorry. Maybe mods could move it in a separate thread -?

quickfur, I don't know what magical powers in viewing things a 4D-er may have. When I, the 3D-er, get into the 4D, my basic visual apparatus remains, and it is conditioned to interpret parallel rays of light that bounce off orthogonally from 3 faces of a cube in 4D as if coming from a 2-dimensional plane. And so when I see a 3d cube in 4D, looking at its vertex, the 3 faces around the vertex are all orthogonal to the 4th dimension. This fact is interpreted by my 3D eyes by a "loss" of a dimension: the 3d-surface appears to me flat like a 2d-plane. If the cube had all its faces colored differently, then this apparent 2d-plane would be divided into 3 segments of different color.

Loss of a dimension in 4D makes it appear similar to the familiar 3D. The extra dimension is there though. As I look at the 3d cube from different angles, this extra dimension appears to "wobble" in and out of existence. At times, to me, the 3 faces of a cube seem to lie on a same plane, which looks to my eyes sometime as 2d and sometime as 3d.

And so as a 3D-er in 4D, I don't see inside this cube. It's basically the same cube it was in 3D, its shape sort of shifts though, appearing to me flat like a pancake at certain angles..

[quickfur]

[...]quickfur, I don't know what magical powers in viewing things a 4D-er may have.

They don't. It's a purely geometric phenomenon.

When I, the 3D-er, get into the 4D, my basic visual apparatus remains, and it is conditioned to interpret parallel rays of light that bounce off orthogonally from 3 faces of a cube in 4D as if coming from a 2-dimensional plane.

And this is where one needs to realize that, from a 4D perspective, the cube's faces occupy as much importance as a (2D) square's edges to us 3D beings. You are correct that the light rays bounce off the square faces as from a 2D plane; but keep in mind that they still occupy non-parallel 2D planes in 4D. So one cannot treat the 2D planes in which the faces lie as though they were the same, because they aren't, even from a purely visual POV.

And so when I see a 3d cube in 4D, looking at its vertex, the 3 faces around the vertex are all orthogonal to the 4th dimension. This fact is interpreted by my 3D eyes by a "loss" of a dimension: the 3d-surface appears to me flat like a 2d-plane.

Ahhhh, I see where you went wrong. Your 4D viewpoint is placed such that it lies in the 3D hyperplane that the cube is in. You are perfectly right that from that angle, the cube appears flat on a 2D plane.

The analogous situation in 3D is that of looking at a polygon, say a hexagon, directly at one of its vertices. Where would the 3D viewpoint be? To truly look "directly" at the vertex, one must, of course, place one's eye along the vector that passes from the center of the hexagon to its vertex. That is, one's eye must lie in the same plane as the hexagon, displaced some distance away. From that viewpoint, the hexagon appears not as a polygon at all, but as a 1D construct consisting of a series of line segments. Even we 3D endowed beings are unable to see the entirety of the hexagon from this viewpoint.

The only time we can see the inside of the hexagon is when our viewpoint is moved off the plane in which the hexagon lies. Preferably, somewhere along the line orthogonal to the center of the hexagon, so that we get the full view of it. Thus, the analogous viewpoint in the case of viewing a 3D cube from 4D would require that you look, not directly at any particular vertex, but at all of them equidistantly. From such a viewpoint, the entirety of the cube is plainly visible to the 4Der, every part interior and boundary point simultaneously. And this visibility isn't confined to this particular special case, either. Just as any viewpoint outside of the 2D plane in which the hexagon lies will let you view the inside of the hexagon to varying degrees, so any viewpoint outside of the 3D hyperplane in which the cube lies will let you see the inside of the cube to varying degrees, the maximum degree being, of course, at the viewpoint described above where your eye is equidistant to all 8 vertices.

(And lest you doubt the possibility of such a viewpoint, I recommend doing a simple calculation: assume the vertices of the cube are all changes of sign of <1,1,1,0>, then take your 4D viewpoint as <0,0,0,5>. Now compute the Euclidean distance of <0,0,0,5> to each point, and see for yourself that they are all equal.)

If the cube had all its faces colored differently, then this apparent 2d-plane would be divided into 3 segments of different color.

Correct.

Loss of a dimension in 4D makes it appear similar to the familiar 3D. The extra dimension is there though. As I look at the 3d cube from different angles, this extra dimension appears to "wobble" in and out of existence. At times, to me, the 3 faces of a cube seem to lie on a same plane, which looks to my eyes sometime as 2d and sometime as 3d.

Correct. This is very commonly observed when visualizing 4D: 3D constructs have varying apparent volume (just as 2D constructs to us have varying apparent area, depending on what angle we look at them). The viewpoint you describe is the 90° viewpoint, where the apparent volume is zero.

And so as a 3D-er in 4D, I don't see inside this cube. It's basically the same cube it was in 3D, its shape sort of shifts though, appearing to me flat like a pancake at certain angles..

OK, hold it right there. If you're talking about a 3D being looking at another 3D object from a point in 4D, then that's a whole different kettle o' fish. A 3Der's eyes don't have enough dimensions to fully capture the scope of 4D vision; you can at the most see only a thin slice of it at a time. That is in no way indicative of what a 4Der can see. (This is, perhaps, a little technical flaw on the part of Edwin A. Abbott in Flatland, when he describes the experience of the Square being lifted into the 3rd dimension and seeing the 2D world laid bare before his eyes. This isn't technically possible unless he also acquires a 3D eye in the process, otherwise his 2D eyes will merely see a slice of this laid-bare view, and he would have to be turned from side to side in order to see all the slices, then piece them together in his head. But then, he would hardly be able to claim that he has "seen" an angle, because to do so requires 3D vision which he does not have, even if he is now displaced somewhere into 3D space!)

But you're perfectly correct that a 3D cube appears flat at certain angles --- specifically, this happens when the 4D viewpoint lies in the same hyperplane as the cube itself. Note that this hyperplane has 3 dimensions, which means there's a lot of freedom in moving your 4D viewpoint around while still remaining on the same hyperplane. You will simply get different projections of the cube into a 2D image when you do so -- perhaps this is what you meant by the "wobbling"?

Once you lift the 4D viewpoint out of that hyperplane, though, things become different. The cube will project to a non-zero volume, and now its internal points are laid bare before your eyes, just as the internal points of a hexagon are laid bare to our eyes once we move our eyes off the 2D plane the hexagon lies in. It's the difference between looking at a piece of paper edge-on versus face-on. The edge-on view is what a 2Der sees; the face-on view is what we (usually) see. Same thing with 4D. The face-on view of a cube is what we 3Ders see; the native 4Der, though, has an inherently different viewpoint: she usually does not look at a 3D construct from within the same 3D hyperplane, but from a point outside. Therefore, she does not (usually) see the face-on view, but the volume-on view. Just as a 2Der would find it hard to comprehend how one could look at a polygon face-on (what's a "face" to a 2Der anyway?), we may have a hard time with the concept of "volume-on". But it's as plain as day to the native 4Der.

[4Dspace]

Thank you quickfur for your explanation of 4D vision.

I sort of understand the concept of how a 4D-er would see the world, but I am a 3D-er who wants to understand 4D. So to me it would be very, very instructive to see the animations that do just that. That's what I meant with my badly worded idea of looking at a normal cube with differently colored faces from various angles in 4D.

And so as I walk around the cube in 4D, looking at it from different angles, it does sort of "wobble", especially when the change in colors (=faces) happens. I see this 2d apparent plane (apparent to my eyes only) divided into 3 colored-segments, and one of the segments is getting smaller (while another one is growing), and then, just before it disappears and another color appears, the real 3D nature of the cube shines through in this wobble into its inherent 3D-ness and wham! it's gone again and where just a sec ago was a cube now lies something flattened, not a thing in itself but an attribute to something else, a decoration on a surface.

So, the conclusion I made regarding higher-D visualizations (for us 3D-ers only, of course) is that an addition of a dimension makes and object actually appear flatter -- quite paradoxically, because I expected the opposite effect. So, if I draw an angle on a 2d plane, say L. Looked at from the 2D perspective, that is "the real corner". Something significant for a 2D-er. Move into 3D and, even though the same lines are seen drawn on the same 2d plane, their "corner-ness" is overshadowed by the linearity of the surface that emerges.

Similarly, what I noticed during my sojourn into 4D is that a cube has volume only in 3D, because in 4D the voluminous corner is substituted by a smooth surface (to my eyes only). To see that volume again, I have to move my POV into the same 3d subspace where the cube lives. From the other 3 3d subspaces, the cube looks flattened, and only the wobble reminds of its volume when I walk through the subspaces around it.

So, now, when I look at the corner of my room, I realize that it will appear to me flat from a certain 4D perspective. Actually, from most 4D perspectives. I suspect that this loss of dimension trick works similarly in n+1 D to infinity, which to me means (I'm still chewing on it, since it's very new to me) that what we got here, the 3D is the ultimate, the best, the most universal, the....

For example, you, the expert on higher dimensions and topology, why do you think there are exactly 3 dimensions? What's so special about them? As opposed to 4, 5, 7, etc? In other words, what do you know about 3D that makes it special?

[quickfur]

Thank you quickfur for your explanation of 4D vision.

I sort of understand the concept of how a 4D-er would see the world, but I am a 3D-er who wants to understand 4D. So to me it would be very, very instructive to see the animations that do just that. That's what I meant with my badly worded idea of looking at a normal cube with differently colored faces from various angles in 4D.

I can't say that it has been helpful to everyone, but many have informed me that they found my 4D visualization document helpful in learning 4D visualization. But there is at least one person who didn't find my approach helpful, so YMMV. The basic thrust of my idea is that the most natural analog of sight in 4D would be 4D->3D perspective projection, just as our own 3D sight is based on 3D->2D perspective projection. But since we are quite intimately familiar with 3D, it should be relatively easy for us to understand the 3D image that forms on the 4Der's retina when she looks at some 4D object. The only missing step then, is the 3D->4D depth inference that the 4Der instinctively does, which can be learned by us via dimensional analogy.

And I'd like to emphasize here that 4D depth inference only works when you start from 3D and go to 4D: it does not work if you start from anything less than 3D. A corollary of this is that staring at a bunch of 2D images on the screen, animated or otherwise, is not going to magically help you see 4D; you first have to reconstruct the full 3D projection image in your mind, and then go from that 3D mental model to 4D by learning how to infer 4D depth. The process is emphatically non-subconscious, because we 3D beings just don't grasp 4D in the same intuitive way we grasp 3D.

Another corollary is that trying to use our 3D-centric vision (which is basically a series of 2D projection images as captured by our eyes) to make the leap into 4D will most likely result in disappointment at best, or just mislead one into a wrong understanding of 4D. You need the entire 3D volume of the projected image for any 4D visualization to work. Trying to visualize 4D from 2D slices (or projections to 2D, etc.) is as futile as trying to visualize 3D by looking at projections of the 3D cube onto a line segment.

And so as I walk around the cube in 4D, looking at it from different angles, it does sort of "wobble", especially when the change in colors (=faces) happens. I see this 2d apparent plane (apparent to my eyes only) divided into 3 colored-segments, and one of the segments is getting smaller (while another one is growing), and then, just before it disappears and another color appears, the real 3D nature of the cube shines through in this wobble into its inherent 3D-ness and wham! it's gone again and where just a sec ago was a cube now lies something flattened, not a thing in itself but an attribute to something else, a decoration on a surface.

I think I'm still unclear as to exactly what process you're going through in your visualizations, because while your description does bear some similarity to how I perceive 4D, it also seems inherently different.

But as to the cube being "flat" in 4D, that's exactly right, in the following sense: in 3D, objects have 2D boundaries -- for example, a cube is bounded by 6 square faces. Now, the interesting thing about 2D surfaces in 3D is that they divide 3D space into two sides: in the case of the cube, its boundary divides 3D space into inside and outside (i.e., the space inside the cube, and the space outside). This isn't limited to finite objects; take an arbitrary 2D plane, for example, that extends in length and width indefinitely. This plane divides 3D space into two: a half-space on one side, and another half-space on the other side. So a 2D surface in 3D space has the property that it divides space.

