Just a simple question, but one that I’ve not yet been able to answer:
Can you go within a surface?
Probably not.
Well, a point is a location that has no extension. A line is a locus of points and has extension in one dimension. A surface is the locus of a line and has extension in two dimensions.
It would seem that a surface has no “within.”
Say I have a surface, plane A. I also have a point, B, some arbitrary distance away from that plane. If I move point B through the plane, as in it starts on one side of it, and ends up on the other side of it, wasn’t it at some point within the surface?
It depends on the shape of the surface. If it’s a 3D enclosed space (like a hollow sphere), you can go within it.
It depends. As panache45 points out, if by “go within” you mean “be encompassed by” then yes. If by “go within” you mean “actually be in the surface itself”, then no.
Although, then again, one could perhaps say that a two dimensional shape can be within another two dimensional shape (ie a blue surface dot on a yellow background could be described as being within the yellow), as long as you are just considering two dimensions.
It all depends on what you mean by “within”.
Isn’t there a difference between going through a thing and actually being within it? Going through something only requires movement from point A to point B, on a flat dimension. But to actually be within a thing requires enclosure, which would seem to imply a spatial reality of at least three dimensions. This is where I get confused, though, and I do not even know if I can explain it well enough for my question to be understood:
The surface of a thing seems as if it must have some amount of depth, however minute, else it could not possibly provide the necessary tension to contain and hold its own matter in place. Yet, once you move within that depth, does it not now have its own surface, so that you are never really within the surface, but always within the confines of what that surface contains?
Are you talking about a physical object or a mathmatical concept? The two likely have different answers.
If we’re talking about a hypothetical 2D universe (as in Edwin Abbot’s Flatland), then it’s impossible for any entity in that universe to be anywhere other than ‘in’ the 2d surface, because the 2D surface encompasses all the places there are to be - just the same way as the 3D space in our world encompasses all the places there are to be, for a solid 3D object.
Indeed. We also need to know, if by “surface,” you mean a plane - which is a theoretical construct of only two dimensions. On the other hand, if by “surface,” you mean the outer boundary of a solid three-dimentionsal object, that may be a different thing altoghether. You could get “inside” the boundary of such an object since the boundary is closed, but you still can’t get “inside” the surface.
Interesting related books:
• Flatland
• Flatterland
• The Planiverse
For a real world answer:
Yes, eventually there is a smallest possible size.
Object > Molecule > Atom > Quark > String
Or something like that.
Any string that is on the outside-facing surface of the quark that is on the outside-facing surface of the atom that is on the outside-facing surface of the molecule that is on the outside-facing surface of the object is really and truly the absolute surface of the object that can’t be physically split any farther.
But as the others have stated, in mathematical terms a surface has no depth. So there you just have to take comfort in the fact that you can’t have negative two apples either. Math is just a tool, not a representation of the real world.
I see what you are saying, but to get technical, you can’t move a point[sup]1[/sup]. A point is an exact location defined by its coordinates in space and can’t move. However, you can speak of a point that is in the set of points defined by the plane. Such a point can be said to be “in” the plane. If you want to use the word “within” I suppose you could, although I don’t know what the OP means by “within.”
- I would reword your situation informally as two points, A and B, on either side of plane P. The line defined by A and B intersects P at point C. Point C is said to be “in” the plane.
I checked that out on Amazon and the reviews are gushing with praise. The funniest one is the computer professor who thought the book was real and came to class so dissapointed when he realized it’s fictional!
I think I may have to pick up a copy. Thanks!
A point can be within a surface, I think this is fairly clear. The only problem with the OP question is that it asks can you go within a surface? Now ‘you’ cannot be a point, but can imagine being a point, this makes answering the OP question a little open ended.
As for whether a point in a plane is within the surface of that plane, I think we must answer unequivically yes. My reason being that we don’t hesitate in saying that our heart is within our body, yet from the point of view of an imaginary four dimensional being our heart is just as exposed to that 4th dimension as any other part of our three dimensional self. A 4th dimensional existance could touch any part within us without traversing our three dimensional surface (our skin) just as we can touch the center of a square without moving our finger through the perimiter of that square.
So to say some point inside a square is not within the square because we can reach it directly from our 3rd dimension point of view is just because of our bias of thinking 3 dimensionaly.