# Meaning of dimension

I was wondering what dimension can mean in physics?

I know it can mean the dimension of the space and time.

But there is dimensional analysis. How is this dimension related to and different from the previous one? How is it related to and different from units (e.g. kilometer)?

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A dimension (in dimensional analysis) is defined by the transformation law of an object under changes in scale. If I have an object which is twice as big, it has 4 times the surface area and 8 times the volume, so the surface area has dimension of length-squared, and the volume has dimension of length-cubed. Dimensional analysis is applied whenever you have a quantity where there is a scale that you can change.

There are two different notions of dimension of space which took a while to be disentangled. The topological dimension is defined inductively by the cutting properties of the space. If a space can be cut in two by a point, it's 1 dimensional. If it can be cut in two by a 1 dimensional shape, it's two dimensional. This type of definition requires care for wild shapes, but it produces an integer dimension of the space.

The scaling dimension, or fractal dimension, is defined differently, in terms of distances on the space. The scaling dimension counts the number of boxes of size A required to cover the space, and sees how this goes up as A gets small. The exponent is the scaling dimension.

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Your definition certainly makes sense, but I am curious how you would apply the transformation law to something like the dimension of an array in a computer language. Similarly, I've heard of equations (particularly in engineering) described as being the dimension of the number of independant variables. How does the transformation law apply when the variables are things like neutron leakage from a reactor core? Is this a dimension for which there is no (imaginable) corresponding second dimension, or is it simply a misnomer? –  AdamRedwine Sep 13 '11 at 16:15
It's not a misnomer--- its just that this is a simpler conception of dimension in terms of "number of independent directions in a particular description", which does not require careful thinking. It is called "dimension" because if you ask what is the "volume" of the CS array, the total memory consumption it goes as the cube of the length of the array. The term dimension is used broadly. –  Ron Maimon Sep 13 '11 at 17:43

Can't help adding a reference to the notion of fractal dimension

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A dimension, as used in physics and engineering, is often used to describe a unique, well-defined attribute to which a quantitative value or function can (at least theoretically) be assigned. Specifically, as described below by the BIPM, this attribute must be able to be described/used within a system of non-contradictory equations relating the relevant system of quantities.

I apologize in advance for the wishy-washiness of this definition, but it is the best that I can come up with. I highly recommend reading section 1.3 of The SI Brochure. It might also help to read an IT perspective like this section on n-dimensional arrays in Numpy.

Dimensions in common parlance refer only to spatial dimensions (and sometimes time) but practically any attribute can be called a dimension depending on the circumstance. Forthermore, the properties of dimensions that are familiar in the spatial case can often be extended to other situations.

For example, if two dimensions are orthogonal, it means that one cannot be described, even in part, by reference to the other. A vector on the xy-plane, for example, can be described in terms of $\hat x$ and $\hat y$; but $\hat x$ and $\hat y$ cannot be described in terms of eachother. This concept is often extended in fields such as quantum mechanics where wave functions can be treated like dimensions and tested for orthogonality.

*edit

I should have included it origianally, but a good source for definitions used in science and engineering is the International Vocabulary of Metrology (VIM). According to this source:

1.7 (1.5)

quantity dimension

dimension of a quantity dimension

expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor

where a "system of quantities" is defined as

1.3 (1.2)

system of quantities

set of quantities together with a set of noncontradictory equations relating those quantities

Clearly the definition of "dimension" (as I understand your question to mean it) is quite broad and might more properly be referred to as "base quantity."

1.4 (1.3)

base quantity

quantity in a conventionally chosen subset of a given system of quantities, where no subset quantity can be expressed in terms of the others

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Thanks! So is dimension same as physics property? –  Tim Sep 13 '11 at 15:22
-1: This is not wishy washy, it is just wrong. –  Ron Maimon Sep 13 '11 at 15:34
@Ron: In that case, perhaps you'd care to provide a correct answer? –  AdamRedwine Sep 13 '11 at 15:42
I suppose I should clarify my reason for downvoting here: "any attribute to which a value or function can (at least theoretically) be assigned" includes many, many things that would never be called "dimensions" in my experience. –  David Z Sep 14 '11 at 12:21
That makes sense, though I have seen the word applied to a great many things that I would have never called dimensions either. In actual usage, in my experience, the word gets used very frequently in a wide variety of circumstances. The question was "what can" it mean, not "what should" it mean. –  AdamRedwine Sep 14 '11 at 13:00
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