The same 2D surface in 4D, however, does not divide space at all. An indefinitely wide and long 2D plane in 4D, for example, hardly presents a bigger barrier to a 4D being who is walking by than a telephone pole presents a barrier to us 3D beings walking on the street. The 4Der simply steps around the plane and passes on, without ever needing to touch it. This may sound strange, but consider the case of us 3Ders walking on the street past telephone poles (or lamp posts, if you prefer). If you're intoxicated you may walk into one, but generally speaking, they don't present any barriers to our movement along the street. If a telephone pole is planted in the middle of the street in front of us, we simply walk around it. Now, the same pole to a 2Der is an entirely different matter. A 2Der walking along the line that is its street (assuming 2Ders are walking on the surface of a very large circle, as we 3Ders walk on the very large sphere that is Earth, their streets can only be line segments), if it encounters a telephone pole, that pole would block its passage forward. It would have to climb over the pole or burrow underneath it in order to keep going. (Corollary: 2D streets don't have telephone poles: that would be like erecting brick walls at intermittent intervals along the freeway in our 3D world -- completely impractical.) Try to explain to the 2Der that we 3Ders can simply walk "around" the telephone pole, and it would be completely baffled. Walk "around"? What's "around"? How does one walk "around" a brick wall without climbing over it? The 1D pole divides 2D into two sides: the near side and the far side, and you can't get from one to the other without going through it.

The same happens in 4D: a 2D surface in 4D is no more than a "telephone pole", so to speak. It does not divide space at all. The 4Der simply walks "around" it.

However, this by no means indicates any magical power on the part of the 4Der: if instead of a 2D barrier, we place a large solid cube in the middle of the 4D road, the 4Der would be unable to walk past it without taking a detour. Make the cube large enough, and it becomes the 4D equivalent of a brick wall: there is no way to pass through it without climbing or burrowing (or using some high explosives to destroy it). So 4D space requires a 3D hypersurface in order to be divided. A 3D hyperplane divides 4D space into two parts, one on either side. Just as a 2D polygon in 3D space is "flat" to us 3Ders, even though to a 2D polygon in 2D space is very bulky from the 2Der's POV, so a 3D object which seems so solid and voluminous to us 3Ders is no more than a "flat" construct in 4D, a hyperplane (or subset thereof) that divides 4D space into two parts. So is it any surprising that a 4Der looking at a 3D object perceives it as being flat? -- for it takes much more bulk than a mere 3D object in order to fill up 4D hypervolume.

So, the conclusion I made regarding higher-D visualizations (for us 3D-ers only, of course) is that an addition of a dimension makes and object actually appear flatter -- quite paradoxically, because I expected the opposite effect. So, if I draw an angle on a 2d plane, say L. Looked at from the 2D perspective, that is "the real corner". Something significant for a 2D-er. Move into 3D and, even though the same lines are seen drawn on the same 2d plane, their "corner-ness" is overshadowed by the linearity of the surface that emerges.

Not sure what you mean by "linearity", but yes, seen from (n+1) dimensions, an n-dimensional object is "flat".

One has to be careful with the interpretation of "flat", though. It does not mean that the object in question somehow "loses" any of its dimensions -- it is still as "bulky" as it was in its native dimension, but it's just that there is now so much more space in (n+1) dimensions that whatever space the object occupied before merely amounts to a "flat" hyperplane. The object itself remains the same, but space has been expanded in a fundamental way such that there is so much more of it, and so much more surrounding the object.

Similarly, what I noticed during my sojourn into 4D is that a cube has volume only in 3D, because in 4D the voluminous corner is substituted by a smooth surface (to my eyes only). To see that volume again, I have to move my POV into the same 3d subspace where the cube lives. From the other 3 3d subspaces, the cube looks flattened, and only the wobble reminds of its volume when I walk through the subspaces around it.

So, now, when I look at the corner of my room, I realize that it will appear to me flat from a certain 4D perspective. Actually, from most 4D perspectives. I suspect that this loss of dimension trick works similarly in n+1 D to infinity,

I got you up to this point...

which to me means (I'm still chewing on it, since it's very new to me) that what we got here, the 3D is the ultimate, the best, the most universal, the...

... but I fail to see how you make this conclusion.

As I said, it's certainly true that an n-dimensional object is "flat" in (n+1)-dimensional space. But this applies in general, not just to 3D/4D. This also happens when you go from 2D to 3D. (A line segment is as bulky as a brick wall to a 2Der; to us, it is but a mere stick.) I don't see how the number 3 got into your line of thought there.

For example, you, the expert on higher dimensions and topology, why do you think there are exactly 3 dimensions? What's so special about them? As opposed to 4, 5, 7, etc? In other words, what do you know about 3D that makes it special?

First, I hardly consider myself an "expert" on higher dimensions, and especially not on topology (which I hardly had any idea of until a few years ago). If you want an "expert" on higher dimensions, you should be talking to Wendy. She's the one who can visualize 6 to 8 dimensions without any mathematical aids.

Second, the fact that the space we inhabit happens to have exactly 3 dimensions is a cosmological/theological question, not a mathematical one. There is nothing about the mathematics of geometry that inherently prefers 3 dimensions. It happens to be one of the most studied spaces, because, well, of its potential practical applications. Studying 4D geometry is all nice and good, but hardly the way to go if you're hoping for practical applications in this lifetime. If we existed in 4D, for example, 4D geometry (esp. practical 4D geometry, such as constructing or assembling objects of different shapes, etc.) would be much more developed than it is today. (I'm leaving 3+1 dimensional Minkowskian space-time out of this, because that's a completely different beast. The time dimension is strange.)

As for what makes 3D "special" in this universe, the reasons have more to do with the specific set of physical laws that govern this place than anything inherent about the mathematics of 3D space. You might want to consult this Wikipedia article for more details. It would be presumptuous to say that it's impossible for a universe with 4D space to exist (perhaps in the form of 4+1 spacetime) under a different set of physical laws. (Note that in the linked article, the assumption is that the familiar physical laws, general relativity and electromagnetism, hold. We don't know about what happens under a different set of physical laws -- because nobody has had a reason to study those, yet. So when it talks about the stability of atoms, etc., it's assuming the n-dimensional generalization of the familiar physical laws, not necessarily speaking of the inherent nature of n-dimensional space.)

[4Dspace]

Thank you , quickfur, for your explanation. Regarding your site, I am sorry if came across as if I did not benefit from it in my understanding of 4D. I did immensely. I told you, your hypercube is the best I've ever seen. And the colors, the text, the shapes -- it's the top of the line. Thank you very much for it.

Simply, in addition to seeing 4D as a 4Der, I'd like to experience 4D as if I am there personally. With my own eyes. Wouldn't that be fun too? And so I have questions regarding that. I am sorry if this appears to you like such a waste, looking at 4D with 3Der eyes, but that's what I want too. I hope you don't mind

Regarding a cube flattening out in 4D, I did as you suggested, with [1,1,1,0] [-1-1-1,0] looking at it from [0,0,0,5] and... What I noticed is a very important feature, which no one mentions, and that is, what SIDES of the faces do I see? Inside or out?

Say, I hang a clock on each face of the cube, facing outwards. It's like you were saying about the dividing plane. In 3D it's a 2d plane, and if I draw a clock on it, its hands will run clockwise. If I look at the same plane from below, the hands of the clock will run counterclockwise.

So, similarly, in 4D looking at this cube with clocks on its 6 sides facing outward, when this cube flattens from 4D POV, how do I see the clock on each face? running clockwise or counterclockwise?

To me it seemed that the top face I see clockwise, the bottom face counterclockwise and the 4 side faces shift, i.e. they flip their... chirality? as I walk past the cube, changing my POV ever so slightly.

Is that right?

I would love to see an animation that shows that. If the inside of the cube is colored differently than outside, that would be very instructive.


Re "Why 3D", it's not an ontological question. The science of topology certainly has an answer to this. I was hoping you would know.

[quickfur]

Thank you , quickfur, for your explanation. Regarding your site, I am sorry if came across as if I did not benefit from it in my understanding of 4D. I did immensely. I told you, your hypercube is the best I've ever seen. And the colors, the text, the shapes -- it's the top of the line. Thank you very much for it.

Thanks for the compliment.

Simply, in addition to seeing 4D as a 4Der, I'd like to experience 4D as if I am there personally. With my own eyes. Wouldn't that be fun too? And so I have questions regarding that. I am sorry if this appears to you like such a waste, looking at 4D with 3Der eyes, but that's what I want too. I hope you don't mind

Well, there's nothing to mind or not mind. I'm just saying that trying to visuailze 4D with 3D vision is of limited utility, and is furthermore prone to error, because a 2D retina is inadequate for capturing the full field of vision in 4D. Whether it's a "waste" or not is subjective, and that's not for me to judge.

Regarding a cube flattening out in 4D, I did as you suggested, with [1,1,1,0] [-1-1-1,0] looking at it from [0,0,0,5] and... What I noticed is a very important feature, which no one mentions, and that is, what SIDES of the faces do I see? Inside or out?

Let's say you were a 2D being, looking at a 2D hexagon. You wouldn't see the inside of the hexagon; you'd only see its edges. And furthermore, because edges divide space in 2D, they have two sides: what you see, looking from outside the hexagon, is the outside of its edges. The other side is where the edges connect with the interior of the hexagon, and are invisible unless the hexagon were transparent, or if you were inside the hexagon looking out.

Now look at the same hexagon from 3D. Which side of its edges do you see? The inside or the outside?

Say, I hang a clock on each face of the cube, facing outwards. It's like you were saying about the dividing plane. In 3D it's a 2d plane, and if I draw a clock on it, its hands will run clockwise. If I look at the same plane from below, the hands of the clock will run counterclockwise.

So, similarly, in 4D looking at this cube with clocks on its 6 sides facing outward, when this cube flattens from 4D POV, how do I see the clock on each face? running clockwise or counterclockwise?

You can figure this out by looking at the 2D->3D case. Let's take our 2D hexagon again, and look at it from a 2Der's POV. We can assign a direction to each edge, for example, so that it always goes left to right, if we're looking at the hexagon from the outside. If we were to look at the same edges from inside the hexagon, we'd see the directions reversed: they go from right to left.

Now, let's climb back up to 3D, and look at the same hexagon. The question is, do the edges of the hexagon run left to right, or right to left?

The answer is something that the 2Der probably has a hard time wrapping his mind around. Looking from 3D, the hexagon's edges aren't restricted to just left-to-right and right-to-left; they have a full 360° freedom. So the correct answer is simultaneously, both and neither. Both, because opposite edges of the hexagon point in opposite directions, and if you orient the hexagon so that a pair of its edges is horizontal, one will be left-to-right and the other will be right-to-left. Neither, because the other edges of the hexagon don't point left-to-right or right-to-left at all; they are slanting upwards/downwards.

But this doesn't mean that the edges lose their directionality, or that the hexagon as a whole loses its directionality; in fact, taken as a whole, we see from our 3D vantage that the edges' directions combine to form a clockwise circuit (or anticlockwise, as the case may be, depending on which direction we look at the hexagon).

In the 3D->4D case, a similar thing happens. What is to us a clear distinction between clockwise and anticlockwise, is to a 4Der merely two opposite directions from a 360° circle of possibilities. So the answer to your question is also simultaneously, both and neither. Both, because if we orient the cube appropriately, two opposite clocks will have opposite orientations. Neither, because there is no longer a binary choice between clockwise and anticlockwise in 4D; you have a full 360° range of possibilities. So the remaining 4 clocks will be "neither clockwise nor anticlockwise". Furthermore, depending on which side of 4D you look at the cube from, the orientations of the clocks will flip, but their relative orientations to each other remains unchanged.

To me it seemed that the top face I see clockwise, the bottom face counterclockwise and the 4 side faces shift, i.e. they flip their... chirality? as I walk past the cube, changing my POV ever so slightly.

Is that right?

I would love to see an animation that shows that. If the inside of the cube is colored differently than outside, that would be very instructive.

You may say they flip their chirality, but it would be more accurate to say that they "rotate" between the two chiralities. Just as the left-to-right and right-to-left distinction in 2D becomes a full 360° range of directions in 3D, so the clockwise/anticlockwise distinction in 3D becomes a 360° range of chiralities in 4D.

(In fact, this is why you can "flip" a 3D object into its mirror image if you rotate it through 4D. What to us is a clear left-handed or right-handed distinction in 3D, is merely a matter of orientation in 4D. Just rotate it and the orientation flips. Or, more accurately, in the intermediate stages of the rotation you traverse a 180° range of chiralities, ending up with the opposite chirality when you're done.)

Re "Why 3D", it's not an ontological question. The science of topology certainly has an answer to this. I was hoping you would know.

I don't see what topology has to do with why 3D is singled out among all other possibilities. The constraint is laid not by the mathematics of it, but the physics. In fact, topology regularly treats spaces in the general sense of allowing any integral dimension (or even non-integral dimension, if you're into exotic spaces).

[4Dspace]

Thank you quickfur for your reply

trying to visuailze 4D with 3D vision is of limited utility, and is furthermore prone to error, because a 2D retina is inadequate for capturing the full field of vision in 4D.

