4D Seeing

[gonegahgah]

Just extrapolating upon some conversations between quickfur and 4DSpace and dragging this into my discussion on 360deg of sideways in 4D...

As quickfur mentions, the 2Der sees the hexagon in line with and via its edges (either from outside from one approach or if they are trapped inside then the inside of all the edges by turning their head up, over and under). Us 3Ders don't tend to look at a hexagon by its edges and instead turn it 90deg to look at it from one side seeing either the 'red' side or the opposite 'green' side.

The same goes for the 4Der who will look at our (let's fall back to) cube at 90deg to our direction of view (a direction we can't physically achieve).
Now we must remember that just as our cube extends into their 4 dimensions; so does the 4Der's eye; though they still alight from a point just as our eyes do.
Their eye, which is looking at a cube (cube by definition as seen by us and them; tainted by our respective perspectives), will not actually be in our 3D space.
Their eye will instead lie outside our 3D space and actually be fairly perpendicular to our 3D space.
This is the same as our eyes not residing in the same plane as the 2Der when we look at their hexagon. We don't look at it edge on like they do.

Further, a 2Der's eyes will bring top through to bottem to the point of their eye.
Our 3Der eyes bring left through right and top through bottom and all 360deg of forward direction to the point of our eye.
A 4Der's eyes will bring a sphere of forward direction to the point of their eye.

Now when we think sphere we tend to think every direction but the 4Der, just like us more limited beings, still sees primarily into just the forward direction.
But, as I mentioned, our forward tends to be 90deg to the forward of a 2Der when it comes to orientating 3D 'flat' objects for our viewing.
Likewise, a 4Der will tend to orientate 4D 'flat' objects (which to us are solids) at 90deg to how we orientate them for viewing.

So a flat cube that we observe (well we don't actually get to see it as flat) will mean the 4Ders eye is not in our 3D space but somewhere off in 4D.

So what do they see?
For a square a 2Der sees two 'flat' lines, for a cube a 3Der sees 3 'flat' faces, for a tesseract a 4Der sees 4 'flat' cubes.
For a square we see one 'flat' face. For a cube a 4Ders sees one 'flat' cube volume face.
A square to a 4Der would almost be like what a line is to us; but not quite because forward, even to a 4Der, is still one direction only; and a square requires 2 directions to be.

So what is their view of the inside - or one of two faces as they call it - of a 'flat' cube.
Well, just as we see a square only from its 90deg outwards direction from either the red or green side; 4Ders only see the 'insides of the 'flat' cubes from two directions.
They can't circle around inside a cube and look at the atoms from every direction just as we can't get inside a flat square and look the atoms from every direction available to us.
Even though we can turn a square around before us; we still only see its two perpendicular outsides.
Except the edge of course. We can look at an edge through a little less than 360deg of angle. The two faces impede a full 360deg of view of an edge.
We could look at a line from a full 360deg around its axis.

So 4Ders are just as limited as us in that sense and can only see the inside of 'flat' cubes from two directions that are perpendicular to our 3D space.
Of course if the 4Der turned the 'flat' cube out of our 3D 'plane' then it would turn instantly into a 'square' or some other 3D flat shape.
If they dragged it out of our 3D 'plane' it would disappear altogether.
Just like our turning a 2Der's square into our plane instantly turns it into a line for them and dragging it out of their 2D plane makes it disappear for them.

When this occurs their 4D vision is no longer perpendicular to our space but is perpendicular to the 'flat' 4D cube.

[quickfur]

Thank you for helping to explain this concept. I was starting to fear that I'm just not getting through.

[4Dspace]

So 4Ders are just as limited as us in that sense and can only see the inside of 'flat' cubes from two directions that are perpendicular to our 3D space.

Agree with you totally. No matter who looks at what, there will always be 2 directions to look at a thing and always 2 sides of it to see. One at a time.

[quickfur]

So 4Ders are just as limited as us in that sense and can only see the inside of 'flat' cubes from two directions that are perpendicular to our 3D space.

Agree with you totally. No matter who looks at what, there will always be 2 directions to look at a thing and always 2 sides of it to see. One at a time.

A 3D being can see both "sides" of an edge (i.e. line segment) simultaneously. Actually, from a 3D perspective, an edge has no two sides, it's just a line segment. 3D space is all around the line segment, unlike 2D space, which gets cut into two pieces, one on the left, one of the right. From us 3D beings' point of view, an edge doesn't have a "left side" or "right side"; its "sides" are all "around" it. (You might consider it as an infinitely thin cylinder, for example, so it doesn't have two sides, just a single curved side.)

