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Re: CR: What Are Dimensions ?
Posted:
Jul 21, 1995 11:15 AM


>One of the questions that was not satisfactorally answered during my >undergraduate career was "What is a dimension?" I'm not sure that I will be >able to dodge the question during the upcoming school year so I was wondering >if anyone on these lists had a good definition. > >Thankyou > marge > MCOTTON@AOL.COM
The most intuitive idea is that if you need D numbers to describe something, then it has dimension D. The problem with this idea is that Cantor gave a oneone (discontinuous) mapping of a line segment to a square, and Peano gave a (manyone) continuous mapping also from a line segment to a square. So can a line segment and a square really be of different dimensions when they can be mapped from one to the other like this? In fact there can be no mapping from one to the other that is both oneone and continuous [Brouwer 1911], so the concept of dimension is not totally meaningless, and the effort to make it precise was one of the driving forces behind the development of topology in the early part of this century.
Euclid's idea was that points are of dimension zero, and lines are of dimension one because they can be cut by points of dimension zero. Planes are of dimension two because they can be cut by lines of dimension one, and so on. But this doesn't really tell you what a dimension is, only how the dimensions are built up from each other.
Another point of view focuses on the fact that as you double the scale, a point stays they same size (2^0 = 1), a line doubles in length (2^1 = 2), a square has four times the area (2^2 = 4), and in general an object of dimension D has its "measure" multiplied by 2^D. (In general m^D if you multiply the scale by m.) Objects were discovered for which this was true, but D was not an integer, so they are of fractional dimension.
There are other concepts too.
The simple answer is that there is no simple answer. The different ideas are related in that they all give the same answer for nice simple Euclidean objects, but they are distinct in that they give different answers for objects of the kind that used to be called pathological, but are now called fractal and are at the forefront of current research, and have been for a hundred years. It is not so often that you can get undergraduates so close to contemporary mathematics.
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