There are some general comments, before we move onto specifics.
1. Euclidean geometry has no notion of 'sides'. What happens is that a 1-sphere has a surface of two points, and any subspace in a space that has a 1-sphere as its orthospace, is capable of dividing the space, and thus "have sides" in that space. In 4D, a hedrix or 2-space does not divide 4-space, and therefore does not have 'sides'.
Also, in euclidean geometry, all-space itself has no sides, since it does not substain a perpendicular vector.
2. The mythical 4D creature, and a POV are identical idioms. The 4D creature does provide the POV.
3. The path to higher dimensions is through a clear word-set. Many of the words in common use in the subject are misplaced idioms.
cells in common parlance means a solid element of a tiling, such as in cellular automata, and board games that are played on a tiled board. The general meaning is the same as 'bubble in a foam', and it is from various presentations of a surface of a 4D polytope that this word migrates. Using it as a specific meaning has the same effect as calling the squares of a tesseract 'cells', because the cells in Conway's game of Life are squares. In the PG, cell is restored to its general meaning.
plane has the general meaning of 'dividing space', specifically, something that supports against gravity etc. Along with
face, this is restored to the meaning of 'dividing space'. Were one to follow the common meaning, 'face' could end up as 2D, while a 'facet' (little face) as 5D, and the 'facing side' as 5D. So go figure. You seem to be caught by the notion that a plane can be both 2D and a dividing space.
hyperplane has a respectable etymology, meaning 'above-plane'. Specifically, the meaning of hyperplane and hyperspace in the PG both expresses the correct etymology and correct use of these. What you are doing with it, is like saying 'the second floor is above us, and so is upstairs', and then applying 'upstairs' to the second floor when you're on the sixth one.
To me this means only one thing, namely that you'are just as unskilled at seeing 4D as he is. Or perhaps you did not understand what the issue is? It is the relationship of any plane --not just 2d-plane, but a 3d-hyperplane as well-- to a line of sight set by a POV, in any number of dimensions.
That's what the issue is. And according to quickfur, and now you, you can do the impossible, namely see planes from their both sides simultaneously in Euclidean spaces, lol. Since you two are the most regular contributors here, the so-called "experts", it makes this forum a sham.
What you and quickfur are saying is impossibility. It is contradictory to the notion of geometry.
If you wish to deal with so-called "experts", then i shall use the terminology of the PG, which is specifically designed for higher dimensions. It teases apart the meanings of words to seperate meanings.
A. By the rules of geometry, a hedrix can only have sides when embedded in a chorix, not in a terix. So there are no sides to a 2d space.
B. No contention exists that supposes that the POV needs to be in the space being viewed. Many of your comments suppose that the viewer must be in a chorix where the subject exists. If the POV exists outside a space, then any ray from the POV to the space only passes through one point of that space. A POV in four dimensions can resolve the entirity of a chorix and anything therein, say a cube, as readily as us, for not being in a hedrix, can see the entire hedrix, and anything therein.
Likewise, a line in 2D, such as a frontier between countries, has two sides. On a map, one is readily able to examine both sides of the line without having to flip it. The surface rendered is a hedrix (specifically, coordinates of the earth), and is perfectly rendered witlout loss on a sheet of paper. A hedrix has no sides, the paper has a hedrix on each side, and perfectly shows the hedrix without the need to flip.
What 4d person? Who are those mythical 4d persons you're talking about here all the time?
It's the same as your POV. We just make it solid relative to four dimensions, to avoid falling in your trap.
The meaning of "side" in the context of a (hyper)plane in relation to the POV is straightforward.
A space has no sides. You need to embed it in a space that it divides for it to 'have sides', and to place the POV in the containing space for the 'far side' to not be observable. It's a pretty simple concept really. In any case, the sides only exist when one is considering the embedding space, and in that case, the embedding space does not have sides.
Who is this 4D who can read all six outer faces of the cube at once? There is a difference between the "sides" of the square and the faces of the cube. 2d-planes are not 1d-lines. Yet for you, in your misleading analogies, one turns to other when you go from one dimension to the next. This is wrong in principle. The fact that we can see a line in its entirety does not change the fact that only one side of a plane is visible to anyone in any dimension at a time. You are misled by the analogies you're using, when going from one D to another. This is the common problem on this forum.