So a 4D being can see both sides of a ridge (e.g., 2D square) simultaneously. Which actually, from the 4D perspective, isn't two sides at all; the square is surrounded by 360° of space (to use gonegahgah's terms.) You can think of it as an infinitely thin cubinder, which only has a single curved side. But she cannot see both sides of a cube simultaneously (the ana/kata sides, that is), just as we cannot see both sides of a square (the +Z and -Z sides) simultaneously.

Do note, however, that "side" from the 4D perspective is completely different from "side" from our 3D perspective, just as "side" from our 3D perspective is completely different from "side" from a 2D perspective. To a 2D being, a hexagon's sides are its 6 edges; but to us 3D beings, the hexagon's sides are its front side and back side, and the 6 edges aren't "sides" at all, they are just edges. Similarly, to us 3D beings, a cube's sides are its 6 faces, but to a 4D being, these 6 squares aren't "sides" at all. They are just the "edges" of the cube. It sees the cube as having a front side and back side (if you wish, an ana side and a kata side, or a +W side and -W side).

So while a 4D being cannot simultaneously see both the +W and -W sides of a cube, it can see all 6 square faces of the cube. Just as we 3D beings cannot see the front/back sides of a hexagon (the +Z and -Z sides, if you will) simultaneously, but we can see all 6 edges of the hexagon all at once. A 2D being cannot do this; it can only see at most 3 edges at a time.

[gonegahgah]

A 3D being can see both "sides" of an edge (i.e. line segment) simultaneously. Actually, from a 3D perspective, an edge has no two sides, it's just a line segment. 3D space is all around the line segment, unlike 2D space, which gets cut into two pieces, one on the left, one of the right. From us 3D beings' point of view, an edge doesn't have a "left side" or "right side"; its "sides" are all "around" it. (You might consider it as an infinitely thin cylinder, for example, so it doesn't have two sides, just a single curved side.)

Just to clarify further, I am talking about a 'solid' square not four line segments describing a square.
So just a slight correction to what you're saying. While looking at one face of a plane shape a 3Der can't see the inside edge as it is inside the shape and forms part of the area. We can only see the outside edge but we can view that outside edge from almost 360deg of perspective for any edge by rotating the plane around the edge before our eyes. We can't do that for the two faces though. We can only see them from each of their perpendicular outfacing directions even when we rotate the plane around an edge and look at the shape from a different perspective. We are still only seeing the outward facing faces and not any other angle of that face.

So a 4D being can see both sides of a ridge (e.g., 2D square) simultaneously. Which actually, from the 4D perspective, isn't two sides at all; the square is surrounded by 360° of space (to use gonegahgah's terms.) You can think of it as an infinitely thin cubinder, which only has a single curved side. But she cannot see both sides of a cube simultaneously (the ana/kata sides, that is), just as we cannot see both sides of a square (the +Z and -Z sides) simultaneously.

Yes. Just as we can't see the inside edge of a 'solid' plane; a 4Der can not see the 'inside' of the faces of a 'flat' (to a 4Der's sense) cube. Actually, they don't see the 'flat' cube from the outside faces either; in contrast to us. As you you say, our 'faces' are their 'edges'. What they do see is every molecule for the 'flat' cube from a direction 90deg to the 'flat' cube. Also they can turn the cube over and look at it from the other 90deg to the 'flat' cube as well.

So they don't see all around the cube as we do for half of it. Instead they see all of the volume filling 'face' molecules of the 'flat' cube from one perpendicular directions.
It's neither the inside nor the outside face of the cube volume they see. Instead, it is the cube volume from one of two perpendicular 'outfacing' (as far as the 4Der is concerned) faces.

Do note, however, that "side" from the 4D perspective is completely different from "side" from our 3D perspective, just as "side" from our 3D perspective is completely different from "side" from a 2D perspective.

I was going to say that a 2Der doesn't have sides; only up-down and forward-back but that's splitting hairs and open to one's perspective.

To a 2D being, a hexagon's sides are its 6 edges; but to us 3D beings, the hexagon's sides are its front side and back side, and the 6 edges aren't "sides" at all, they are just edges. Similarly, to us 3D beings, a cube's sides are its 6 faces, but to a 4D being, these 6 squares aren't "sides" at all. They are just the "edges" of the cube. It sees the cube as having a front side and back side (if you wish, an ana side and a kata side, or a +W side and -W side).

Yep, I agree too.

So while a 4D being cannot simultaneously see both the +W and -W sides of a cube, it can see all 6 square faces of the cube. Just as we 3D beings cannot see the front/back sides of a hexagon (the +Z and -Z sides, if you will) simultaneously, but we can see all 6 edges of the hexagon all at once. A 2D being cannot do this; it can only see at most 3 edges at a time.

Again, I agree. Just to slightly take this further, again like above, they see all 6 square faces but not from the 'outside' in our sense of perspective (nor from inside). They see the 6 square faces simply as part of the cube (like we see a line and not a series of dots; or a square and not a series of lines laid side-by-side) and they see those faces all from their same 'outward' direction/perspective. Whereas we would think of the cube as having 6 outwards directions; they see it as only having two outward directions.

[quickfur]

A 3D being can see both "sides" of an edge (i.e. line segment) simultaneously. Actually, from a 3D perspective, an edge has no two sides, it's just a line segment. 3D space is all around the line segment, unlike 2D space, which gets cut into two pieces, one on the left, one of the right. From us 3D beings' point of view, an edge doesn't have a "left side" or "right side"; its "sides" are all "around" it. (You might consider it as an infinitely thin cylinder, for example, so it doesn't have two sides, just a single curved side.)

Just to clarify further, I am talking about a 'solid' square not four line segments describing a square.
So just a slight correction to what you're saying. While looking at one face of a plane shape a 3Der can't see the inside edge as it is inside the shape and forms part of the area. We can only see the outside edge but we can view that outside edge from almost 360deg of perspective for any edge by rotating the plane around the edge before our eyes. We can't do that for the two faces though. We can only see them from each of their perpendicular outfacing directions even when we rotate the plane around an edge and look at the shape from a different perspective. We are still only seeing the outward facing faces and not any other angle of that face.

You're right, thanks for this clarification.

[...]
Yes. Just as we can't see the inside edge of a 'solid' plane; a 4Der can not see the 'inside' of the faces of a 'flat' (to a 4Der's sense) cube. Actually, they don't see the 'flat' cube from the outside faces either; in contrast to us. As you you say, our 'faces' are their 'edges'. What they do see is every molecule for the 'flat' cube from a direction 90deg to the 'flat' cube. Also they can turn the cube over and look at it from the other 90deg to the 'flat' cube as well.

So they don't see all around the cube as we do for half of it. Instead they see all of the volume filling 'face' molecules of the 'flat' cube from one perpendicular directions.
It's neither the inside nor the outside face of the cube volume they see. Instead, it is the cube volume from one of two perpendicular 'outfacing' (as far as the 4Der is concerned) faces.

Yes, this is exactly what I was trying to say. Thanks for clarifying!

Do note, however, that "side" from the 4D perspective is completely different from "side" from our 3D perspective, just as "side" from our 3D perspective is completely different from "side" from a 2D perspective.

I was going to say that a 2Der doesn't have sides; only up-down and forward-back but that's splitting hairs and open to one's perspective.

Well, if you want to use the "upright" model of 2D beings (i.e., standing on circular planets) then you could say that there's "front" and "back" but no "sides". But yeah, that's splitting hairs.

[...] Just to slightly take this further, again like above, they see all 6 square faces but not from the 'outside' in our sense of perspective (nor from inside). They see the 6 square faces simply as part of the cube (like we see a line and not a series of dots; or a square and not a series of lines laid side-by-side) and they see those faces all from their same 'outward' direction/perspective. Whereas we would think of the cube as having 6 outwards directions; they see it as only having two outward directions.

Thanks for expressing what I was trying to say in such a clear way!

[gonegahgah]

You're welcome. It's enjoyable to reach this level of 'seeing' that I'm achieving for myself - with help of course - by re-explaining and clarifying.

The other interesting thing out of it is that a 4Der can see something that we can't; just as we can see something that a 2Der can't.

The 2Der can look at their square and think that the 3Der sees directly into their square.
They may get a cutter and chop into their square and think the stuff they reveal is some of what the 3Der sees.

