I need some guidance to explain the difference between two and three dimensions!
If you have some answer please give that to me because I think that I am in a fix, and not able to explain clearly what is the difference between the two. Also if there is any web site please refer me about that as well.
Thank you in advance, to all the distinguished members.
Zaiq.

I must admit, I was about to go to bed but I'll have a think.
There are a couple of interesting differences between your normal 2 and 3 dimensional spaces that I can think of now.

How about-

If you reflect an object in two dimensional space in some mirror line through the origin and then reflect again in some different line you end up having rotated the object. In 3 dimensions if you reflect twice in two planes through the origin you do not generally get a rotation.
Think that's right, anyway.

There are all sorts of regular two dimensional shapes in 2 dimensions- eg. the regular n-gon has all sides equal length, all angles equal. But in 3 dimensions the only solid shapes with the property that all faces are the same and all edges are the same length etc are the five Platonic solids.
So there are numerous groups (finite) of rotations of flat space (just rotate by some small rational fraction of (2 x pi) enough times) but if we insist on rotation groups of 3-space not to just fix one dimension then there are only five. (rotations of the Platonic solids)

If we start a particle at the origin in a 2 dimensional lattice and say that each second it moves left/right/up/down with probability (1/4) then the expected time for it to return back to the origin is finite. In > or = 3 dimensions this is not true; there is a non-zero probability of it being at the origin at infinitely many times but the expected time for it to return is infinite. This says that there is qualitatively 'more room' for a wandering person to get lost in 3 rather than 2 dimensions.

Um, I'm not sure about this one but I think that Newton's equations of gravity only work in 3, not 2 dimensions.

I'm not sure if that's the sort of thing you want but I hope it's all reasonably true and really interesting.


John.

Hi John!

I think that in fact the expected time for the return is infinite in 2-D and even 1-D! However the probability that it eventually returns is one. In 3-D the probability of a return once is greater than zero but less than one. Shouldn't this mean that the probability of returning to the origin infinitely often is zero?

(By the way there is more on random walks in this thread .)

Many thanks,

Michael

Yep. Think you're right.
Also I think I just made up the bit about gravity.
Maybe only > = 3 dimensions works or something??
I'll have a look at your question , but I daresay you're right again !
John.

Newton's model for gravitation could easily be applied to a theoretical two dimensional world, simply by saying that the attractive force between two bodies is inversely proportional to the square of the distance between them. In fact, you could apply this law to any space with a metric (a distance function).

You will apply Newtonian gravitation to two dimensional spaces all the time in A-level maths and physics; problems in mechanics are often modelled two dimensionally.

Similarly, Einsteins four-dimensional space time model is often simplified to its three or two dimensional analogue when it is introduced for the first time in, say, a special relativity university course.

For a really interesting account of life in a two dimensional world you should read "Flatland" by Edwin A. Abbott. It tells the story of a square from a two dimensional world who travels through higher dimensions. It is a kind of "Gulliver's Travels" for mathematicians, and is great.

Harry Smith

However in a true 2-D world (not the kind met in physics or mechanics) it would almost certainly be an inverse law (not an inverse square law). I'm not sure (so correct me if I'm wrong) but most quantum gravity theories have gravity caused by the influence of gravitons. These can eminate from a point source. Therefore their density is inversely proportional to the area over which they have spread. Therefore the attractive force is proportional to 1/r² . This would change if it was only spreading out in 2-D.

If it was an inverse law then the escape velocity of an object from a field due to a point source would be infinite - the potential energy difference between a point and a point at infinity would be infinite.

Of course there is absolutely no problem with solving 3-D questions by only considering two dimensions, because any two body orbit takes place in just one plane.

Michael

Zaiq

I have to admit that the above discussion is very interesting, but it doesn't give you any information about the exact mathematical difference between two and three dimensional space. In case that was what you were interested in, here is my attempt at explaining it.

Dimension is a fairly fundamental mathematical concept, and it is typical in the sense that mathematicians have looked at a property of the > real world, worked out what the bare bones of it are, defined it in an abstract way and then used it much more generally than it is ever found in real life.

What I mean is this: we all know that the world we live in is three dimensional, because there are always three "mutually perpendicular" directions (by which I mean that each direction is at right angles to BOTH the others) - up, forward and right, for example. The important property of these directions is that they are "linearly independent". That means (roughly speaking) that you can't go in any one of the three directions by moving along any combination of the other two (e.g. you won't go up, however you move in the directions forward and right). [In fact, this is only true when the directions are mutually perpendicular, which is not necessary for linear independence, but we don't need to worry about that]. This may well seem pretty obvious - it is, but only because the 3-D world is so familiar. The second vital property is that if you use all three directions, you can (in principle) go anywhere (of course, you may need to go backwards in any of the directions as well - down as well as up, left as well as right, etc.).

Now think about two dimensions, e.g. a sheet of paper. How many linearly independent directions are there? There are in fact two - e.g. towards (or away from) the top of the paper, and towards (or away from) the right hand edge of the paper. These are "linearly independent": you can't get any closer to the right hand edge of the paper by > going towards the top, and you can't get any closer to the top by moving towards the right hand edge. But you can get to anywhere on the paper by using both directions.

Now you should be able to analyse a one- dimensional space - i.e. a line. There is now only one possible direction, which is the direction of the line, and you can get anywhere on the line by moving forwards or backwards along the direction of the line (that really should be obvious!)

So it turns out that the dimension of a space (real or abstract) is exactly the number of linearly independent directions that there are. Now mathematicians use the idea of direction much more widely than normal people, by defining and then generalising the concept of "vectors". But it is all inspired by this fundamental property of spaces that we all know about. Interestingly, it is possible to have spaces that are very similar > to the one we live in, but which have MORE than three dimensions. There is no mathematical reason for just sticking at three.

I hope that sheds some light on things.

Just to add more confusion to this thread, it was proved in my Dynamics lecture course last year that planetary orbits are unstable in spaces of more than 3 dimensions. Also, you need more than 2 dimensions for humans to be able to exist (think about eating food, if we were 2D the path of the food would cut us in half), so our universe must be 3 dimensional.

I remember reading an interesting article by David Singmaster (I think) in which he proposed a method for tubes and tunnels in two dimensions which doesn't "cut us in half". It involved a series of 'valves' which would open to allow liquid (or food) through, but close again afterwards, maintaining the solidity of the structure.

In response to Michael's point, Newton derived his law of gravitation by considering a particle at infinity and integrating - I'm not sure whether it is valid to talk about it in quantum theoretical term. It was, after all, superceded as a model by general relativity long before bosons and gravitons were even dreamt of.

But we're way off the point - sorry. Save this one for another time maybe.

Harry

Zaiq,

Most of the messages above are about strange mathematical things that happen in different numbers of dimensions. This is all very well, but doesn't answer your question of how to explain what the difference is. I'll try to give a simple explanation.

Three dimensional space is what we live in. There are three essentially different directions. These are forwards, left and upwards. All other directions can be made out of these. For example, you can go forwards a bit, then left a bit to get somewhere diagonally in front of you. You can think of going right as going a negative distance left and so on.

A piece of paper is two dimensional (if we ignore its thickness). If you start out at a point on the paper you can only go up or left (or combinations or negatives of these).

Similarly a straight line is one dimensional as you can only go along it; there is nowhere else to go!

Please let me know whether or not this has been helpful, and if you want me to explain anything more clearly.

There is a very good book called "Flatland" by Edwin Abbott which explains dimensions very well and is very funny.

Jonathan