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Full Description
This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's 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.
Contents
1. The origin of error-correcting codes; an introduction to coding; the work of Hamming; the Hamming-Holbrook patent; the Hamming codes are linear; the work of Golay; the priority controversy; 2. From coding to sphere packing; an introduction to sphere packing; the Leech connection; the origin of Leech's first packing in E24; the matrix for Leech's first packing; the Leech lattice; 3. From sphere packing to new simple groups; is there an interesting group in Leech's lattice?; the hard sell of a simple group; twelve hours on Saturday, six on Wednesday; the structure of 0; new simple groups; Appendix 1. Densest known sphere packings; Appendix 2. Further properties of the (12,24) Golay code and the related Steiner system s (5,8,24); Appendix 3. A calculation of the number of spheres with centers in 2; Appendix 4. The Mathieu group M24 and the order of M22; Appendix 5. The proof of lemma 3.3; Appendix 6. The sporadic simple groups; Bibliography; Index.