But, this is not really true. We actually see the third side (or 6 including opposites) of the inside stuff that the 2Der can't see.
They can only see it from two sides (or 4 including opposites) but we can see their cut out stuff from 3 sides.

The same goes for us. We can't see the fourth side of things.
We can do the same operation and cut into our cube and say that this stuff is what the 4Der sees.
But, again it is not totally true because they can see that same stuff from four sides (or 8 including opposites).

But, if we assume that things have a consistency in their granularity then we can assume that the grains will have a similar pattern even into the 4th direction.

That's not to say we can simply extrapolate our everyday objects into the 4th dimension.
No, we still need to think about the natural organic growth under those circumstances and re-calibrate for that.

Our trees for example branch out in 360deg of sideways with a general up direction (except for willowy branches and hangers).
A tree in 4D has a whole sphere of sideways-ness to branch into while still growing up.

But, again the line comes to the for with branches describing lines in a general sort of random sideways but upwards direction.
And again the same holds true even in 4D. Separate branches will grow in a whole sphere of sideways directions with a general upward trend.

The grain of the wood follows this branch line direction trend in both our worlds.
So their logs will still be lines; though theirs will be lines of spheres whereas ours are lines of circles.

So making a log house might be more of a challenge for a 4Der.
We simply have to build a vertical wall of logs to form a wall and then have 4 of these to form the side walls.

If a 4Der were to do this you could simply walk around the wall in the other available sideways direction; that we don't have in our 3D world.
Instead a 4Der would have to build what we think of as a cube; but in reality that 'cube' is till open to the sky and has the other extra side: the floor.

Too the 4Der it wouldn't look like what we see as a cube. It wouldn't look like what we see as a tesseract either to them.
This is because they have the advantage of being able to see from the four sides.

[quickfur]

You're welcome. It's enjoyable to reach this level of 'seeing' that I'm achieving for myself - with help of course - by re-explaining and clarifying.

Yeah, when it first "clicks", it's like wow, so that's how it works!! Of course, then you find out that there's more to come, and so the "journey into 4d", so to speak, begins.

The other interesting thing out of it is that a 4Der can see something that we can't; just as we can see something that a 2Der can't.

The 2Der can look at their square and think that the 3Der sees directly into their square.
They may get a cutter and chop into their square and think the stuff they reveal is some of what the 3Der sees.

But, this is not really true. We actually see the third side (or 6 including opposites) of the inside stuff that the 2Der can't see.
They can only see it from two sides (or 4 including opposites) but we can see their cut out stuff from 3 sides.

I've often thought about this. It's an interesting question, how an n-dimensional being sees an (n-1)-dimensional object.

Mathematically speaking, an (n-1)-dimensional object in n-space is infinitely thin; in fact, it has zero thickness in the nth direction. If any point in it can be seen at all, it can only appear the same as it appears to the (n-1)-dimensional being, because, after all, you're looking at the same mathematical point. In this respect, a "true" (n-1)-dimensional object actually only has one "side", from the n-dimensional POV.

But this simplistic analysis misses an important practical consideration: in "real-life", objects with zero thickness can't interact meaningfully with the surrounding world. A square only 1 atom thick, for example, would quickly crumple and fall apart from all the air molecules striking it, or at the slightest touch. Needless to say an actual zero width object -- it would crumple and cease to exist in any meaningful way. Besides, all our experience with the physical world is with objects that have non-zero thickness, even if that thickness is so small that to us, it looks like zero thickness. So to draw meaningful analogies with lower dimensions, it makes sense to consider (n-1)D objects as being very thin prisms of the mathematical zero-thickness object.

Assuming non-zero thickness has important consequences; because the 2D square that we look at from 3D is actually a very thin cube, so it has two distinct faces (the other 4 are so thin that, practically speaking, they are just 4 edges). These two faces are not visible to the 2Der at all. In fact, we could think of it as the 2D world being a very very flat cubic space, such that objects in this space are very flat prisms (so 2D triangles are very flat triangular prisms, 2D circles are very flat cylinders, etc.).

So let's say 2D atoms (which are approximately circles) are actually very flat cylinders. So what the 2Der sees is only the curved side of these cylinders; they cannot see the cylinder's lids. But a 3Der looking at the 2D objects practically only sees these lids (because the cylinders are so thin, you practically can't see them). So this creates a very interesting situation. The light that the 2Ders see interacts with these atoms by bouncing off their curved sides. But the light that we 3Ders see isn't bouncing off the curved sides; it's bouncing off the lids of the cylinders. That means what the 2Der sees is fundamentally different from what the 3Der sees. If the cylinders reflected blue light from their curved sides, and red light from their lids, then what appears blue to the 2Der actually appears red to the 3Der.

