No account of the symmetric group can be complete without mentioning the remarkable fact that the symmetric group of degree n (finite or infinite) has an outer automorphism if and only if n=6. Here are the definitions. An automorphism of a group G is a permutation p of the group which preserves products, that is,…
This manual has been available on this site since about 1996, with improvements taking place frequently. The current version has been published as a book of about 350 pages by Cambridge University Press. By agreement with the Press, however, it will remain posted on this web site. Many improvements in the current version over previous ones are due to the (anonymous) referees of the Press, whom I wish to thank heartily. I also wish to thank Lauren Cowles, of the New York office of the Press, for much help with preparing the original version for publication. The paper edition appears also in Duotone red and black. For information on obtaining the paper edition, take a look at the Cambridge Press catalogue.
To make a density map from a 2D set of data, the first step is to compute values for the third dimension. (Gnuplot has no facilities for computing these values automatically.) The simplest way is to make a 2D histogram; the plot is divided in small 2D regions, and the z-values are proportional to the number of points inside these regions. The following Python script will make an histogram from a time series of two dihedral angles.
The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful detail. It's known as a 'fractal' - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you 'zoom' into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).
It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal.