Exceptional Objects

One of my favourite topics in mathematics is the existence of exceptional objects. There have been many articles written on the web about this subject - particularly by John Baez. But I recently came across this article by John Stillwell, in American Mathematical Monthly, that also gives a historical perspective. The connection with classical geometry - Desargue's theorem and Pappus's hexagonal theorem - was also new to me.

Very briefly: mathematics is full of classification theorems. What frequently happens is that objects of a certain type are classified into a bunch of series with a handful of exceptional objects left over. For example the simple finite groups have all been classified and besides the series (such as the cyclic groups of prime order) there are 26 'sporadic' simple groups such as the Monster. The classification of compact Lie groups gives a similar pattern and even the classification of regular polyhedra is similar. Bizarrely, exceptional objects from different branches of mathematics are often related to each other. In fact, the entire classifications from different branches of mathematics are often closely related. And for some reason Dynkin diagrams and octonions seem to play a special role in many of these classifications. The article suggests that the octonions are the "mother of all exceptional objetcs". Anyway, read the article for more information.

I also found this article by the same author on the history of the 120-cell, one of those exceptional regular polyhedra.