A line is not a plane. Going from one dimension to another does not change the underlying structures of the object under examination. A plane remains a plane and a line remains a line. A plane does not turn into a line or vise versa, as your analogies on this forum often imply.
Suppose one has a coordinate system in three dimensions, where the POV is (2,0,0). The square is a conventional coordinates in y,z, gives (0,±1,±1), Any line from the POV to the plane x=0 will pass through only point, and the only notion that the edges of the square has sides, comes from recognising the object as having that number of sides. Likewise, the POV (2,0,0,0), viewing the conventional cube (0,±1,±1,±1), can render all six sides of the cube, and the 'inside' and 'outside' of the cube in exactly the same manner.
In this sense, the sides of the cube only have meaning when these are rendered in a chorix. Since the cube is already chorid itself, it is feasable that it can both have 12 sides like the dodecahedral coin and heads-and-tails sides.
Again, here you confuse points, which are dimensionless entities with planes, which are not. You cannot see points. You can see lines and planes. And while lines are one-dimensional and have no sides (something quickfur, by the way said many time, i.e. that lines have sides, lol) planes and hyperplanes do have sides, and they show only one to a POV.
The meaning of words like 'deadline', 'line in a sand', 'front line', 'toe the line', all suggest that a line has sides, which has different conditions on each side. This is because line exists in a space where it divides, and the meaning is what is being divided. A dead line is something you step over and you're dead. A line in the sand is also a line to be crossed: that is, something is different on the other side.
In 4D, this sense is not preserved, since space, divided by gravity and a surface division, can never give a line.
I do not play on the meaning of side. I emphasize that a plane has sides, which are actually its directions, just like a line has directions. (line does not have sides, though). The 4 sides of a square are completely different thing. Those are the boundaries of a plane.
Lines do have bounding extents - they're called 'ends' or 'vertices'. Lines, like planes, are not vectors. They do not have directions, You need something else to provide a specific item. If you're going to be pedantic on keeping the same dimensional terminology throughout, if a square has sides in 3D, then it has sides in 4D, even though it is a side of a cube in 3D and 4D. In this sense, you are tying side to the meanings variously held by 'face' and 'margin' in the PG.
Again, you confuse yourself with wrong analogies. There is hardly any difference between a sheet of paper (presumably a representation of an infinite 2d plane) and a square, which is just a particular segment of the same plane. That the square is bounded and the sheet, presumably, not, is an irrelevant detail in our context. We are talking about the fact that any plane has 2 directions. Both a sheet and a square are planes.
The fact that a square can be rendered completely on one side of a sheet of paper blows apart your theory that a plane has two arounding-sides. It is sufficient to see the full extent and detail of a square from a single POV, even though the paper has two sides. The other side of the same paper can contain an entirely different view of a different plane. A plane, thus, has no side, as is evident from endoanalysis.
Yes, this is obvious and is not the source of confusion, at least not for me. quickfur, going from 2D to 4D, forgot about the 3 orthogonals in 3D. All 3 merged into the line of sight of his "4Der", thus enabling his mythical creature to capture rays of light going in 3 orthogonal directions simultaneously. He called it "4Der vision". Lacking POV, and thus able to see everything from all sides simultaneously is the hallmark of his "4Der vision". Unfortunately, a vision without a POV is not a vision, but perhaps echo-location at best.
The assumption here is that the POV is in the same chorix as the object. It isn't.
Not sure what you mean here. I find frequent mention of mythical beings on this forum unnecessary. There are spaces of various dimensions though and a POV, to which the light converges. This is much simpler and straightforward in my opinion.
A tennent of Euclidean geometry is that it is possible to draw a line between any two points. Analysis of this is that if one point lies in a space, and the second does not lie in that same subspace, then there is only one point in the subspace in the line. The whole point of moving the POV into hyperspace is to achieve the same effect that the view of a plane in 3d can be seen from some point in the hyperplane (ie chorix) that is not in 3D.
Likewise, the information in a map is a perfect representation of a hedrix, has no 'front and back', and is amply served on a single side of a page.
