# What does the dimension mean?

1. Jun 30, 2005

### shekoofeh

Has the Dimension a mathematical defination?

2. Jun 30, 2005

The most frequent context of the term 'dimension' in mathematics is denoted to vector spaces. We can define the dimension of a vector space as the maximal number of linear independent vectors. Vectors should not always be considered as 'typical' vectors - for example, the set of all continous functions on a segment is a vector space too.
An easy example of the dimension of a 'more obvious' vector space is the dimension of the vector space of all radius-vectors in 3-d Euclidean space. That dimension is equal 3 ( $$dimV_{3}=3$$), because there are 3 linear independent vectors there, what means that you can write any other vector from that space as a linear combination of these three. Also, these three vectors cannot be shown as linear combinations of one another.

3. Jul 1, 2005

### gravenewworld

To make his point more clear--dimension refers to the number of vectors in a basis (which are inherently linearly independent) for a vector space.

4. Jul 1, 2005

### HallsofIvy

Staff Emeritus
More generally, "dimension" is used to tell how many numbers are needed to identify something of interest. If we working with problems on a single line or curve, each point can be identified by a single number (simplest way: choose 1 point on the curve to be 0, identify every other point by its distance along the curve from that point, one way being +, the other -)- dimension 1. If we are working with problems on a plane or other surface, set up a coordinate system (i.e. parametric equations for the surface which will involve two parameters) and we can identify each point by two numbers- dimension 2. The extension to 3 dimensions is obvious. Physicists work with "events"-things that happen at a particular point in space and a particular time. That requires 3 numbers to identify the point and one to identify the time: physicists work in 4 dimensions.
If I were doing a study of all possible spheres in space, I might record each sphere by the position of its center (3 numbers) and its radius (1 number). That is, again, 4 dimensional.
In Thermodynamics, we might have to imagine a cloud of N particles, each with a given position (3 numbers) and momentum (3 numbers) as well as time (1 number). That would be 6N+ 1 dimensional! (Which is why thermodynamics uses statistical methods.)

5. Jul 1, 2005

### Borogoves

As a non physicist, I ask you, how do you physicists define time ?

6. Jul 1, 2005

### uart

I don't think I've ever seen a definition of time that's not to some extent circular. My personal favorite is :

Time : That property of the universe which prevents everything from happening all at once. :)

7. Jul 1, 2005

### Borogoves

Which is meaningless. Too vague !

8. Jul 1, 2005

time is a continuum...a very simple definition :)

9. Jul 1, 2005

### The Rev

Sweet.

:rofl:

The Rev

10. Jul 2, 2005

### mathwonk

dimension is often defined inductively, based on the idea that the boundary of something has dimension one less than the something, boundary in the sense of boundary of a manifold, not topological limit points.

this is explicitly used as a definition by poincare in his essays. he defines a continuum as having dimension one if it can be separated into more than one connected component by removing a finite set of points, say as a circle can. so a circle is one dimensional.

then a continuum is 2 dimensional if it can be separated by the removal of a one dimensional continuum.

etc, etc.. this actually occurs in riemann, as a "fragment on analysis situs", some 50 or 60 years earlier, as apparently almost everything does.

these ideas are justified as theorems by the results in another thread on the jordan brouwer separation theorem. i.e. there the dimension of a manifold is defined in terms of saying th dimension of R^n is n, and that any manifold, locally homeo to R^n also has dimension n. then it is a therem that a compact n-1 dimensional manifold separates R^n.

the lovely little book by hurewicz and wallman, dimension theory, says something like this for more general topological spaces:

a space has dimension zero at a point, if there is a basis of neighborhoods at that point consisting of sets with empty topological boundary, i beieve.

so the rationals are zero dimensional for example.

then one goes on up inductively. i.e. finite sets are zero dimesnional too.

so then the reals are one dimensional everywhere since each point has a basis of nbhds with finite boundary.

in algebra and algebraic geometry, dimension of a domain is defined in terms of "krull dimension", i.e. the maximal length of a chain of prime ideals, or transcendence degree of the field of fractions. the transcendence degree allows the definiton to be relative. i.e.th field of fractions of a complex plane curve has dimension one over the complex field C, but dimension zero over the field C(z) of rational functions in one variable z.