So the 2Der never actually sees the "inside" of any 2D object at all, like you said, cutting out the object to get at its insides still only lets the 2Der see the curved sides of the atoms, not the circular lids which we 3Ders see. We fundamentally see area, but the 2Der fundamentally sees only edges.

The only thing the 2Der can do is to mentally imagine what area is, but he can never see area the same way we do.

So in the same way, a 4Der looking at a 3D object (which may be thought of as a very very thin prism in the 4D sense) sees the inside volume of the object in a fundamentally different way than we imagine. The atoms in the cube can be thought of as very thin spherinders (extruded spheres). We only ever see the light that bounces off the curved sides of these atoms; but the 4Der directly see the sphere-shaped ends of the spherinders, which we can never see. The 4Der sees the volume of the cube directly, but we can only imagine it (our mental image of a cube is inextricably bound with the image of its 6 bounding square faces -- we just can't think of volume apart from its boundary!)

[...] That's not to say we can simply extrapolate our everyday objects into the 4th dimension.
No, we still need to think about the natural organic growth under those circumstances and re-calibrate for that.

Well, when first starting off with 4D visualization it's helpful to use direct extrapolations of familiar objects, so that we're not overwhelmed by a completely different world. It's a different matter altogether when considering 4D objects natively. For example, on my website I frequently use the extrapolation of the Earth's poles and equator when describing the positions of cells on a 4D polytope, because that's something very familiar, and not too hard to understand once you learn how to extrapolate it to 4D.

However, as the topic on 4D spheres recently showed, a true 4D planet does not have two poles and a spherical equator as a direct extrapolation would have us believe; due to the way 4D rotations work, a native 4D planet is much more likely to have ring-shaped poles with toroidal climate zones. But to use toroidal zones in describing the position of cells in a 4D polytope would make the description rather unhelpful, because the readers would have a hard time visualizing the toroidal zones in the first place, much less use them to form a coherent mental picture of where those cells are on the surface of the polytope.

Of course, the 4D planet with toroidal zones is itself based on a rather shaky assumption: that planets would even exist in the first place. 4D orbits are unstable, so it's not likely that a real, native 4D universe would actually sport life growing on a 4D planet orbiting a central star. All of these are extrapolations from 3D, and they probably don't really work in a real 4D universe. But they are useful to help us understand 4D, because we're familiar with the concept of orbits and planets, so by observing how these extrapolated objects behave, we can get a good idea of how 4D space works.

Our trees for example branch out in 360deg of sideways with a general up direction (except for willowy branches and hangers).
A tree in 4D has a whole sphere of sideways-ness to branch into while still growing up.

I've always entertained the idea of a 4D tree where branch points have icosahedral symmetry, so you'd have 12 branches going outwards (and upwards). You can make all sorts of pretty shapes this way.

[...] The grain of the wood follows this branch line direction trend in both our worlds.
So their logs will still be lines; though theirs will be lines of spheres whereas ours are lines of circles.

If 4D trees exist, I'd expect their logs to show concentric "growth spheres", just like our logs show concentric growth circles.

So making a log house might be more of a challenge for a 4Der.
We simply have to build a vertical wall of logs to form a wall and then have 4 of these to form the side walls.

If a 4Der were to do this you could simply walk around the wall in the other available sideways direction; that we don't have in our 3D world.
Instead a 4Der would have to build what we think of as a cube; but in reality that 'cube' is till open to the sky and has the other extra side: the floor. [...]

One thing about 4D is that to build a structure, you need a lot of materials. A wall needs to cover a 3D volume before it can function as a wall, and furthermore whereas in 3D we only need 4 perpendicular walls to enclose a house/room, in 4D you need 6 walls. So the amount of wood you need to build a log house is much, much more than in 3D. Not only you need 2 more walls, but each wall needs to be filled up in one whole dimension more.