Yeah, I was wondering about the nature of the "visual" apparatus of this 4Der of yours and... it's not visual in the sense that info is delivered to it by light rays to a specific POV. Instead, in order to "see in volumes" as you call it, it appears, the 4Der is sort of omnipresent in the 4D space, which makes her POV sort of irrelevant. Here is why:

1. If I divide 4D space into 2 sides with a giant cube whose 3 faces extend into infinity, a 4Der can see only the 3 bounding faces of this cube. Together with the vertex, they form the summit of a pyramid. The clocks on all 3 bounding faces of this cube appear to her to run clockwise. All 3 of them.

2. The other interesting feature of this giant cube, dividing the 4D space into 2, is that no matter in what direction and how far the 4Der moves along this great divide, trying to find an embrasure to get to the other side of the 4D space, the summit of the pyramid formed by the 3 bounding faces of this cube is always beneath her feet (or in front of her <-- this detail is irrelevant). And the 3 clocks continue to run clockwise.

Now, to help her out, we shrink this immense cube, so that the 4Der can walk around it. What does she see? According to what you said above, she can see simultaneously both inside and outside the cube. But this implies that she is "looking" at it simultaneously from both sides of "the great divide" spoken of above. This appears self-contradictory: on one hand, a 4Der cannot get to the other side of the divide, on the other hand her vision is such that it affords her the view of the cube from both sides. -?

In order for her to see inside and outside of the cube simultaneously (the little cube this time ), she gotta be present on its both sides, and also inside it, which, in fact implies that a 4Der is omnipresent in 4D. Ah?

In the 3D->4D case, a similar thing happens. What is to us a clear distinction between clockwise and anticlockwise, is to a 4Der merely two opposite directions from a 360° circle of possibilities. So the answer to your question is also simultaneously, both and neither. Both, because if we orient the cube appropriately, two opposite clocks will have opposite orientations. Neither, because there is no longer a binary choice between clockwise and anticlockwise in 4D; you have a full 360° range of possibilities. So the remaining 4 clocks will be "neither clockwise nor anticlockwise". Furthermore, depending on which side of 4D you look at the cube from, the orientations of the clocks will flip, but their relative orientations to each other remains unchanged. [emphasis by 4Dspace]

Yes, I agree with you. The orientations of the clocks should flip together simultaneously (my analysis in my previous post about the faces of the cube was wrong).

But. The 4Der cannot see the far side of the cube, because you said it yourself, a 4Der cannot cross to the other side of the "divide". This "divide" marks the direction from which you look at the cube and, you can see only one side of it at any given time ---OR--- the "great 3d divide" that separates the 4D space into two halves is not a real barrier for a 4Der, since it's not the impediment to her "omnipresent vision". See? something does not compute here. The place ain't logical.

You may say they flip their chirality, but it would be more accurate to say that they "rotate" between the two chiralities. Just as the left-to-right and right-to-left distinction in 2D becomes a full 360° range of directions in 3D, so the clockwise/anticlockwise distinction in 3D becomes a 360° range of chiralities in 4D.

Rotate between the 2 chiralities... that's an interesting vision. Thank you for it

I thought about them and... in our 3D world the word is turn. They turn their orientation. But it was the rigid cube we started with... so, all its faces gotta turn in unison.

(In fact, this is why you can "flip" a 3D object into its mirror image if you rotate it through 4D. What to us is a clear left-handed or right-handed distinction in 3D, is merely a matter of orientation in 4D. Just rotate it and the orientation flips. Or, more accurately, in the intermediate stages of the rotation you traverse a 180° range of chiralities, ending up with the opposite chirality when you're done.)

Yeah... except that I thought that chirality is a fundamental thing in topology. Chirality that traverses a 180° range is something else altogether.

I don't see what topology has to do with why 3D is singled out among all other possibilities. The constraint is laid not by the mathematics of it, but the physics. In fact, topology regularly treats spaces in the general sense of allowing any integral dimension (or even non-integral dimension, if you're into exotic spaces).

No, each space has its own properties, limitations and advantages. The "great divide" that divides any space greater than 2d into 2 unequal sides (unequal, because chirality differs) is what.... somehow here I feel lack of terms... but in primitive language ...and one always has to start somewhere... somehow, in this simple fact the... inescapability? from the basic 3D setup is what makes the 3D "special". I can't word it yet, but I feel it.

[4Dspace]

PS
and so about the difference in our vision of 4D, now it became clear to me: all the objects in your view are transparent (thus you see their insides and their far sides). But I see surfaces, like in our real world.

[quickfur]

Thank you quickfur for your reply

trying to visuailze 4D with 3D vision is of limited utility, and is furthermore prone to error, because a 2D retina is inadequate for capturing the full field of vision in 4D.

Yeah, I was wondering about the nature of the "visual" apparatus of this 4Der of yours and... it's not visual in the sense that info is delivered to it by light rays to a specific POV. Instead, in order to "see in volumes" as you call it, it appears, the 4Der is sort of omnipresent in the 4D space, which makes her POV sort of irrelevant.

And here is where you slip up by assuming that vision is 2D. This is one of pitfalls I was alluding to. My 4Der has no omnipresence or omniscience at all.. This is purely a geometric phenomenon.

To avoid confusion and make sure we're both on the same page, let's be concrete. Let's place a 3D cube at the origin (0,0,0,0) with its corners at (±1, ±1, ±1, 0). Now let's say we are 3D beings confined to the hyperplane where the last coordinate is 0. To be concrete, let's say we're standing at (0,0,5,0) and looking towards the origin.

So what parts of the cube can we see? Let's take some test points. The center of the cube, for example, lies at (0,0,0,0). So our line of sight is from (0,0,5,0) to (0,0,0,0). A little calculation will show that the line of sight intersects the cube at (0,0,1,0), which means that the front face of the cube is obscuring the origin, so we cannot see (0,0,0,0) at all; we only see (0,0,1,0). Can we see (1,1,1,0)? If you do the calculation, you will see that the line of sight from (0,0,5,0) to (1,1,1,0) is unobstructed, so we can see that corner of the cube. What about (1,1,-1,0)? If you to the calculation, you'll see that the line of sight intersects the cube somewhere at (x,y,1,0) for some x and y. Which means that corner is obscured; we can't see (1,1,-1,0) from this POV.

If you collect all the points that can be seen from this viewpoint, you'll see that only a single square face of the cube is visible to the 3Der. Every other point is obscured.

Now, let's go to 4D. If the 4Der were to look from the same point (0,0,5,0), then no doubt she would see exactly the same thing: just a single square face, because that face obscured everything else behind it. But what if she looks from (0,0,0,5)? Let's take some test points. Can she see the center of the cube at (0,0,0,0)? A simple calculation shows that yes the center of the cube is visible. Which corners of the cube can she see? If you do the calculations, you'll discover that all corners are visible, something that is impossible in 3D. Can she see other points internal to the cube, say (1/2, 1/2, 1/2, 1/2)? Again a simple calculation shows that yes, this point is visible too. In fact, every point in the cube, both inside and outside, are completely unobstructed from this 4D viewpoint. It does not take any magic, omnipresence, or omniscience for the 4Der to see the cube in its entirety. This is purely geometrical.

Here is why:

1. If I divide 4D space into 2 sides with a giant cube whose 3 faces extend into infinity, a 4Der can see only the 3 bounding faces of this cube. Together with the vertex, they form the summit of a pyramid. The clocks on all 3 bounding faces of this cube appear to her to run clockwise. All 3 of them.

You're making the mistake of assuming that 4D vision is the same as 3D vision. It's analogous, but definitely not the same. She is not looking from the same hyperplane as the cube lies in, but from a point outside. Therefore, clockwise/anticlockwise is ambiguous: she sees them not as a 2D projection like we do, where we can unambiguously assign a clockwise direction; she sees the entire thing in 3D in some particular orientation. Remember that clockwise/anticlockwise is meaningless in 3D unless you're looking at the thing from another point in 3D. The 4Der is outside the 3D hyperplane.

2. The other interesting feature of this giant cube, dividing the 4D space into 2, is that no matter in what direction and how far the 4Der moves along this great divide, trying to find an embrasure to get to the other side of the 4D space, the summit of the pyramid formed by the 3 bounding faces of this cube is always beneath her feet (or in front of her <-- this detail is irrelevant). And the 3 clocks continue to run clockwise.

They are clockwise/anticlockwise only if you're looking at them from 3D.

Now, to help her out, we shrink this immense cube, so that the 4Der can walk around it. What does she see? According to what you said above, she can see simultaneously both inside and outside the cube. But this implies that she is "looking" at it simultaneously from both sides of "the great divide" spoken of above.

You're confusing yourself by equating inside/outside with the two sides of the great divide. They are completely unrelated things.

Take our favorite example of a hexagon. From a 2Der's point of view, the hexagon consists of 6 edges surrounding an inside area. When looking from outside the hexagon, the 2Der sees the "outside" of the edge. When standing inside the hexagon, the 2Der sees the "inside" of the edge. For a 2Der, an edge always has two sides, since edges divide 2D space. But look at an edge from a 3D point of view. Where are its two sides? There aren't two sides, because in 3D, the edge is surrounded by space. Edges don't divide 3D space.

But there's another aspect to this: when we 3Ders look at a hexagon, we see that it has two sides, like the two sides of a coin. If the hexagon were made of paper, for example, we could paint one side red and the other green. Now go back to the 2Der. Which "side" of the hexagon does the 2Der see, the red side or the green side? The answer is ... neither. The 2Der can only see edges, it cannot see area! So let's say we cut the hexagon in half. What does the 2Der see, red or green? Neither, because it's looking at the new edge introduced by the cut. It doesn't see the two "sides" of the hexagon at all! In fact, what the 2Der considers as "the" inside of the hexagon, is actually in itself a 2-sided surface to us 3Ders -- we just painted one side red and the other green.

Long story short, what the 2Der considers as "inside" and "outside" are completely different from what we 3Ders consider as "this side" and "that side".

Now, let's look at the 4D case. To us 3Ders, a cube consists of an "outside" -- its surface, consisting of 6 squares, and an "inside" -- the volume enclosed inside the 6 faces. Does this have any correspondence with what the 4Der considers as the two "sides" of the cube? Not at all. What divides 4D space is not the inside and outside of the cube; it is the volume of the cube itself. In other words, from the 4D point of view, the cube's volume actually has two sides -- if you like, there are two "copies" of the cube's volume, one facing one side of the divided 4D space, and the other facing the other side.

This appears self-contradictory: on one hand, a 4Der cannot get to the other side of the divide, on the other hand her vision is such that it affords her the view of the cube from both sides. -?

It's not contradictory, you just confused yourself over what "side" means. The inside/outside of a cube has nothing to do with the two sides of the dividing cube-shaped wall. Terms like "side" are ambiguous, because to us 3Ders it refers to one thing, but to the 4Der it refers to something else.

[...] But. The 4Der cannot see the far side of the cube, because you said it yourself, a 4Der cannot cross to the other side of the "divide". This "divide" marks the direction from which you look at the cube and, you can see only one side of it at any given time ---OR--- the "great 3d divide" that separates the 4D space into two halves is not a real barrier for a 4Der, since it's not the impediment to her "omnipresent vision". See? something does not compute here. The place ain't logical.

To clear this up, let's say we are standing on a road with a big hexagon resting upright in front of us. We can only see one of its sides, obviously. But we can also see all 6 edges and its hexagonal interior at the same time, no?

Tell this to a 2Der, and it will be totally puzzled. What? If you can only see one side of the hexagon at a time, how can you claim that you can see both the inside and the outside of the hexagon?

The problem is that the 2Der is thinking of "side" in terms of travelling in a direction in the plane of the hexagon, whereas we 3Ders are thinking of "side" in terms of travelling in a direction perpendicular to the plane of the hexagon.

Similarly, your confusion arises because you're thinking in terms of directions within the 3D hyperplane that the cube lies in, but the 4Der is looking at the thing from a direction perpendicular to this hyperplane.

[...]

(In fact, this is why you can "flip" a 3D object into its mirror image if you rotate it through 4D. What to us is a clear left-handed or right-handed distinction in 3D, is merely a matter of orientation in 4D. Just rotate it and the orientation flips. Or, more accurately, in the intermediate stages of the rotation you traverse a 180° range of chiralities, ending up with the opposite chirality when you're done.)

Yeah... except that I thought that chirality is a fundamental thing in topology. Chirality that traverses a 180° range is something else altogether.

No, chirality only arises when you embed a shape into a suitable ambient space. If you embed a n-dimensional object in n+1 space, there is no chirality since both chiralities would be equivalent to each other via a rotation.