this allows the concept of families of curves, i.e. a familiy of curves parametrized by a variety V is a variety mapping to V and whose function field has tr. deg. one over that of V. i.e. a surface S mapping to a curve V, can be viewed simply as a curve over V.

it is natural to restrict dimension to domains since domains correspond to spaces having only one irreducible component, and dimension should be defiend separately on each component.

or again, actually dimension should be defined locally at each point, so one uses the krull dimension of the local ring at a point.

one can also proceed as in manifold theory, saying the standard object, i.e. affine space k^n has dimension n, and then defining an appropriate family of maps that preserve dimension, in this case finite maps, amd defining the dimension of an affine variety as the minimum dimensionl affine space to which one can map your variety by a finite map. finite maps are defined either as proper maps with finite fibers, or maps inducing on pullback of rings of functions, a finite module map.

there are also homological notions of dimension. i.e. the homology of a compact n manifold vanishes above dimension n, but also in lower dimensions for non orientable manifolds. so one ahs again a local homological definition interms of the homology of a deleted neighborhood of a point.

also there is a homological definiion in ring theory, pioneered by hilbert, and also developed by auslander and buchsbaum, serre and others.

i.e. the length of a projective resolution. e.g. the ring of polynomials in n variables, has transcendence degree n, and also admits a projective resolution of length n and no shorter. however auslander buchsbaum showed that a local ring which is not regular has infinite homological dimension. so homological dimension only works for regular local rings.

the number of generatros of a maximal ideal is another approach but must be defined carefully.

so this is such an important idea it has appropriate definitions in every setting.

for exmple a pid is nothing but a one dimensional ufd in ring theory.

a dedekind domain is a one dimensional "regular" local ring, i.e. the local ring of a smooth curve.

Last edited: Jul 2, 2005
11. Jul 2, 2005

### Borogoves

I can see that you've deliberately deviated from the question which I posed, namely, how physicists define time.

Furthermore, the response you gave to this gentleman is way too advanced.
I think what HallsoIvy said sums it up nicely.

12. Jul 2, 2005

### Icebreaker

13. Jul 8, 2005

### Gokul43201

Staff Emeritus
Wow, that's about as presumptuous a statement as I've seen in a bit. Why do you think mathwonk should be responding to your question? I think it's pretty obvious that he is responding to the OP's question.

Too advanced for whom ? Please let the OP (whom you call "this gentleman") determine what is too advanced for him/her. Personally, I found mathwonk's post extremely enlightening.

If anything, you caused this thread to deviate from its original purpose. If you have a fresh question to ask, please start a new thread.

14. Jul 9, 2005

### mathwonk

a definition that is nice and simple that i only alluded to above is the "krull dimension " of a ring. this is analogous to the idea of defining the dimension of a vector space as the maximal number of non zero subspaces one can find in a chain, each contained in the next.

for algebraic varieties the concept corresponding to subspaces is irreducible subvarieties. and for rings, prime ideals correspond to irreducible subvarieties, so the krull dimension of a ring is the maiximal number of non zero prime ideals one can find in a chain of prime ideals in that ring.

15. Jul 9, 2005

### mathwonk

another nice notion of dimension, i believe it is due to lebesgue, is called covering dimension. it is the smallest number n such that every open cover has a refinement in which at most n+1 open sets meet.

e.g. on the real line every open cover has a refinement by intervals such that at most 2 intervals overlap at any one point.

if you think about the way bricks are usually laid you can imagine that for the plane you can always refine until at most three sets overlap, etc....

with this definition it follows immediately that cech chomology vanishes above the dimension of the space, with any coefficients, since a cech r-cocycle is a family of functions from r+1 fold overlaps of an open cover, to the coefficient sheaf.

this leads insigth into the otherwise wildly unintuitive notion of cohomological dimension, as the smallest number n such that the sheaf cohomology groups above dimension n always vanish with all coefficient sheaves.

Last edited: Jul 9, 2005
16. Jul 9, 2005

### Borogoves

Why should he respond ? for the same reason that he chooses to respond to other members, whatever that reason may be. It's called equality.