[gonegahgah]

...The light that the 2Ders see interacts with these atoms by bouncing off their curved sides. But the light that we 3Ders see isn't bouncing off the curved sides; it's bouncing off the lids of the cylinders. That means what the 2Der sees is fundamentally different from what the 3Der sees. If the cylinders reflected blue light from their curved sides, and red light from their lids, then what appears blue to the 2Der actually appears red to the 3Der.
So the 2Der never actually sees the "inside" of any 2D object at all, like you said, cutting out the object to get at its insides still only lets the 2Der see the curved sides of the atoms, not the circular lids which we 3Ders see. We fundamentally see area, but the 2Der fundamentally sees only edges.
The only thing the 2Der can do is to mentally imagine what area is, but he can never see area the same way we do.

Cool, that's an even clearer way of explaining it, thanks.

So in the same way, a 4Der looking at a 3D object (which may be thought of as a very very thin prism in the 4D sense) sees the inside volume of the object in a fundamentally different way than we imagine. The atoms in the cube can be thought of as very thin spherinders (extruded spheres). We only ever see the light that bounces off the curved sides of these atoms; but the 4Der directly see the sphere-shaped ends of the spherinders, which we can never see.

Or we could even preferably think of atoms as 'glomes' to give them evenness all around though it probably makes the example more difficult.
They are interesting though with how the 4Der will see a glome. A 2Der sees a circle as a line that is a hump. We see a sphere as a circle that is a hump. A 4Der sees a glome as a sphere with a hump.
We already think of a sphere as having a hump but what we actually only see is a circle that has shade to give the hump impression. The 4Der sees the glome actually as a flat sphere from one of the glome's two perpendicular outfaces. To a 4Der a sphere alone has no shading (shading that we in fact see) just as to us a circle by itself has no shading (shading that a 2Der in fact sees).
Although it gets easier, it is still hard for us to imagine a flat sphere; and then to put a hump on that flat sphere to form the glome as the 4Der sees it.

However, as the topic on 4D spheres recently showed, a true 4D planet does not have two poles and a spherical equator...

Some food for my thoughts tomorrow I think...

If 4D trees exist, I'd expect their logs to show concentric "growth spheres", just like our logs show concentric growth circles.

Cool.

One thing about 4D is that to build a structure, you need a lot of materials. A wall needs to cover a 3D volume before it can function as a wall, and furthermore whereas in 3D we only need 4 perpendicular walls to enclose a house/room, in 4D you need 6 walls. So the amount of wood you need to build a log house is much, much more than in 3D. Not only you need 2 more walls, but each wall needs to be filled up in one whole dimension more.

That's a lot of trees!

[quickfur]

[...] A 2Der sees a circle as a line that is a hump. We see a sphere as a circle that is a hump. A 4Der sees a glome as a sphere with a hump.
We already think of a sphere as having a hump but what we actually only see is a circle that has shade to give the hump impression. The 4Der sees the glome actually as a flat sphere from one of the glome's two perpendicular outfaces. To a 4Der a sphere alone has no shading (shading that we in fact see) just as to us a circle by itself has no shading (shading that a 2Der in fact sees).

The shading on the sphere that we see is only on the surface of the sphere. From the 4D viewpoint, this is just boundary of flatly shaded spherical volume. The inside volume of the sphere is flatly shaded, but we 3Ders can't see that. We only see the "humped shading" on the sphere's surface.

Although it gets easier, it is still hard for us to imagine a flat sphere; and then to put a hump on that flat sphere to form the glome as the 4Der sees it.

Yeah, one of the original chapters of my 4D visualization document was entitled "the bulge of the hypersphere", i.e., the hump of the glome. I wanted to draw the contrast between seeing a glome vs. seeing a spherinder's lid. One has a "bulge" (in the center of the projected spherical image, if you're into projections like I am), the other doesn't (the spherical image has "flat" shading" -- not the surface of the image, but the volume inside, of course). I was supposed to rework it to incorporate nice povray-rendered images and added as a new chapter, but so far I haven't gotten around to it yet.

[...]

One thing about 4D is that to build a structure, you need a lot of materials. A wall needs to cover a 3D volume before it can function as a wall, and furthermore whereas in 3D we only need 4 perpendicular walls to enclose a house/room, in 4D you need 6 walls. So the amount of wood you need to build a log house is much, much more than in 3D. Not only you need 2 more walls, but each wall needs to be filled up in one whole dimension more.

That's a lot of trees!

Yeah, the 4D tree activists must be out in force!