I don't see what topology has to do with why 3D is singled out among all other possibilities. The constraint is laid not by the mathematics of it, but the physics. In fact, topology regularly treats spaces in the general sense of allowing any integral dimension (or even non-integral dimension, if you're into exotic spaces).

No, each space has its own properties, limitations and advantages. The "great divide" that divides any space greater than 2d into 2 unequal sides (unequal, because chirality differs) is what.... somehow here I feel lack of terms... but in primitive language ...and one always has to start somewhere... somehow, in this simple fact the... inescapability? from the basic 3D setup is what makes the 3D "special". I can't word it yet, but I feel it.

When you can word it, let me know. I don't know how to answer something based on your subjective feeling, which is unknown to me.

[4Dspace]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

[quickfur]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

If you have a hexagon painted blue with yellow edges, what would you, a 3Der, see? You'd see both blue and yellow.

So the 4Der would see both blue and yellow.

Of course, there's another aspect to this that should be cleared up: a 2Der looking at our painted hexagon can only ever see the edges of the hexagon, since a filled hexagon occupies area, and in 2D, area fills up space. From our 3D point of view, the hexagon has two sides (like the two sides of a coin), but these two sides are inaccessible to the 2Der. In fact, the very concept is foreign to them, because to see these two sides requires the 3rd dimension, which the 2Der cannot access. So the idea that the two sides of the hexagon can be painted with two different colors is a completely strange concept to them -- they wouldn't be able to wrap their mind around the idea that what to them is a single area (the area inside the hexagon) having two elusive "sides".

Similarly, we can only ever see the faces of a solid cube, since volume occupies space. We can't actually see volume (we think we do, but actually what we see are only 2D surfaces; volume is inferred by our brain subconsciously). From a 4D perspective, the volume inside the cube has two sides -- and just as the hexagon's two sides have nothing to do with its edges, so the two "sides" of the cube from the 4Der's point of view have nothing to do with the cube's faces. It's the volume itself that has two sides -- and unsurprisingly, this is a completely foreign concept to us 3Ders, because to perceive these two sides we have to see it from 4D, but we can't. To the 4Der, however, this fact is as plain as a coin having two sides. She can paint the two sides of the cube with different colors, but that cannot be seen from our 3D perspective, because we're only ever looking at the faces of the cube; we can't see the volume. It's the volume itself that has two sides, with respect to the 4th direction.

[4Dspace]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

If you have a hexagon painted blue with yellow edges, what would you, a 3Der, see? You'd see both blue and yellow.

So the 4Der would see both blue and yellow.

Let's label the XY plane Z, XZ Y and YZ, X, so we have 6 faces: -X,+X,-Y,+Z,+Y,-Z
Could you please assign the colors to these planes as seen from (0,0,0,5) POV?
Thanks

Of course, there's another aspect to this that should be cleared up: a 2Der looking at our painted hexagon can only ever see the edges of the hexagon, since a filled hexagon occupies area, and in 2D, area fills up space. From our 3D point of view, the hexagon has two sides (like the two sides of a coin), but these two sides are inaccessible to the 2Der. In fact, the very concept is foreign to them, because to see these two sides requires the 3rd dimension, which the 2Der cannot access. So the idea that the two sides of the hexagon can be painted with two different colors is a completely strange concept to them -- they wouldn't be able to wrap their mind around the idea that what to them is a single area (the area inside the hexagon) having two elusive "sides".

I disagree. 2Ders have a good grasp on a concept of clockwise and counterclockwise direction, because it is accessible to their experience on the plane. So, they should understand that one side of a plane differs from its other side and that this difference can be represented by different colors.

...From a 4D perspective, the volume inside the cube has two sides --and just as the hexagon's two sides have nothing to do with its edges, so the two "sides" of the cube from the 4Der's point of view have nothing to do with the cube's faces. It's the volume itself that has two sides -- and unsurprisingly, this is a completely foreign concept to us 3Ders, because to perceive these two sides we have to see it from 4D, but we can't.

In 4D space, as 3Ders, yes we can and do see the two sides. There is always 2 sides that mark 2 opposite directions. The descriptions of 4Der vision which you give appear to violate the very basic principle, and that is: you cannot look in two opposite directions at once. But for 4Ders --the way you describe it-- this constitutes "seeing in volumes". I think you too would benefit greatly from seeing animations that show surfaces in 4D. Seeing transparent objects obscures these important details

[quickfur]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

If you have a hexagon painted blue with yellow edges, what would you, a 3Der, see? You'd see both blue and yellow.

So the 4Der would see both blue and yellow.

Let's label the XY plane Z, XZ Y and YZ, X, so we have 6 faces: -X,+X,-Y,+Z,+Y,-Z
Could you please assign the colors to these planes as seen from (0,0,0,5) POV?
Thanks

Let's label the faces of a square (±1,±1,0) with 4 labels, according to the direction they face: -X, +X, -Y, +Y. Now color the outside of the square red and the inside blue. So what colors should be assigned to each of these planes, when you look at this square from the 3D viewpoint (0,0,5)?

Of course, there's another aspect to this that should be cleared up: a 2Der looking at our painted hexagon can only ever see the edges of the hexagon, since a filled hexagon occupies area, and in 2D, area fills up space. From our 3D point of view, the hexagon has two sides (like the two sides of a coin), but these two sides are inaccessible to the 2Der. In fact, the very concept is foreign to them, because to see these two sides requires the 3rd dimension, which the 2Der cannot access. So the idea that the two sides of the hexagon can be painted with two different colors is a completely strange concept to them -- they wouldn't be able to wrap their mind around the idea that what to them is a single area (the area inside the hexagon) having two elusive "sides".

I disagree. 2Ders have a good grasp on a concept of clockwise and counterclockwise direction, because it is accessible to their experience on the plane. So, they should understand that one side of a plane differs from its other side and that this difference can be represented by different colors.

No. They understand the difference between clockwise and anticlockwise, but that is a property of the edges, not of the area inside the hexagon. So tell me, how many areas are inside the hexagon? Clearly only one. From the 2D point of view, this is the inside of the hexagon, and its boundary, consisting of 6 edges, is the outside. So are you saying that 3D space is divided by the inside and outside of the hexagon?

From the 2Der's point of view, the plane is the universe, there is no "two sides". Remember, they have no experience with 3D, and no way to access the 3rd direction. As far as they can tell, the 2D plane comprises the entirety of the universe. Therefore, saying that the plane has two sides is completely meaningless to them. How would you explain where those two sides are? If you were a 2Der, can you point to where those two sides are? If you can't, then how can such a thing be real? -- you need access to the 3rd dimension for this to make any sense.

Let's take it even further. What about a 1Der? A 1D being lives on a line, and in its experience, that comprises the entire universe. The only directions it knows are forward and backward. Objects are line segments, and line segments have two endpoints. The stuff between these two endpoints are the "inside", and the two endpoints are the "outside". There is no such thing as sideways. Now let's look at a line from the 2Der's point of view. To them, a line divides space, so a line has two sides, one side for one half of the divided space, and one side for the other. So let's say the 2Der paints one side of the line red, and the other side blue. Now how would the 2Der explain this to the 1Der?

2Der: I have a line segment L with endpoints A and B. I painted one side red, and one side blue.
1Der: Oh, you mean A is red and B is blue?
2Der: No, one side of L is red, and the other is blue.
1Der: Huh? What is a "side"?
2Der: A line has two sides, the near side and the far side.
1Der: You mean the space in front of A and the space behind B?
2Der: No, no, I mean one side faces me, and the other faces away from me.
1Der: Right, so if you're in front of A, then A is one side and B is the other side?
2Der: No, I'm not standing on the line that L sits in. I'm standing to one side of the L.
1Der: Huh? You mean you're standing inside L?
2Der: No, I'm standing to the left of L.
1Der: What's "left"?
2Der: arghhh....

Now a 3Der comes into the picture.

2Der: Hey 3Der, I painted one side of L red and the other side blue.
3Der: Yes, I can see that.
2Der: Which color do you see?
3Der: I see both red and blue.
2Der: What?! How can you possibly see both sides at the same time? Are you omnipresent?!
3Der: No, I'm just looking at L from the 3rd direction.
2Der: What?! What's a "3rd direction"?

...From a 4D perspective, the volume inside the cube has two sides --and just as the hexagon's two sides have nothing to do with its edges, so the two "sides" of the cube from the 4Der's point of view have nothing to do with the cube's faces. It's the volume itself that has two sides -- and unsurprisingly, this is a completely foreign concept to us 3Ders, because to perceive these two sides we have to see it from 4D, but we can't.

In 4D space, as 3Ders, yes we can and do see the two sides. There is always 2 sides that mark 2 opposite directions. The descriptions of 4Der vision which you give appear to violate the very basic principle, and that is: you cannot look in two opposite directions at once.

The 4Der does not see in opposite directions at once. It's merely looking from a direction at 90° angle to the 3D hyperplane.

But for 4Ders --the way you describe it-- this constitutes "seeing in volumes". I think you too would benefit greatly from seeing animations that show surfaces in 4D. Seeing transparent objects obscures these important details

Again, you're confusing yourself by the ambiguous meaning of "surface". A surface to us 3Ders is a 2D construct. A cube's surface consists of squares. An icosahedron's surface consists of triangles. A sphere's surface consist of a curved 2D sheet. But you have to understand that in 4D, these things are not surfaces at all. They are mere ridges. A "surface" to a 4Der is a 3D construct. A tesseract's surface consists of 8 cubes. NOT merely the surface of the cubes, each of which consists of 6 squares, but the cubes' volumes. Take any of the 8 cubes that lie on the surface of a tesseract. Every point of the interior of the cube lies on the surface of the tesseract, not the inside. The inside of the tesseract is a hypervolume, whose boundary consists of 8 cubical volumes.

Find this confusing or strange? Let's go back to the 2D->3D case. A 2D being's concept of "surface" is not a 2D construct, but a 1D construct. A square's surface consists of 4 edges. A triangle's surface consists of 3 edges. A hexagon's surface consists of 6 edges. A 2Der's sight is only 1D: it cannot see an entire square at once, but it can only see, at the most, 2 edges at a time. So as far as the 2Der is concerned, it is impossible to see all 4 edges of a square simultaneously. You'd have to look from two opposite directions at the same time, which is impossible. Now comes along a 3Der, who claims that he can see all 4 edges of a square simultaneously. How can this be possible? Surely the 3Der must be omnipresent, because seeing all 4 edges of the square involves looking at it from two directions simultaneously, which is impossible.

Or, take the example of a line segment. Let's be concrete here. Say the line segment starts from (1,0,0) and goes to (-1,0,0). A 2D being standing at (0,1,0) will see one side of the line segment, whereas the other side is obscured from his view. So he says, let me paint this side of the line red, and then I'll walk around and paint the other side blue. Now, from his point of view, it's impossible to see both red and blue at the same time, because that would require looking at the line segment from two directions (+Y and -Y) simultaneously, which requires omnipresence.

But here comes along a 3Der, who's standing at (0,0,5). The 2Der asks, which color do you see, red or blue? The 3Der replies, I see both. The 2Der says, What?! How can you see both at the same time? Are you omnipresent? How can you possibly look from two directions (+Y and -Y) at the same time? The 3Der replies, I'm not. I'm just looking from a direction perpendicular to the plane you're in.

The 2Der is confused, and thinks the 3Der is talking BS, so to test him, the 2Der removes the line and puts a square in its place, with vertices (±1,±1). Then he stands at (0,5).

2Der: So, this square has 4 edges, facing the directions +X, -X, +Y, -Y. Now I'll paint the outside of these edges red, and the inside blue. So tell me, what colors do you see?
3Der (standing at (0,0,5)): I see red on the outside and blue on the inside.
2Der: What?! You're lying! How can you possibly see two colors at the same time? OK, let's call the edge on the +X side of the square +X, and the edge on the -X side of the square -X, and similarly for the +Y and -Y edges. Can you please assign colors to each of +X, -X, +Y, -Y, according to how you see them?
3Der: Erm, the edges are red on one side and blue on the other.
2Der: You're bluffing! How can you possibly see both sides of an edge simultaneously? How can you look simultaneously from +Y and -Y?
3Der: I'm not. I'm looking from +Z.
2Der: What on earth is a Z?! (pause)
3Der: I can see the entire area of the square, along with all of its edges.
2Der: What?! Are you saying you see in areas? That's impossible! Everybody knows we only see line segments!
3Der: ...