The trouble is, the question itself is rather vague, and therefore is prone to various interpretations. It was certainly not my intention to cause this thread to deviate from its original purpose. On the contrary.

Since you have 'answered' in place of mathwonk, would you care to answer my question yourself ?

17. Jul 9, 2005

### Hurkyl

Staff Emeritus
If you would like to ask a question, you can certainly start a thread asking it, rather than trying to hijack someone else's.

18. Jul 10, 2005

### mathwonk

so we have: (with some references)
1) minimum cardinality of vector basis. (linear algebra)

2) minimum cardinality of transcendence basis. (field theory)

3) maximal length of chain of linear subspaces. (vector geometry)

4) maximal length of chain of subvarieties. (algebraic geometry)

5) maximal length of chain of prime ideals. (commutative algebra, krull)

6) minimal number of generators for an ideal such that the maximal ideal at a point is its radical. (commutative algebra, algebraic geometry, shafarevich)

7) 1 more than dimension of "boundary". (topology of manifolds)

8) maximal dimension of non zero homology group. (sheaf theory, topology, grothendieck)

9) minimal length of projective resolution. (homological algebra, hilbert, auslander buchsbaum,)

10) dimension of topological boundary of basis of neighborhoods at a point. (abstarct topological spaces, hurewicz - wallman)

11) 1 more than dimension of space that separates or disconnects the given one (riemann, poincare).

12) minimum n such that there is an admissble map from R^n onto the space. ("naive definition", includes Halls definition when admissible means smooth. this was exploded for continuous maps by examples of cantor.)

(not all equivalent of course.)

others?

Last edited: Jul 10, 2005
19. Jul 10, 2005

### EnumaElish

Dimensionality of an object in the Cartesian space is the number of different (nonidentical) points you need to identify that object uniquely, minus one. Thus:

You need only a point (itself) to identify a point in the Cartesian space.
Dimensionality of a point is = 1 - 1 = 0. A point is dimensionless.

You need two points to identify a line. Stated differently, a unique line goes through two nonidentical points in space.
Dimensionality of a line is 2 - 1 = 1. A line is one-dimensional.

You need 3 points to identify a plane. Stated differently, a unique plane goes through 3 nonidentical points in space.
Dimensionality of a plane is 3 - 1 = 2. A plane is two-dimensional.

You need 4 points to identify a cube. Stated differently, a unique cube goes through 4 nonidentical points in space.
Dimensionality of a cube is 4 - 1 = 3. A cube is three-dimensional.

Last edited: Jul 10, 2005
20. Jul 10, 2005

### mathwonk

you have not defined the term "identify" but one definition would be "span", hence this is a nice geometric variant of the vector spanning definition.

(actually there are some problems with your statements about planes and cubes, i.e. they are not true as they stand. but anyway, how would you continue to define a 4 dimensional cube?)

others due to riemann are these:

1) a one dimensional manifold is obtained by moving a point continuously from one position to another. on the resulting locus, a point can only move in two directions, forward or backward.

by moving a one dimensional manifold from one location to come to lie on another one dimensional manifold, one sweeps out a two dimensional manifold, i.e. a surface.

etc....

this is essentially defining an n manifold as the image of an injective continuous map from R^n into the given target space.

2) in the reverse direction, one takes a given n dimensional manifold and defines a maiximal rank smooth function on it with real values. then the inverse image of a single real number constitutes an n-1 dimensional manifold.

thus in all cases for riemann an n - manifold is something compunded of an n-1 dimensional manifold and a one dimensional manifold.

(taken from his lecture "on the hypotheses underlying geometry".)

notice also the following extract of a translation of euclids elements:

Definition 1.
A point is that which has no part.

Definition 2.

Definition 3.
The ends of a line are points.

Definition 4.
A straight line is a line which lies evenly with the points on itself.

Definition 5.
A surface is that which has length and breadth only.

Definition 6.
The edges of a surface are lines.

notice that although definitions 1,2, and 5 of lines, points and surfaces, make no sense alone, definitions 3 and 6 which relate them, amount to an inductive definition of dimension as given above.

Last edited: Jul 10, 2005