gonegahgah wrote:[...]However, gravitational force is actually calculated combinationally. F = G(m1m2))/r2.
So both the bulk of the hyperplanet and the body contribute to the combined gravitational force.
Yes, and in 4D, you have F = G(m
1m
2)/r
3.
But we're talking about gravity on the surface of the planet here, so r and m
1 are fixed. So that amounts to F = Km
2 where K is (approximately) a constant and m
2 is the mass of the object on the planet. In this sense, it's identical to the 3D case. Things only become different when the value of r is significantly different, say when you're flying out into orbit or something.
I think you also have to remember that no matter how big an object is it will still fall at the same speed which is purely dependent upon the object it is falling towards.
It's bulk does not contribute to its own falling speed.
What it's own bulk does contribute to is the speed the planet falls towards it so they will actually meet sooner if one or both are bulkier.
You mean the
acceleration is the same. The speed at which an object falls changes over time. But you're right that if two objects, one heavy and the other light, were to be dropped from the same height, they would hit the ground at the same time. That's because they are experiencing exactly the same gravitional force, which gives them identical acceleration. The bulkier object feels more force, but it's directly in proportion to its mass, so the resulting acceleration is equal.
One way to think about it is that the atoms in the object are identical (or, if you like, the subatomic particles or whatever it is at the most basic level), so they all fall down with the same acceleration and reach the ground at the same time. Just because a small number of atoms happen to comprise a light object doesn't change this, and just because a large number of atoms happen to comprise a heavy object doesn't change this either.
Also, although we can see all sides of a square when it is planewards towards us we can not see the opposite inside or those same sides from the other side.
That assumes the square has non-zero thickness. In other words, it's actually a very flat cube. In a truly 2D universe, a square has zero thickness, and 3D light would pass right through it (i.e. it will be invisible). But
if we can see it somehow, we'd be able to see all of it at once, because its "other side" is actually its "near side" too.
If we turn a square so that it is edge on to us and we close one eye than it becomes hard to see any of the other edges or either square inside.
Again, that assumes that the square is actually a very flat cube. If it's a true 2D square, it would literally disappear when viewed edge-on because it will have 0 thickness. In this case we won't even be able to make assumptions about magically seeing it regardless, because it will project to a line of 0 width in our eye, which cannot be seen. But
if we make the additional assumption that we can somehow see a mathematical line segment, then that's what we will see: just that one edge of the square (or 2 if you look at it from an angle).
The same goes for a 4Der who's eyes can see all 'faces' of the cubes when it is at the right 'angle' to them. They can 'rotate' the cube so that they are looking at it 'edge' on as well; the edge being a plane.
Yes, and I see this in projection all the time when looking at polytopes with equatorial facets. Facets like cubes show up as squares or hexagons when seen from that angle.
The 4Der doesn't have eyes that surround the cube; instead they have eyes that exist like hyperspheres. The back of a cube to us is like the back of a tesseract to them. If something has a back in the 4D world then the 4Der will no more see that back then we see the back of anything. What they will see is the front spread through 3 of their 4 dimensions; whereas what we see is the front of things spread through 2 dimensions. So I strongly disagree that they will see cubes as having a top, bottom, back, front, and two sides. They will not see an enclosed cube at all.
OK, I still can't figure out if you're talking about 4D cubes or 3D cubes. A 4Der who looks at a 3D cube sees all 6 faces at once, plus the entire inside of the cube.
When they see a tesseract, they only see at most 4 of its cubical cells at once. (Unless it's transparent, of course.) They have to turn it around to see the other 4 cubes. But nevertheless, in their mind's 4D model of the tesseract, there will be a top cube, a bottom cube, and 6 lateral cubes.
I don't think it is simply a case of translating the 3rd dimension into the 4th. The behind of a cube in our world remains part of the behind of a tesseract in their world. To them a cube would be a completely different beast to our cube. Our cube does not share the same 'space' as their cube.
OK, you're using very confusing terminology. I can't figure out where "cube" means 3D cube and where "cube"' means 4D cube. But they are very different beasts.
The behind of
our 3D cube is completely laid bare before the 4Der's eyes. They can see all of it, back faces, front faces, insides, everything. In fact, to them there is no such thing as a front face or a back face; all 6 faces are equivalent. It's only from our 3D-centric view that there's a difference between a cube's front face and its back face.
A 4D cube, on the other hand, can only be seen one side at a time from the 4Der's eyes. If they look at the 4D cube from one of its facets, all they would see is a 3D cube. This 3D cube, of course, has no "front" or "back" faces from their point of view; they can see all of it simultaneously. However, this 3D cube obscures their view of the other 7 cubes in the 4D cube. If they were to turn the 4D cube a little, then they might see a second 3D cube come into view (and the other 6 remain hidden from their view in the 4th direction). Or if they turn it a different way, they may see three 3D cubes (and the remaining 5 remain hidden from view). Or, if they turn it so that they are looking at one of its 16 corners, then they would see 4 3D cubes simultaneously. (And by that I mean that
all 4 3D cubes along with all of their faces, edges, and vertices, are
all visible
at the same time.) The other 4 remain out of sight. And this is the best they can do: they simply cannot see more than 4 3D cubes on the tesseract at one time. They have to rotate the tesseract around to its far side to be able to see the other 4 3D cubes, at which point, these 4 3D cubes will no longer be visible.