[4Dspace]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

Let's label the XY plane Z, XZ Y and YZ, X, so we have 6 faces: -X,+X,-Y,+Z,+Y,-Z
Could you please assign the colors to these planes as seen from (0,0,0,5) POV?
Thanks

Let's label the faces of a square (±1,±1,0) with 4 labels, according to the direction they face: -X, +X, -Y, +Y. Now color the outside of the square red and the inside blue. So what colors should be assigned to each of these planes, when you look at this square from the 3D viewpoint (0,0,5)?

That's not fair. I posited a concrete question. You reply with another question. Forgot, what that technique is called?

The point of my question was: fundamental property of a 2d plane is preserved, no matter how high in number of dimensions you go. 4Der can NOT see both sides of a 2d plane simultaneously. He can see only one side at a time. Just like us the 3Ders.

For you, once you get into 4D the fundamental properties of 2d seem to disappear and are replaced by other concepts, only because that would seem to make sense to you in a progression you're making from 1d -> 2d -> 3d -> 4d. But each higher in number of dimensions space can be deconstructed to its constituent subspaces. And they better hold their properties no matter in how high a space they end up.

No. They understand the difference between clockwise and anticlockwise, but that is a property of the edges, not of the area inside the hexagon. So tell me, how many areas are inside the hexagon? Clearly only one. From the 2D point of view, this is the inside of the hexagon, and its boundary, consisting of 6 edges, is the outside. So are you saying that 3D space is divided by the inside and outside of the hexagon?

clockwise being a property of an edge? I disagree. That's direction of a vector orthogonal to a plane, I'm pretty sure about that. Plane is the principal player there, for without it, the concept of chirality would not made sense. And you keep on lapsing to either 2Der or 4Der's "eyes" whenever asked a concrete question. Yes, analogies can be helpful, but they can also be misleading. It's easy to make a wrong analogy going from one n-dimensional space to another. I'm afraid, with your 4Der vision, you are making such an error. Even though, I must say, this is only error in the way you describe the 4Der vision. The animations and projections on your site are done in the traditional and the only correct way: they are formed by projections onto planes and display one side of an object at a time. I thought it would be interesting to go a step further and see how real solid objects would look in 4D, but to you, for whatever reason, this means going backwards, not up.

From the 2Der's point of view, the plane is the universe, there is no "two sides".

You're mistaken. only a point does not have 2 sides. Absolutely everything else does.

Remember, they have no experience with 3D, and no way to access the 3rd direction. As far as they can tell, the 2D plane comprises the entirety of the universe. Therefore, saying that the plane has two sides is completely meaningless to them. How would you explain where those two sides are? If you were a 2Der, can you point to where those two sides are? If you can't, then how can such a thing be real? -- you need access to the 3rd dimension for this to make any sense.

Again you lapse into the Flatland, when asked a concrete question about 4D space. Your argument above sounds very much like when people thought that Earth was flat. The argument against its roundness was that the antipodes would certainly fall off, not to mention that oceans would drain in no time. But I'm pretty sure that Flatlanders form a map of their world in their head, just like we do of ours. Why, I can hold the whole Universe in my head, and not only its present, but also its past and future. So, I am pretty sure a Flatlander has a very good grasp of what a plane is and that a circle on it can be drawn, going in 2 opposite directions (which constitute chirality).

The 4Der does not see in opposite directions at once. It's merely looking from a direction at 90° angle to the 3D hyperplane.

Totally agree with you.

Again, you're confusing yourself by the ambiguous meaning of "surface". A surface to us 3Ders is a 2D construct. A cube's surface consists of squares. An icosahedron's surface consists of triangles. A sphere's surface consist of a curved 2D sheet. But you have to understand that in 4D, these things are not surfaces at all. They are mere ridges.

I'm afraid, confusion lies with you. I am very clear on what a surface is. And one of the things about any surface is that you can see only one side of it at a time. When you look at ridges, you seem to forget that they are made of bona fide planes. Those building blocks of which spaces are made seem to turn to mush in your 4Der eyes.

A "surface" to a 4Der is a 3D construct. A tesseract's surface consists of 8 cubes.

A "surface" to a 4Der is a 3d ridge, which is very easy to visualize exactly like the term implies: the outlines of mountains as seen from the airplane. And all 8 cubes of a tesseract's surface can be seen from only one direction, and any view will show only a half the object.

NOT merely the surface of the cubes, each of which consists of 6 squares, but the cubes' volumes. Take any of the 8 cubes that lie on the surface of a tesseract. Every point of the interior of the cube lies on the surface of the tesseract, not the inside. The inside of the tesseract is a hypervolume, whose boundary consists of 8 cubical volumes.

Find this confusing or strange?

No, I rather think that you made a wrong analogy in your transition from 3D to 4D, but, as I said, it lies in words only. The action, and it is judged by your excellent site, shows the projections to a POV. Yes, all objects you show are transparent, and this is because you want to emphasize the volumes. However, seeing transparent objects could be confusing even in 3D. So I thought that having both ways of seeing things in 4D could be even better. For one thing, it would stop this sort of arguments between us. Seeing is believing

[quickfur]

About that cube with corners at (±1, ±1, ±1, 0), looked at from (0,0,0,5). If we paint the inside of the cube blue and outside, yellow, what colors the 4Der sees?

Let's label the XY plane Z, XZ Y and YZ, X, so we have 6 faces: -X,+X,-Y,+Z,+Y,-Z
Could you please assign the colors to these planes as seen from (0,0,0,5) POV?
Thanks

Let's label the faces of a square (±1,±1,0) with 4 labels, according to the direction they face: -X, +X, -Y, +Y. Now color the outside of the square red and the inside blue. So what colors should be assigned to each of these planes, when you look at this square from the 3D viewpoint (0,0,5)?

That's not fair. I posited a concrete question. You reply with another question. Forgot, what that technique is called?

I was trying to get you to consider an analogous situation in lower dimensions that will help you get the right understanding in higher dimensions. But it seems you refuse to even consider it. I was not trying to play mind games with you.

The point of my question was: fundamental property of a 2d plane is preserved, no matter how high in number of dimensions you go. 4Der can NOT see both sides of a 2d plane simultaneously. He can see only one side at a time. Just like us the 3Ders.

That is your own incorrect presumption, and frankly, at this point I don't care anymore. You can go on believing whatever you like. That doesn't change the facts, which are plain to anyone who would consider it fairly. The flaw in your argument is obvious if you apply it to the 2D case. Look at the 2D plane. Draw a line through it. That divides the 2D plane into two sides, no? From the point of view of a 2Der, then, the fundamental property of lines is that they have two sides, and you certainly cannot see both sides at the same time.

So look at this picture from your 3D point of view. What do you see? You see the same line, cutting a plane in half. Can you see both sides of the line? Surely you can. It's as plain as the day. The left side and right side of the line are plainly visible from the 3D point of view. Why? Because you're looking at the thing from the 3rd direction, from the Z axis. The 2Der, however, cannot understand this, because from its point of view, you can either look at the line from the left, or you can look at it from the right, and there's all there is to it. It's eminently clear that a line has exactly two sides. Only, when you look at a line from a 3D point of view, you realize that there are not two sides, but an infinitude of sides, because in 3D, lines are surrounded by space, all around it.

The same thing happens when you go from 3D to 4D. Take a 3D hyperplane, and divide it into two with a 2D plane. Well, obviously, then, the plane must have two sides, corresponding with the two halves of the hyperplane. As far as we 3Ders are concerned, that's the end of the story. But a 4Der would say, the plane has two sides only with respect to the hyperplane it's contained in. If you consider the same plane in the context of the entire 4D space, you'll realize there's actually space all around it; there aren't just 2 sides.

You probably don't believe me, which is fine. Let the mathematics speak for itself. For the 2D case, say your line corresponds with the X axis, with the equation y=0. So it divides the plane into two halves, y>0 and y<0. Based on this, one could say that this line has two sides, one on the +Y side, one on the -Y side. And if 2D is as far as we go, then this is true.

Now, when we add the 3rd dimension, we now have another axis, Z. The line then is described by the pair of equations y=0, z=0. So what happens to the two "sides" of this line? The +Y and -Y directions obviously still exist. But now there's also the +Z and -Z direction (for example, (0,0,1) lies in the +Z direction relative to the line, and (0,0,-1) lies in the -Z direction from the line). So there are at least 4 sides to the line now, corresponding with +Y, -Y, +Z, -Z. What about the point (0,1,1)? This lies in the direction +Y+Z. It's not +Y only, nor +Z only, but it's both +Y and +Z. So it's a new direction. And this point isn't on the line either, so it must represent another "side" of the line.

Keep going, and you'll see that the line in 3D now has a whole lot of "sides" to it, an infinitude of them, in fact, comprising all the points in 3D space that surrounds the line.

OK, so you cry foul, because this is Flatland rehashed, and doesn't apply to 4D, right? Let the mathematics speak for itself.

Start with a plane in 3D, say z=0, corresponding with the XY plane. Obviously, this divides 3D space into two parts, the part with z>0, and the part with z<0. So this plane obviously has two sides, corresponding with the +Z direction and the -Z direction. As far as we 3D beings are concerned, this is all there is to it, and there's nothing more to say.

Now let's look at this plane in 4D space. There's now a new axis, W, and the plane is now described by the pair of equations z=0, w=0. As before, the +Z and -Z directions don't disappear; points like (0,0,1,0) and (0,0,-1,0) still lie on two opposite "sides" of the plane. But now there's something new. There's the point (0,0,0,1), which lies in the +W direction from the plane. And there's also its opposite, (0,0,0,-1), in the -W direction from the plane. So the plane must have at least 4 sides, right? Well, what about (0,0,1,1)? Or (0,0,1,-1)? Or (0,0,1/2,3/2)? If you look at all the points where the last two coordinates aren't both zero, you'll see that they don't lie on the plane.

Furthermore, consider this set of points: (0, 0, cos T, sin T), where T ranges from 0 to 2*pi. The last two coordinates are never zero at the same time, so none of these points lie on the plane. Moreover, for any point (0,0,cos T, sin T), there is a corresponding point (0,0,cos (T+pi), sin(T+pi)) which lies directly opposite. If you draw a line between these two points, you'll see that the line intersects our plane (z=0, w=0) exactly at the origin (0,0,0,0). You should recognize by now that (0,0,cos T, sin T) describes a circle around the origin. Every point of this circle is equidistant to the plane. So this is a circle that wraps around the plane. (If you don't believe me, just take any sampling of points on the plane, and see for yourself that none of them touch the circle.)

Do you see the analogy now? Just as the 1D line, which the 2Der thought had two well-defined sides, acquired a whole lot of space all around it when transplanted into 3D space, so the 2D plane, which we 3Ders thought has two well-defined sides, acquired a whole 360° of "sides". An entire circle of them.

So it's meaningless to contend over which "side" of a plane the 4Der sees. Do you think it makes sense for us 3D beings to ask, which side of a line do you see? You may argue that it depends on whether we see it from the +Y side or the -Y side. But think about this again carefully. If I give you a line in 3D space, which direction is +Y and which is -Y? You can arbitrarily assign one perpendicular direction as +Y and its opposite as -Y, but what if I look from +Z? Or from -Z? Or from (0,1,1), say? Would you consider +Z and -Z as "sides" too? Ultimately, this is meaningless, because the line's "sides" has lost their relevance once you get to 3D. Similarly, a 2D plane's two "sides" lose their relevance in 4D, because there are now an infinitude of "sides", all around the plane. (Yes I know you can't imagine how a plane could possibly have space "around" it. Welcome to 4D.)

For you, once you get into 4D the fundamental properties of 2d seem to disappear and are replaced by other concepts, only because that would seem to make sense to you in a progression you're making from 1d -> 2d -> 3d -> 4d. But each higher in number of dimensions space can be deconstructed to its constituent subspaces. And they better hold their properties no matter in how high a space they end up.

They do not disappear. They merely become irrelevant. But, each to his own opinion. The mathematics speaks for itself.

No. They understand the difference between clockwise and anticlockwise, but that is a property of the edges, not of the area inside the hexagon. So tell me, how many areas are inside the hexagon? Clearly only one. From the 2D point of view, this is the inside of the hexagon, and its boundary, consisting of 6 edges, is the outside. So are you saying that 3D space is divided by the inside and outside of the hexagon?

clockwise being a property of an edge? I disagree.

I never said such a thing. Clockwise is a property of a circuit of edges. Please at least read what I wrote before you start shooting at strawmen.

That's direction of a vector orthogonal to a plane, I'm pretty sure about that.

Oh, really now? OK, let's let the mathematics speak for themselves. Consider the 2D plane in 4D defined by the equations z=0, w=0. It consists of all points (x,y,z,w) where the last two coordinates are zero. Correct?

