Question on lower dimensions.

[siromar]

Is it mathematically possible to stack a number of 2-dimensional planes on top of each other? What exactly would happen? Would it become 3-dimensional? Would they overlap? Merge?

 

[emperor_of_localgroup]

Is it mathematically possible to stack a number of 2-dimensional planes on top of each other? What exactly would happen? Would it become 3-dimensional? Would they overlap? Merge?

Good question.
My answer is they will merge. First, you can not stack (a 3D action) one 2D plane on another, you can push one 'on' another causing a merger.

Hope someone has a different explanation.

 

[siromar]

Is it mathematically possible to stack a number of 2-dimensional planes on top of each other? What exactly would happen? Would it become 3-dimensional? Would they overlap? Merge?

Good question.
My answer is they will merge. First, you can not stack (a 3D action) one 2D plane on another, you can push one 'on' another causing a merger.

Hope someone has a different explanation.

But can a 2d-plane exist in a 3d universe? If so, how could we make sense of 3d actions on 2d planes? My understanding, from hyperdimensions and superstring books, is that you can bend a 2d plane in a 3d universe, which would give the appearance of force to any 2-dimensional beings living in that 2d universe. I cannot visualize how a 2d plane can exist in our universe. But if it can, then its plausible that the planes can also be stacked.

 

[ramparts]

It's perfectly legitimate to think of a 3D space as an infinite number of 2D planes (or other surfaces) "stacked" on top of each other. Alternatively (but completely equivalently) you can think of taking 2D "slices" of a 3D space. In fact, our everyday 3D space is the same thing, a slice of 4D spacetime taken at a particular time.

Siromar - of course 2D surfaces can exist in a 3D world! Think of the Earth's surface - a curved 2D space. Of course, we humans are 3D beings so we can see above and below it, but mathematically speaking that's a perfectly valid 2D manifold. You might be interested in reading Edwin Abbott Abbott's book "Flatland." It's a very entertaining and short read about 2D beings living in a 2D world (called Flatland, of course), written from the perspective of a Flatlander who traveled into our 3D world and is telling us about life in 2D. For those beings, "up" and "down" would be completely meaningless concepts.

 

[siromar]

But if a 2D surface has a thickness of 0, then wouldn't even an infinite stack of them have a height of 0? How can "stacking" be achieved? In other words, how would you stack several surfaces of earth, without first giving it a value for an additional dimension?

 

[Jerromy]

I don't think it is fair to give a 2D plane a height of "zero". If it were truly 0 then the 2 dimensions would have no substance to say they exist... giving a plane a 3D depth of 1 allows for the plane itself to exist as far as 3D reference is concerned. Give the 2D plane a dimension of time and "things" could move around each other just never "over or under".

 

[ramparts]

I don't think it is fair to give a 2D plane a height of "zero". If it were truly 0 then the 2 dimensions would have no substance to say they exist... giving a plane a 3D depth of 1 allows for the plane itself to exist as far as 3D reference is concerned. Give the 2D plane a dimension of time and "things" could move around each other just never "over or under".

Exactly. Infinities are funny. A 2D plane is more or less equivalent to an infinitely thin (but non-zero) slice within a 3D manifold. Multiply that by infinity.... well, it works out :lol:

If it helps, you can think of how a line (curved or otherwise) is made of an infinite number of points. It's the same thing, except instead of 2D slices of a 3D space, the points are 0D "slices" of a 1D space, the line. Say you have the curve y=2x, a straight line. This line can be defined by, at every point x (there are an infinite number of these), plotting a y value of 2x above the axis.