[gonegahgah]

The shading on the sphere that we see is only on the surface of the sphere. From the 4D viewpoint, this is just boundary of flatly shaded spherical volume. The inside volume of the sphere is flatly shaded, but we 3Ders can't see that. We only see the "humped shading" on the sphere's surface.

Yeah I get that. It's like when we have a rectangle the edges are lighter on the sun side edges and darker on the opposite side; so in some respect we get a close impression of what a 2Der must see. We just ignore the evenly shaded area and think about seeing just what edges we can see with our eyes inline with the edges from only one direction at a time. For a circle with its curved edges this is a progressive change in shading again giving us a good impression of what the 2Der would see by doing the same. So it should be the same for the 4Der. They would hopefully get a similar impression of the sphere that we see by looking at a 4D 'flat' sphere and ignoring the evenly shaded volume then seeing the shadows that are cast by what they see as the sphere as 'edge' (for a very thin spherinder that is).

[quickfur]

The shading on the sphere that we see is only on the surface of the sphere. From the 4D viewpoint, this is just boundary of flatly shaded spherical volume. The inside volume of the sphere is flatly shaded, but we 3Ders can't see that. We only see the "humped shading" on the sphere's surface.

Yeah I get that. It's like when we have a rectangle the edges are lighter on the sun side edges and darker on the opposite side; so in some respect we get a close impression of what a 2Der must see. We just ignore the evenly shaded area and think about seeing just what edges we can see with our eyes inline with the edges from only one direction at a time. For a circle with its curved edges this is a progressive change in shading again giving us a good impression of what the 2Der would see by doing the same. So it should be the same for the 4Der. They would hopefully get a similar impression of the sphere that we see by looking at a 4D 'flat' sphere and ignoring the evenly shaded volume then seeing the shadows that are cast by what they see as the sphere as 'edge' (for a very thin spherinder that is).

Yeah, they could look from the side of the spherinder and see a cylinder-shaped volume that has "progressive shading" that indicates its spherical shape. This "progressive shading" would be the extruded version of what we see as the shading on the sphere's surface.

[4Dspace]

This is such an interesting discussion! Thank you for it For me it was ..as if I was trying to discern something in a dimly lit room and then wham! the lights went on. gonegahgah, you vision is so crisp, thank you for sharing it. And then quickfur pointed out some interesting aspects of it.

I wish I could contribute meaningfully to the discussion, but at the moment I can only sit in the back of the upper gallery, with popcorn falling out of my mouth, lol, with my jaw floored, like they say it here in America.

I realize now that "nDer vision" is a code for analysis of properties of n-spaces and their relationships between each other. How very interesting! I need to reread your whole conversation again ..and think it over.

There was one thing that jumped at me right away. And that was in regard to this:

Mathematically speaking, an (n-1)-dimensional object in n-space is infinitely thin; in fact, it has zero thickness in the nth direction.

We have such "objects" in our world, and that is Maxwell's "physical lines of force". The magnetic lines we always see edge-on. So they must have 0-thickness in our world, and so they are invisible to us, except when they are revealed in how iron shavings fall on a sheet of paper with a magnet beneath.

[gonegahgah]

I think it even gives us a greater insight into our own world. I would never have thought before of a sphere as a humped circle.
Magnets are interesting in that they reduce in force approaching a cubed rate whereas gravitational force reduces at a squared rate.
This is what allow us to have maglev trains.

[gonegahgah]

Don't ever press Ctrl-R. Very bad!!! Crap!

[gonegahgah]

As I was saying... I just had a moment of clarity. Just as you said QuickFur it is a journey.

Basically, 4Ders can not access things that we can not access from our direction. To them our directions are the edges.
If we can not access the inside of a cube then they can not access the inside either from our directions.
A 4Der can only access things we can't from their extra fourth direction.

This helps to explain how things can be on the road in four dimensions (which the projection model can otherwise confuse our understanding of).

If we have a 3D road then a 4Ders tyre can not exist inside our road just as ours can't.
It can only exist on our road or be somewhere off in the fourth direction. Even then it would still be on the road in other adjacent 3D 'planes'.
It would be on the road across multiple adjacent 3D 'planes'.

You can hark back to the 2D<->3D environment to see this as well. Exactly the same situation would exist there.
As an example, if a 2Der has a mountain then we can put a tunnel through it without them even knowing.
However if we want to put a road from their direction of access then we are going to have to flatten their mountain and the 2Der will see this alteration.