OK. Can you tell me what is the vector orthogonal to this plane? Welp, let's see. (0,0,1,0) is orthogonal to it. So obviously (0,0,-1,0) is, too. So obviously the plane has two sides, right? One in the direction of +Z, the other in the direction of -Z.

What about (0,0,0,1)? Erm... that vector is also orthogonal to the plane! As is (0,0,0,-1). So obviously, the plane must have four sides, corresponding with +Z, -Z, +W, -W.

OK, what about (0,0,1,1)? Go ahead, do the calculations. Wait a minute, (0,0,1,1) is also orthogonal to the plane!

Are you ready? Let's look at the set of points (0,0,cos T, sin T), where T ranges from 0 to 2*pi. I leave it up to you to convince yourself that all of these points are orthogonal to the plane. Yes, an entire circle of vectors are all orthogonal to the plane. These orthogonal vectors make all sorts of angles with each other, too. Everything from 0° to 360°.

So now let me ask. How many sides does this plane have? If I paint one side red and the other blue, which side would you see? (The catch being, what's "one" and the "other" when there is an infinitude of sides here?)

[...] The animations and projections on your site are done in the traditional and the only correct way: they are formed by projections onto planes and display one side of an object at a time.

You're wrong. They are projections from 4D into 3D. But since our eyes can't see 3D images directly, I further projected them from 3D to 2D so that they can be displayed on a 2D screen. There are two viewpoints here, a 4D viewpoint which is used for the 4D->3D projection, and a 3D viewpoint used for projecting 3D -> 2D screen.

I thought it would be interesting to go a step further and see how real solid objects would look in 4D, but to you, for whatever reason, this means going backwards, not up.

If that's what you want, then all you would see is the projection envelope. Which is fine by me, if you like to think of cubes as hexagonal envelopes, and tesseracts as rhombic dodecahedral envelopes. To each his own. *shrug*

From the 2Der's point of view, the plane is the universe, there is no "two sides".

You're mistaken. only a point does not have 2 sides. Absolutely everything else does.

Oh? So given a line in 3D, what are its "two sides"?

Remember, they have no experience with 3D, and no way to access the 3rd direction. As far as they can tell, the 2D plane comprises the entirety of the universe. Therefore, saying that the plane has two sides is completely meaningless to them. How would you explain where those two sides are? If you were a 2Der, can you point to where those two sides are? If you can't, then how can such a thing be real? -- you need access to the 3rd dimension for this to make any sense.

Again you lapse into the Flatland, when asked a concrete question about 4D space. Your argument above sounds very much like when people thought that Earth was flat. The argument against its roundness was that the antipodes would certainly fall off, not to mention that oceans would drain in no time. But I'm pretty sure that Flatlanders form a map of their world in their head, just like we do of ours. Why, I can hold the whole Universe in my head, and not only its present, but also its past and future. So, I am pretty sure a Flatlander has a very good grasp of what a plane is and that a circle on it can be drawn, going in 2 opposite directions (which constitute chirality).

You're attacking strawmen again. Of course the flatlander knows that you can draw a circle clockwise or counterclockwise. What has that got anything to do with the plane having two sides?

[...]

Again, you're confusing yourself by the ambiguous meaning of "surface". A surface to us 3Ders is a 2D construct. A cube's surface consists of squares. An icosahedron's surface consists of triangles. A sphere's surface consist of a curved 2D sheet. But you have to understand that in 4D, these things are not surfaces at all. They are mere ridges.

I'm afraid, confusion lies with you. I am very clear on what a surface is. And one of the things about any surface is that you can see only one side of it at a time. When you look at ridges, you seem to forget that they are made of bona fide planes. Those building blocks of which spaces are made seem to turn to mush in your 4Der eyes.

You seem to be convinced that 2D planes only ever have a single vector perpendicular to them. As I've shown above, this is only true in 3D. In 4D, there is an entire circle of vectors that are perpendicular to any given plane. In 5D, there's a sphere of them.

Not to mention, in 2D, there is no vector perpendicular to the plane. Don't believe me? That vector is outside the 2D universe, and therefore, as far as a 2Der is concerned, there is no such vector. Still don't believe me? OK, let the 2D universe consist of all points (x,y), where x and y can be any real number. So let (a,b) be the coordinates of the vector that's perpendicular to the 2D plane. Please tell me, what are the values of a and b?

So you see, your preconception that 2D planes always have a unique perpendicular direction is true only in 3D. It's not even true in 2D, for crying out loud, so why should it be true in any other dimension?

A "surface" to a 4Der is a 3D construct. A tesseract's surface consists of 8 cubes.

A "surface" to a 4Der is a 3d ridge, which is very easy to visualize exactly like the term implies: the outlines of mountains as seen from the airplane. And all 8 cubes of a tesseract's surface can be seen from only one direction, and any view will show only a half the object.

Are you sure you understand what "ridge" means in this context?

And yes, the 8 cubes on the tesseract's surface can only be seen from one direction. That direction is perpendicular to all of each cube's square faces.

NOT merely the surface of the cubes, each of which consists of 6 squares, but the cubes' volumes. Take any of the 8 cubes that lie on the surface of a tesseract. Every point of the interior of the cube lies on the surface of the tesseract, not the inside. The inside of the tesseract is a hypervolume, whose boundary consists of 8 cubical volumes.

Find this confusing or strange?

No, I rather think that you made a wrong analogy in your transition from 3D to 4D, but, as I said, it lies in words only. The action, and it is judged by your excellent site, shows the projections to a POV. Yes, all objects you show are transparent, and this is because you want to emphasize the volumes. However, seeing transparent objects could be confusing even in 3D. So I thought that having both ways of seeing things in 4D could be even better. For one thing, it would stop this sort of arguments between us. Seeing is believing

You're confusing the images on the 2D screen with the actual 3D projection image. The 4D viewpoint produces a 3D image. The 2D images shown on the site are for the benefit of us 3D beings, who don't happen to have 3D retinas that can capture the entire 3D image directly. The 2D images are produced by a separate viewpoint, in order to help us understand what is going on in the 4Der's eyes. Every point in the 3D volume is plainly visible to the 4Der.

Anyway, I've already explained everything as clearly as I can. If you still don't get it, then I can't help you. Like I said, these are plain mathematical facts. The mathematics speaks for itself. You can go on believing whatever it is you want to believe, it doesn't bother me, but the plain fact is that every part of a cube, including every point in its volume, is plainly visible to a 4D observer standing outside the hyperplane that the cube lies in. I'm not going to try to convince you anymore. You just have to do the calculations yourself, and see for yourself. I've repeated myself enough, and I think I should just shut up now.

[4Dspace]

The point of my question was: fundamental property of a 2d plane is preserved, no matter how high in number of dimensions you go. 4Der can NOT see both sides of a 2d plane simultaneously. He can see only one side at a time. Just like us the 3Ders.

That is your own incorrect presumption ... The flaw in your argument is obvious if you apply it to the 2D case. Look at the 2D plane. Draw a line through it. That divides the 2D plane into two sides, no? From the point of view of a 2Der, then, the fundamental property of lines is that they have two sides, and you certainly cannot see both sides at the same time.

The flaw in your argument is that you draw wrong analogies in your analysis of spaces. The correct way to do it is to break up a higher-D space into its constituent subspaces (going all the way down to 1D, if you wish). What you are doing instead, is you substitute a nD object with (n-1)D object and claim in effect that.. say, a plane becomes a line in a higher-dimensional space. That's where the flaw of your argument lies. And you don't see that such reductionist approach is wrong in principle.

So look at this picture from your 3D point of view. What do you see? You see the same line, cutting a plane in half. Can you see both sides of the line?

Of course, strictly speaking, line does not have sides. What it has is 2 distinct directions, such that anything in the world, no matter in how many-dimensional-space, can be viewed looking in only ONE direction at a time and => seen from one side only.

The same thing happens when you go from 3D to 4D. Take a 3D hyperplane, and divide it into two with a 2D plane. Well, obviously, then, the plane must have two sides, corresponding with the two halves of the hyperplane. As far as we 3Ders are concerned, that's the end of the story. But a 4Der would say, the plane has two sides only with respect to the hyperplane it's contained in.

Exactly, just like the hyperplane divides the 4D into 2 sides. Nonetheless, a 4Der has a unique location in 4D, which constitutes his POV. No matter where he is, this point has a relationship with all the 2d planes present in 4D. From this unique POV, each 2D plane, as well as hyperplane, is intersected at a certain angle. And even though this angle may be such that the 2d plane will appear as a line segment, or even a point, it still will be crossed from one of its 2 sides at a time.

If you consider the same plane in the context of the entire 4D space, you'll realize there's actually space all around it; there aren't just 2 sides.

Yes, but you do not look at the object from all around . At any given moment you have a unique POV, which determines the direction, which in turn determines what side you see.The 4Der sees each plane, from a given POV, from one side. Thus was a question to you above.

You probably don't believe me, which is fine. Let the mathematics speak for itself. For the 2D case, say your line corresponds with the X axis, with the equation y=0. So it divides the plane into two halves, y>0 and y<0. Based on this, one could say that this line has two sides, one on the +Y side, one on the -Y side. And if 2D is as far as we go, then this is true.

again you talk about lines when the talk was about planes. You interchange the concepts freely going from one space to another. This is wrong in principle. Line is always a line and a plane is always a plane. The relationships between them in different spaces may differ, but not their own unique properties.

Start with a plane in 3D, say z=0, corresponding with the XY plane. Obviously, this divides 3D space into two parts, the part with z>0, and the part with z<0. So this plane obviously has two sides, corresponding with the +Z direction and the -Z direction. As far as we 3D beings are concerned, this is all there is to it, and there's nothing more to say.

Now let's look at this plane in 4D space. There's now a new axis, W, and the plane is now described by the pair of equations z=0, w=0. As before, the +Z and -Z directions don't disappear; points like (0,0,1,0) and (0,0,-1,0) still lie on two opposite "sides" of the plane. But now there's something new. There's the point (0,0,0,1), which lies in the +W direction from the plane. And there's also its opposite, (0,0,0,-1), in the -W direction from the plane. So the plane must have at least 4 sides, right?

Wrong. The line that is defined by the unique POV will cross both planes, W and Z at unique angles, such that (POV angle with W) will be perpendicular to the (POV angle with Z), but in both cases, POV line will cross each plane, W and Z, from one side and => will show only one side of each plane.

This is elementary geometry.

Well, what about (0,0,1,1)? Or (0,0,1,-1)? Or (0,0,1/2,3/2)? If you look at all the points where the last two coordinates aren't both zero, you'll see that they don't lie on the plane.

Yes, but the POV will align with only ONE specific angle. Granted, this angle may make a plane appear just a line, when looked from the edge, and this happens in 3D all the time, but change the POV ever so slightly and you'll see either one side of the plane. Only ONE side at a time. This is a fundamental property of a plane in Euclidean spaces.

Do you see the analogy now? Just as the 1D line, which the 2Der thought had two well-defined sides, acquired a whole lot of space all around it when transplanted into 3D space, so the 2D plane, which we 3Ders thought has two well-defined sides, acquired a whole 360° of "sides". An entire circle of them.

As I said above, strictly speaking, a line does not have sides, but has directions. A line can separate a plane into 2 sides though, which is not the same thing. By going back and forth between 2D, 3D and 4D spaces, you're making wrong analogies. Instead, you have to break higher Ds into their subspaces and see that the basic property of each subspace holds no matter how high up (or low down) you will put it. The relationships between them will change and that will reflect the property of the higher space itself.

For you, once you get into 4D the fundamental properties of 2d seem to disappear and are replaced by other concepts, only because that would seem to make sense to you in a progression you're making from 1d -> 2d -> 3d -> 4d. But each higher in number of dimensions space can be deconstructed to its constituent subspaces. And they better hold their properties no matter in how high a space they end up.

They do not disappear. They merely become irrelevant.

Irrelevant? It's like you make the house out of bricks and then, in 4D, the bricks become irrelevant. What will happen to your house? The persistent properties of the underlying spaces is what makes the science of topology meaningful. In your statement that I put in bold face, is where your error lies.

[...] The animations and projections on your site are done in the traditional and the only correct way: they are formed by projections onto planes and display one side of an object at a time.

You're wrong. They are projections from 4D into 3D. But since our eyes can't see 3D images directly, I further projected them from 3D to 2D so that they can be displayed on a 2D screen. There are two viewpoints here, a 4D viewpoint which is used for the 4D->3D projection, and a 3D viewpoint used for projecting 3D -> 2D screen.

