From Error-Correcting Codes Through Sphere Packings to Simple Groups (Carus Mathematical Monographs)

This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940s electro-mechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels led to discoveries of extremely efficient lattice packings of equal-radius balls especially in 24-dimensional space. In turn this highly symmetric lattice with each point neighbouring exactly 196 560 other points suggested the possible presence of new simple groups as groups of symmetries. Indeed new groups were found and are now part of the Enormous Theorem - the classification of all simple groups whose entire proof runs to some 10 000 pages. And these connections along with the fascinating history and the proof of the simplicity of one of those sporadic simple groups are presented at an undergraduate mathematical level.