And so it took another week-worth of posts to make him understand the basic tenet of Euclidean geometry, namely that only one side of a (hyper)plane is shown to a given POV, in any N-space. Which, to this moment, quickfur was not able to admit. Which means that he will stubbornly continue to give out wrong answers to the unsuspecting visitors of the site in regard to the aspects of the 4D space and 4D visualization.
A plane object, such as a map, plan or picture, in 3d is perfectly rendered on a single side of paper, with little regard to what's on the other side of the same page, so it is inappropriate to speak of 'the other side' of a plane in the sense you are trying to do.
What is stubborn, is that you refuse to admit that the POV of space, for being in hyperspace, yields the entirity of space without any transcept of it.
No, I use the terms as they are used by most. A plane is a 2d plane, and a hyperplane is a 3-d plane. That's how it is used by most.
Which means that you have not got around to either using words in their etymological meanings, and have not got around to untangling meanings. You claim that only '(hyper)planes' can have sides. Firstly, this is incorrect, as i have shown that both points (eg point of no return), and lines can have bounding meanings and sensible meanings of 'sides', and that the meaning of 'side' exists only when the side exists in a space where its orthospace is a line: that is M-1 relative to M. A plane of 2d does not divide 4space.
No, I don't. That's what you ascribe to me. What I am saying, merely, is that a plane (or a hyperplane) has 2 directions, which can be expressed as its chirallity, and that a (hyper)plane can show its only ONE SIDE to a given POV. Do you disagree with this statement?
You confuse the function of hedrix (plane) and chorix (plane) in 4D. This equates to confusing the meaning of line and hedrix in 3D. Both of these are 'sides' in the sense that when gravity rules,a line suffices to divide the ground (which is what a line in the sand, a dead line, a front line do), and a side divides the faces of a cube, yet the cube has six sides, since the six sides divide it from the external space. In short a side supports an outvector.
A dividing space is only one side. There are two spaces on either side of it that may have different natures. Likewise, one does not have to use both sides of a page to represent a plane object, so it can hardly have more than one side from any POV.
you confuse yourself with these sides. A cube has 6 faces. A square has one face. A cube is made of 6 planes, a square is made of one plane. Hope this is clear for you as it is for me.
A cube is a chorid, which supports outvectors pointing in six different directions. In the chorix that it defines, these six boundaries are sides. A square is a hedrid, with four sides, or divisions in the hedrix it supports. This much is evident from endoanalysis. The hedric faces of a cube fail to support the chorid nature of it, as much as the choric nature of the cube fail to support the terid tesseract it is a face of.
This was the mistake quickfur was making all the time. Going from 1 D to another, lines turned to planes and planes to lines, faces to edges and back. This is crazy. No wonder you people here are so confused. You are confused by overuse of your analogies. An object remains what it is in whatever N-space you put it.
Get used to it, hon. The analogies used by quickfur are pretty much those used by Hutton and others.
If you read my discussion with quickfur, you'd know that what I meant by the side of a plane was, first of all, its chirality, which is the plane's direction.
A plane has no chirality. In any case, if a plane did have chiralty, so does a line and all space. In fact, this device is used in endoanalysis to construct a pseudovector which is normal to the chirality of the object, the vector has no real meaning.
"faults with the higher dimensions"? I do not assign faults with dimensions. The faults lie with faulty assumptions and lack of coherence when going from one D to another, which is a bad habit for many people on this board.
You suppose that both a hedrix and a chorix have sides in 4D, and that a POV in 4D is incapable of rendering a chorix without having to look through other parts of it. Further, you suppose that a plane, for dividing space, has several "sides". Yet recourse to simple examples will show that all of these are wrong, unless you suppose the 'view of 4D', is renedered in the same chorix that the POV exists. The terminology all suggest this.
For one that is supposed to view four dimensions, i suspect something entirely different is going on (like what is suggested above).
The lack of coherence is from taking words with loaded meanings, and applying them with gay abandon in higher dimensions, and to suppose that all of the loaded meanings continue to work. The standard terminology uses words with loaded meanings where the meanings no longer apply.