Yes, and the 3D viewpoint is derived from the 4D viewpoint. I know exactly how the stuff is rendered, since I made similar programs back in school. Just as I said, first you project onto 3d hyperplane and from it onto a 2d plane. Which gives you a projection of an object from a unique POV. Which in turn determines what side of the thing you're looking at. That's the point of our argument.

I thought it would be interesting to go a step further and see how real solid objects would look in 4D, but to you, for whatever reason, this means going backwards, not up.

If that's what you want, then all you would see is the projection envelope. Which is fine by me, if you like to think of cubes as hexagonal envelopes, and tesseracts as rhombic dodecahedral envelopes. To each his own. *shrug*

Well, the point of your site is to help people see 4D. Our discussion showed that to really appreciate 4D, rather than seeing 4D objects in 4D, it is far more instructive to see our familiar 3D objects in 4D. In deformations of our familiar 3d objects as seen from a 4D POV is where 4D shows itself to us. I thought it would be a quick project for you, since you have the software handy and ready to go, just plug in different numbers and voila. Ah?

You seem to be convinced that 2D planes only ever have a single vector perpendicular to them. As I've shown above, this is only true in 3D. In 4D, there is an entire circle of vectors that are perpendicular to any given plane. In 5D, there's a sphere of them.

See? Couple of days ago you said that 3D has nothing special about it. And now this!

So you see, your preconception that 2D planes always have a unique perpendicular direction is true only in 3D. It's not even true in 2D, for crying out loud, so why should it be true in any other dimension?

I never --EVER!-- said that. I said that any line, from whatever dimension, can cross a plane at a unique angle (even if that angle is 0 or 180 or whatever. And! I said that, since a line has 2 directions, it will show the plane it crosses from ONE SIDE. That's all I've been saying all along

Every point in the 3D volume is plainly visible to the 4Der.

Well, we already established that some POVs do not allow 4Der to see all points at the same time. More than that, the sampling points cross the planes that make up the cube from various angles, thus showing only one side of each plane at a time

Anyway, I've already explained everything as clearly as I can. If you still don't get it, then I can't help you.

Thank you for your time. You did help me a lot. I hope that my little input was of help to you too

[quickfur]

[...]

So you see, your preconception that 2D planes always have a unique perpendicular direction is true only in 3D. It's not even true in 2D, for crying out loud, so why should it be true in any other dimension?

I never --EVER!-- said that. I said that any line, from whatever dimension, can cross a plane at a unique angle (even if that angle is 0 or 180 or whatever.

So from which side of the plane does the line cross it when the angle is 0?

And! I said that, since a line has 2 directions, it will show the plane it crosses from ONE SIDE. That's all I've been saying all along

So let's say the plane is defined in 4D by y=0 and z=0 (where 4D coordinates are taken as (w,x,y,z)). The 4D viewpoint is (0,0,0,5). From which side of the plane does the line of sight cross it? What about when viewed from (0,0,5,5)?

Every point in the 3D volume is plainly visible to the 4Der.

Well, we already established that some POVs do not allow 4Der to see all points at the same time. [...]

Which are the POVs where the 4D viewpoint lies in the same hyperplane as the 3D object. You keep evading the case where the 4D viewpoint lies outside, which, incidentally, is where the interesting stuff happens, because otherwise you might as well just stick with 3D to begin with.

Anyway, since you're so smart, please prove me wrong. Let's say our cube's vertices are (±1,±1,±1,0), and our 4D viewpoint is (0,0,0,5). Do you agree that every point in the cube (both on its surface and in its interior) has an unobstructed path to the 4D viewpoint? If so, then it follows that every point of the cube is visible from that viewpoint, correct? Therefore, the entirety of the cube is visible to the 4Der. But if not, then please give one example of a point in the cube that does not have an unobstructed path to the viewpoint. Prove me wrong.

[4Dspace]

Sorry for the delay. Don't know where you are, but it's holiday time here had to schmooze.

So from which side of the plane does the line cross it when the angle is 0?

Oh come on! It means you're looking at the plane edge-on => you see the edge.

So let's say the plane is defined in 4D by y=0 and z=0 (where 4D coordinates are taken as (w,x,y,z)). The 4D viewpoint is (0,0,0,5). From which side of the plane does the line of sight cross it? What about when viewed from (0,0,5,5)?

I'm not sure how a plane is defined by y=0 and z=0... what about other points? I'm confused Which side of the plane, i.e. its chirality, is the property of the plane, not POV. POV reveals just one side of it.

You keep evading the case where the 4D viewpoint lies outside, which, incidentally, is where the interesting stuff happens, because otherwise you might as well just stick with 3D to begin with.

I do not evade this POV. Agree with you. It is very interesting to view a 3d object from various POV in 4D. I'd love to see the animation walking around a cube in 4D. I keep begging you make one, but you play hard to get

Anyway, since you're so smart,

alas, not as smart as you

Let's say our cube's vertices are (±1,±1,±1,0), and our 4D viewpoint is (0,0,0,5). Do you agree that every point in the cube (both on its surface and in its interior) has an unobstructed path to the 4D viewpoint? If so, then it follows that every point of the cube is visible from that viewpoint, correct? Therefore, the entirety of the cube is visible to the 4Der. But if not, then please give one example of a point in the cube that does not have an unobstructed path to the viewpoint. Prove me wrong.

The way you pose the question, is not quite correct. You're right from conventional point of view. But in topology --the way I understand it-- you do not see points. In this concrete case, you see planes. And you see them from a specific angle. I thought it would be easier for you to simply plug in the numbers in your software than for me to sit down and try to figure it out. My first impression of what I see from 0,0,0,5 I described in the post above, but afterward I had doubts. Please do make the animation so that we all could learn.

Then, a very, very interesting variation on this cube colored differently inside/outside (only because this way it is easier to see the chirality of its planes), would be to construct a cube that is made up of, say, 27 cubes, also colored to mark the sides of their planes. This would represent the inside "points" you're talking about above. How will these inner cubes be seen? How the sides of their planes line up? I think that would be phenomenally interesting My guess is that we would see something like a chess board, i.e. the planes sides would alternate -? but I'm not sure.

[quickfur]

[...]

So let's say the plane is defined in 4D by y=0 and z=0 (where 4D coordinates are taken as (w,x,y,z)). The 4D viewpoint is (0,0,0,5). From which side of the plane does the line of sight cross it? What about when viewed from (0,0,5,5)?

I'm not sure how a plane is defined by y=0 and z=0... what about other points? I'm confused Which side of the plane, i.e. its chirality, is the property of the plane, not POV. POV reveals just one side of it.

A plane is defined by y=0 and z=0 in that it consists of all the points (w,x,y,z) in which y=0 and z=0, and w and x are free to vary. IOW the points on the plane are (w,x,0,0) for all possible values of w and x.

You keep evading the case where the 4D viewpoint lies outside, which, incidentally, is where the interesting stuff happens, because otherwise you might as well just stick with 3D to begin with.

I do not evade this POV. Agree with you. It is very interesting to view a 3d object from various POV in 4D. I'd love to see the animation walking around a cube in 4D. I keep begging you make one, but you play hard to get

Huh? I already have one on my website:



More than one, in fact, they're all in the 4D visualization document.

[...]

Let's say our cube's vertices are (±1,±1,±1,0), and our 4D viewpoint is (0,0,0,5). Do you agree that every point in the cube (both on its surface and in its interior) has an unobstructed path to the 4D viewpoint? If so, then it follows that every point of the cube is visible from that viewpoint, correct? Therefore, the entirety of the cube is visible to the 4Der. But if not, then please give one example of a point in the cube that does not have an unobstructed path to the viewpoint. Prove me wrong.

The way you pose the question, is not quite correct. You're right from conventional point of view. But in topology --the way I understand it-- you do not see points. In this concrete case, you see planes. And you see them from a specific angle.

Huh?? So you kept on deriding me for a wrong interpretation of 4D vision, and then when I ask you to show me where I went wrong, you changed your mind and said I'm right "from a conventional point of view"?? What's a "conventional point of view", and what other point of view are we discussing?? And what do you mean by "in topology you do not see points"? I'm baffled by your strange logic here. No wonder we can't seem to agree on the most basic matters. We've been talking at cross purposes. I'm on Mars and you're on Venus, and something isn't quite going through in the communication.

I thought it would be easier for you to simply plug in the numbers in your software than for me to sit down and try to figure it out.

I wish you would sit down and figure it out for yourself, because if you would just do that, perhaps you might actually understand what I'm trying to say here. I didn't ask you that question just because I felt like handing out random math exercises, you know. It was because I was hoping that when you sit down and do the calculations yourself, you will finally understand what I've been trying to say.

My first impression of what I see from 0,0,0,5 I described in the post above, but afterward I had doubts. Please do make the animation so that we all could learn.

The animation has already been made a long time ago, and has been available on my website for a long time now. Some people on this forum found it helpful, but obviously not everybody did, otherwise we wouldn't be having this discussion.

Then, a very, very interesting variation on this cube colored differently inside/outside (only because this way it is easier to see the chirality of its planes), would be to construct a cube that is made up of, say, 27 cubes, also colored to mark the sides of their planes. This would represent the inside "points" you're talking about above. How will these inner cubes be seen? How the sides of their planes line up? I think that would be phenomenally interesting My guess is that we would see something like a chess board, i.e. the planes sides would alternate -? but I'm not sure.

OK, are we talking about seeing the cube with 4D eyes, or just 3D eyes? Because if you're talking about just 3D eyes, I'm afraid you'll find the "animation" even more perplexing, because most of the time it would just appear to be an unmoving square (a cross-section of the cube, incidentally), and at exactly two points in the rotation the cube will appear, once in its "normal" orientation, and once flipped into its mirror image -- and then disappear again. (This is because 3D eyes can only see along a single hyperplane at a time, and most of the time the cube --- or the bulk of it except a single square cross section --- is outside this hyperplane.)

Anyway, none of this is new, and I feel like I'm just repeating myself all over again. And we both know how well that went before. So I think I really should shut up, like I said I would. My explanations clearly aren't helping you. Perhaps you'd like to read the other topic, by gonegahgah; maybe his style of description is more helpful to you, I don't know. I think I'm done repeating myself.

[4Dspace]

Huh? I already have one on my website:

]

Yes, thank you, now I remember. This is very interesting, but not that helpful to me. It looks like a cube is turning itself inside out, as seen from a basically fixed POV. I'd like to see how the cube would behave from a POV that keeps changing and sometimes "sees" the cube from its 3d subspace and then leaves it again to 4d POV proper.

So you kept on deriding me for a wrong interpretation of 4D vision, and then when I ask you to show me where I went wrong, you changed your mind and said I'm right "from a conventional point of view"?? What's a "conventional point of view", and what other point of view are we discussing?? And what do you mean by "in topology you do not see points"? I'm baffled by your strange logic here. No wonder we can't seem to agree on the most basic matters. We've been talking at cross purposes. I'm on Mars and you're on Venus, and something isn't quite going through in the communication.

I was never --ever!-- deriding your interpretation of 4D vision. It's just that our purposes of seeing 4D differ. You are interested in static objects (thus conventional POV) and I am interested in their structure. From the POV of structure of space, which way things "flow" is very important. A vector points to a specific direction; a plane too has a direction. Thus, if you are looking at the same plane at one POV and see it going clockwise, then if you see the same plane at another POV and see that there it seems to go counterclockwise, this tells you a lot: If your vector POVs are parallel in a give subspace, then this plane is twisted in the same subspace. Or, if you decide that the plane gotta be flat, then the POV vectors in question are themselves sit in a twisted subspace. I'll tell you how I "deconstruct" space on the other thread, if you don't mind.

My first impression of what I see from 0,0,0,5 I described in the post above, but afterward I had doubts. Please do make the animation so that we all could learn.

The animation has already been made a long time ago, and has been available on my website for a long time now. Some people on this forum found it helpful, but obviously not everybody did, otherwise we wouldn't be having this discussion.

Yes, it is very good, but I am slow and wish there was a button that would stop its rotation at any given time, so that I could examine it closely. And, as I said above, I wish I could change the POV by, say, lowering it to the level of the cube and then leave that plane again, rather than seeing it rotate in one spot from a fixed POV.

OK, are we talking about seeing the cube with 4D eyes, or just 3D eyes?

I thought we already established that there is no "4D eyes". The projections that create the final rendering in 4D work in the same way as in 3D, and in the same way in 2D. So, there is only one way to see, really. What your animations show is just that: a series of projections for a given POV, as the object rotates around a fixed point in space. Changing POVs in 4D is not the same as seeing a rotating thing from the same POV. I guess I have to write such a progam myself, but all I have now is PostScript interpreter, and it renders one frame at a time, plus I have to do everything by hand -- I don't have time for this now. What software do you use? Can you recommend some easy to use shareware I can download?

