The Mathematics of Complicated Polymer Structures Fractals

1 A Bit More About Maths in Physics How Does a Physicist Determine the Dimensionality of a Space  

A good starting point for another very interesting yet unfinished story is Brownian motion. As you remember, the displacement of a Brownian particle (or the end-to-end distance of a polymer chain) is proportional to the square root of the time traveled (or the contour length of the chain). Surprising as it may seem in a book on polymers, the story is about the dimensionality of a space. Mathematicians have already been studying this topic for nearly one hundred years, and know quite a bit about it. However, it seemed of no particular relevance for physics until very recently, after two books by B. Mandelbrot appeared in 1977 and 1982 [47]. We shall avoid too much maths here, and basically talk about the physics side. 

The space we live in is, of course, three-dimensional. We know this because three coordinates, e. g. x, y, and z, are needed to describe any position. You might have also heard that time is often regarded as the fourth coordinate. Thus, space-time is four-dimensional. A two-dimensional space is merely a plane, a one-dimensional space is a straight line. However, it turns out that there are also objects with fractional dimensionality  

The quantity df defined by this formula is known as the dimensionality of the object. More precisely, it is the so-called fractal, scaling, or Hausdorff dimensionality. (In maths, you may hear of lots of others, e.g. metric, topological, etc., but we shall not talk about them.) 

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