What Are the Different Definitions of Dimension in Mathematics?

[shekoofeh]

Dimension,
what does this mean?!
and Has anyone heared about 2-d matter?

 

[Mariko]

SOURCE OF DEFINITION~Merriam Webster dictionary online*(AOL)

Main Entry: [1]di·men·sion
Pronunciation: d&-'men(t)-sh&n also dI-
Function: noun
Etymology: Middle English, from Middle French, from Latin dimension-, dimensio, from dimetiri to measure out, from dis- + metiri to measure —more at MEASURE
Date: 14th century
1 a (1) : measure in one direction; specifically : one of three coordinates determining a position in space or four coordinates determining a position in space and time (2) : one of a group of properties whose number is necessary and sufficient to determine uniquely each element of a system of usually mathematical entities (as an aggregate of points in real or abstract space) <the surface of a sphere has two dimensions>; also : a parameter or coordinate variable assigned to such a property <the three dimensions of momentum> (3) : the number of elements in a basis of a vector space b : the quality of spatial extension : MAGNITUDE, SIZE c : a lifelike or realistic quality d : the range over which or the degree to which something extends : SCOPE — usually used in plural e : one of the elements or factors making up a complete personality or entity : ASPECT
2 : obsolete : bodily form or proportions
3 : any of the fundamental units (as of mass, length, or time) on which a derived unit is based; also : the power of such a unit
4 : wood or stone cut to pieces of specified size
5 : a level of existence or consciousness
- di·men·sion·al /-'mench-n&l, -'men(t)-sh&-n&l/ adjective
- di·men·sion·al·i·ty /-"men(t)-sh&-'na-l&-tE/ noun
- di·men·sion·al·ly /-'mench-n&-lE, -'men(t)-sh&-

 

[fliptomato]

Dimension,
what does this mean?!
and Has anyone heared about 2-d matter?

(1) See Mariko's post, especially 1a and 3, which are the contexts in which physicists use the term dimension.

(2) Perhaps you are thinking about 2-d systems, which crop up in condensed matter physics a lot? For example, you can have a 2-d lattice of atoms in a material such that that the electrons living on this lattice effectively live in this 2-d sheet. (That's a really rough way of thinking about it.) Note that the 'matter' itself is not two dimensional--atoms have definite volume in 3-space, it's just that the extension in the third dimension is pretty small causing the system to behave in a way where there are only 2 relevant directions.

 

[pervect]

Dimension has a large number of closely related definitions in mathematics.

http://en.wikipedia.org/wiki/Dimension

In mathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space E n. The point E 0 is 0-dimensional. The line E 1 is 1-dimensional. The plane E 2 is 2-dimensional. And in general E n is n-dimensional.

In the rest of this article we examine some of the more important mathematical definitions of dimension.

You might also want to look at the following old thread in the math forum

https://www.physicsforums.com/showthread.php?t=41627&highlight=dimension

I rather like the Lebesque covering dimension, because it defines the concept in very primitive spaces, called topological spaces, that need only the concept of a point and it's neighborhood. This is a much more basic concept (IMO) than a vector space.

The idea that a n-dimensional object is the boundary of a n+1 dimensional object is also an appealing way to define dimension, and is discussed somewhat in the above thread. Thus a line bounds a point, if you take a point out of a line it divides it into two parts. A line bounds a plane, etc, etc.