Because if you're talking about just 3D eyes, I'm afraid you'll find the "animation" even more perplexing,..

Let it be my problem. I want to be perplexed. I need to see how, in effect, a 3D subspace may appear "deformed" from some 4D POVs. I have a very good grasp of how 3d things look and behave in 3D. Seeing them misbehave in 4D, would do tons for my understanding of the 4D space.

... because most of the time it would just appear to be an unmoving square (a cross-section of the cube, incidentally), and at exactly two points in the rotation the cube will appear, once in its "normal" orientation, and once flipped into its mirror image -- and then disappear again. (This is because 3D eyes can only see along a single hyperplane at a time, and most of the time the cube --- or the bulk of it except a single square cross section --- is outside this hyperplane.)

Would love to see this with my own eyes

[quickfur]

Huh? I already have one on my website:

]

Yes, thank you, now I remember. This is very interesting, but not that helpful to me. It looks like a cube is turning itself inside out, as seen from a basically fixed POV. I'd like to see how the cube would behave from a POV that keeps changing and sometimes "sees" the cube from its 3d subspace and then leaves it again to 4d POV proper.

You'd see exactly the same thing.

Well, except that this is actually what a 4D eye would see. A 3D eye would see nothing except a square cross-section most of the time, and only see the cube when it's inside the same 3D subspace as the cube. So the cube would just appear and disappear, once in its "right orientation", and once as its mirror image.

[...] I was never --ever!-- deriding your interpretation of 4D vision. It's just that our purposes of seeing 4D differ. You are interested in static objects (thus conventional POV) and I am interested in their structure. From the POV of structure of space, which way things "flow" is very important. A vector points to a specific direction; a plane too has a direction.

A plane has two directions. What we think of as "the direction" is actually an orthogonal direction, the uniqueness of which is a peculiarity of 3D. It doesn't generalize to any other dimension. But we've been through this before.

Thus, if you are looking at the same plane at one POV and see it going clockwise, then if you see the same plane at another POV and see that there it seems to go counterclockwise, this tells you a lot: If your vector POVs are parallel in a give subspace, then this plane is twisted in the same subspace. Or, if you decide that the plane gotta be flat, then the POV vectors in question are themselves sit in a twisted subspace. I'll tell you how I "deconstruct" space on the other thread, if you don't mind.

Well, I've tried to tell you before that this analysis pre-assumes 3D, but that discussion went nowhere. But since you insist, let's just be mathematically correct, and say that a plane defines a pair of directions (just pick any two vectors whose span is the plane). To make things simpler, let the two vectors be perpendicular to each other, then impose an ordering on them. So in that way you have a clockwise and anticlockwise "direction", so to speak (you can define clockwise to be the rotation that starts with the first vector and rotates into the second). So far so good.

However, once you start talking about POVs, the clockwise/anticlockwise split only works in 3D. The reason for this is that we 3Ders see in 2D projections, and in 2D, the projected vectors are (almost) always non-parallel, so by following the ordering imposed on them you can assign a clockwise/anticlockwise direction to them. However, this fails in every higher dimension. A 4Der sees in 3D projections -- the two chosen vectors of the plane project into two 3D vectors. The problem is, given two 3D vectors, how do you assign "clockwise" and "anticlockwise". The two projected vectors define an arbitrary plane in 3D, which can be in any orientation relative to the coordinate axis. So how do you assign "clockwise" and "anticlockwise"? You can't, unless you arbitrarily pick a random vector, and do another 3D->2D projection along that vector, and then use the 2D projection to decide between them. (The 4D line-of-sight vector isn't going to help you here, because it projects to the zero vector.) But introducing this extra vector means that you have, in effect, made the decision between clockwise and anticlockwise yourself, because the plane may be "clockwise" if you choose +X as the extra vector, but I could just as easily choose +Y as my extra vector and end up with "anticlockwise". So there's no unique clockwise/anticlockwise assignment unless we both agree on the same arbitrary vector. In which case, you have the additional problem of, what should this extra vector be when you're talking about space-time, since there is no fixed frame of reference for you to consistently choose it. (Besides the problem that, by introducing that extra vector, you have basically broken the very important assumption in physics that physical laws appear the same from every frame of reference -- which they no longer do since the clockwise/anticlockwise assignment changes depending on where that extra vector points relative to a given frame of reference.)

My first impression of what I see from 0,0,0,5 I described in the post above, but afterward I had doubts. Please do make the animation so that we all could learn.

The animation has already been made a long time ago, and has been available on my website for a long time now. Some people on this forum found it helpful, but obviously not everybody did, otherwise we wouldn't be having this discussion.

Yes, it is very good, but I am slow and wish there was a button that would stop its rotation at any given time, so that I could examine it closely. And, as I said above, I wish I could change the POV by, say, lowering it to the level of the cube and then leave that plane again, rather than seeing it rotate in one spot from a fixed POV.

Actually, you'd see exactly the same thing if the cube was fixed in place, and the POV was rotating around it. The two are equivalent.

OK, are we talking about seeing the cube with 4D eyes, or just 3D eyes?

I thought we already established that there is no "4D eyes". The projections that create the final rendering in 4D work in the same way as in 3D, and in the same way in 2D. So, there is only one way to see, really.

I'm not going to argue with you about this anymore. I've already repeated myself too many times, that there are two different projections working here, the first of which is 4D->3D, which is the 4D vision. But since the computer screen is only 2D, I can't render a 3D image on it without doing a further projection from 3D->2D. This second projection is a post-processing step to make the 3D image visible to us poor 3D beings. A 4Der needs no such thing, since she sees the 3D image directly. But anyway. This has been argued to the death, and I'm not going to bother with it anymore.

[...]Let it be my problem. I want to be perplexed. I need to see how, in effect, a 3D subspace may appear "deformed" from some 4D POVs. I have a very good grasp of how 3d things look and behave in 3D. Seeing them misbehave in 4D, would do tons for my understanding of the 4D space.

You will not see any deformation. The 3D subspace will appear as a 2D plane for every view angle except 0° and 180°, at which point the entire subspace comes into view, once in its normal orientation, and once as its mirror image. That's all. You cannot see any deformation at all, because there is none. The so-called deformation that you see is an artifact of 4D->3D projection, which only happens for 4D eyes. 3D eyes can only see a 3D slice of things at a time, and most of the time, this slice is just a 2D cross-section of the 3D subspace. But since you don't believe in the possibility of 4D eyes, that means there is no such thing as 4D->3D projection, so you'll only ever see 3 things: a 2D cross-section, the entire 3D subspace, and the mirror image of the 3D subspace.

... because most of the time it would just appear to be an unmoving square (a cross-section of the cube, incidentally), and at exactly two points in the rotation the cube will appear, once in its "normal" orientation, and once flipped into its mirror image -- and then disappear again. (This is because 3D eyes can only see along a single hyperplane at a time, and most of the time the cube --- or the bulk of it except a single square cross section --- is outside this hyperplane.)

Would love to see this with my own eyes

As you can see, this is very insightful indeed. *shrug*

[4Dspace]

Thank you for your reply

... let's just be mathematically correct, and say that a plane defines a pair of directions (just pick any two vectors whose span is the plane). To make things simpler, let the two vectors be perpendicular to each other, then impose an ordering on them. So in that way you have a clockwise and anticlockwise "direction", so to speak (you can define clockwise to be the rotation that starts with the first vector and rotates into the second). So far so good.

However, once you start talking about POVs, the clockwise/anticlockwise split only works in 3D. The reason for this is that we 3Ders see in 2D projections, and in 2D, the projected vectors are (almost) always non-parallel, so by following the ordering imposed on them you can assign a clockwise/anticlockwise direction to them. However, this fails in every higher dimension.

In my analysis, clockwise/anticlockwise direction is the property of a given plane, however this plane happened to be oriented in whatever n-space. So, from the POV of the given plane, it has only 2 well-defined directions. However these directions will appear from the POV of another (sub)space, will define the relationship of that subspace to this plane.But the planes and vectors are the bricks that build the structure of space from bottom up. They may seem differently in higher spaces. But there is a difference between what seems and what is.

In your analysis, it appears, whoever looks, decides on what it is. But things are not what they seem

A 4Der sees in 3D projections -- the two chosen vectors of the plane project into two 3D vectors. The problem is, given two 3D vectors, how do you assign "clockwise" and "anticlockwise".

Why, you can do the same trick you did with 2 vectors that define a direction of a plane. In 3D we have a sphere spinning around an axis, which could be defined as a vector that tells which way it spins according to the "handed" rule (same as with "handed" vector -- you need only 1 to indicate chirality, according to the same rule).

The two projected vectors define an arbitrary plane in 3D, which can be in any orientation relative to the coordinate axis. So how do you assign "clockwise" and "anticlockwise"? ...

Well (anti)clock-wise is the direction of a plane. A sphere spins and its equator traces a rotating plane. In this sense, there is hardly any difference, really, between a plane and a sphere. Now a 4d-sphere spinning -- that's an interesting thing to see! I guess it looks like an 8 or a torus with the direction of the spin into its "hole" -?

...You can't, unless you arbitrarily pick a random vector, and do another 3D->2D projection along that vector, and then use the 2D projection to decide between them. (The 4D line-of-sight vector isn't going to help you here, because it projects to the zero vector.) But introducing this extra vector means that you have, in effect, made the decision between clockwise and anticlockwise yourself, because the plane may be "clockwise" if you choose +X as the extra vector, but I could just as easily choose +Y as my extra vector and end up with "anticlockwise". So there's no unique clockwise/anticlockwise assignment unless we both agree on the same arbitrary vector.

How things progress from one subspace to another, can be made into a "rule" according to some parameters, say |value| of vectors and their direction, the angle between them, etc.

... In which case, you have the additional problem of, what should this extra vector be when you're talking about space-time, since there is no fixed frame of reference for you to consistently choose it. (Besides the problem that, by introducing that extra vector, you have basically broken the very important assumption in physics that physical laws appear the same from every frame of reference -- which they no longer do since the clockwise/anticlockwise assignment changes depending on where that extra vector points relative to a given frame of reference.)

Ahhh... Here you touched upon the difference between the space, in which we live and observe physicals laws, and what I call "functional space". Clearly, there is a huge difference, which is plainly obvious to you, the person dealing with spaces all the time. There is a reality underlying what seems. It seemed to us for a long time that Earth was flat and stationary. Nowadays we know that it is round, spinning, spiraling and hurling through space with mind-boggling speed.

Our POV of the observers is such, that we always see the same thing, no matter how fast and in what direction we go. That's the POV I am talking about. From that POV it is very important, how things appear. What side of a plane we see makes a difference between seeing, say, an electron and an anti-electron, and between matter and anti-matter.

Yes, it is very good, but I am slow and wish there was a button that would stop its rotation at any given time, so that I could examine it closely. And, as I said above, I wish I could change the POV by, say, lowering it to the level of the cube and then leave that plane again, rather than seeing it rotate in one spot from a fixed POV.

Actually, you'd see exactly the same thing if the cube was fixed in place, and the POV was rotating around it. The two are equivalent.

Really? And I thought that the thing is rotating around a vertical axis the angle of which to my POV is fixed. I want to be able to change that angle. .. as my POV goes up and down.

You will not see any deformation. The 3D subspace will appear as a 2D plane for every view angle except 0° and 180°, at which point the entire subspace comes into view, once in its normal orientation, and once as its mirror image. That's all. You cannot see any deformation at all, because there is none. The so-called deformation that you see is an artifact of 4D->3D projection, which only happens for 4D eyes.

As you can see, this is very insightful indeed. *shrug*

Ah... I see now what you mean by 4Der eyes. So, you omit the 4D-> 3D projection entirely... Why? That's pretty silly, and that's not what I was talking about.

Now, look at the turning inside out cube from your site and freeze it for a moment:



Please note that at any given moment you see it only from a specific POV, which determines, as I have been saying all along, which side of each face you see... Indeed, this has been discussed to death and it turns out what you called "3Der eyes" I, 3Der never even considered as a possibility, which consisted in discarding the most important projection of them all: 4d->3d. All along I thought there was only one way to see, and it turned out that's what you call "4Der eyes". Funny.

[4Dspace]

Now that I stopped laughing at our misunderstanding... actually, my understanding of you. The fault lies entirely with me My apologies! So, the "4Der eyes" is the projection of 4d->3d? Granted, it is still not delivered to us in its full glory, because, after that, it has to be projected 3d->2d. Now I understand what you meant!

Thank you for all our efforts and all your time