gonegahgah wrote: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.
4Dspace wrote:gonegahgah wrote: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 wrote: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.)
quickfur wrote: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.
quickfur wrote: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.
quickfur wrote: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).
quickfur wrote: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 wrote:quickfur wrote: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.
[...]
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.
quickfur wrote: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.
[...] 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.
gonegahgah wrote: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.
[...] 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.
[...] 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. [...]
quickfur wrote:...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.
quickfur wrote: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.
quickfur wrote:However, as the topic on 4D spheres recently showed, a true 4D planet does not have two poles and a spherical equator...
quickfur wrote:If 4D trees exist, I'd expect their logs to show concentric "growth spheres", just like our logs show concentric growth circles.
quickfur wrote: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 wrote:[...] 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.
[...]quickfur wrote: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 wrote: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.
gonegahgah wrote:quickfur wrote: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 wrote:Mathematically speaking, an (n-1)-dimensional object in n-space is infinitely thin; in fact, it has zero thickness in the nth direction.
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