Softcover ISBN:  9781470454609 
Product Code:  CAR/21 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
eBook ISBN:  9781614440215 
Product Code:  CAR/21.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
Softcover ISBN:  9781470454609 
eBook: ISBN:  9781614440215 
Product Code:  CAR/21.B 
List Price:  $105.00 $80.00 
MAA Member Price:  $78.75 $60.00 
AMS Member Price:  $78.75 $60.00 
Softcover ISBN:  9781470454609 
Product Code:  CAR/21 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
eBook ISBN:  9781614440215 
Product Code:  CAR/21.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
Softcover ISBN:  9781470454609 
eBook ISBN:  9781614440215 
Product Code:  CAR/21.B 
List Price:  $105.00 $80.00 
MAA Member Price:  $78.75 $60.00 
AMS Member Price:  $78.75 $60.00 

Book DetailsThe Carus Mathematical MonographsVolume: 21; 1983; 228 ppMSC: Primary 94; Secondary 01; 11; 20; 52Recipient of the Mathematical Association of America's Beckenbach Book Prize in 1989!
This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's electromechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels lead to discoveries of extremely efficient lattice packings of equalradius balls, especially in 24dimensional space. In turn, this highly symmetric lattice, with each point neighboring 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 some 10,000+ pages—and these connections, along with the fascinating history and the proof of the simplicity of one of those "sporatic" simple groups, are presented at an undergraduate mathematical level.

Table of Contents

Chapters

Chapter 1. The origin of errorcorrecting codes

Chapter 2. From coding to sphere packing

Chapter 3. From sphere packing to new simple groups

Appendix 1. Densest known sphere packings

Appendix 2. Further properties of the (24, 12) Golay code and the related Steiner system $S(5,8,24)$

Appendix 3. A calculation of the number of spheres with centers in $\Lambda _2$ adjacent to one, two, three and four adjacent spheres with centers in $\Lambda _2$

Appendix 4. The Mathieu group $M_{24}$ and the order of $M_{22}$

Appendix 5. The proof of lemma 3.3

Appendix 6. The sporadic simple groups


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This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's electromechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels lead to discoveries of extremely efficient lattice packings of equalradius balls, especially in 24dimensional space. In turn, this highly symmetric lattice, with each point neighboring 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 some 10,000+ pages—and these connections, along with the fascinating history and the proof of the simplicity of one of those "sporatic" simple groups, are presented at an undergraduate mathematical level.

Chapters

Chapter 1. The origin of errorcorrecting codes

Chapter 2. From coding to sphere packing

Chapter 3. From sphere packing to new simple groups

Appendix 1. Densest known sphere packings

Appendix 2. Further properties of the (24, 12) Golay code and the related Steiner system $S(5,8,24)$

Appendix 3. A calculation of the number of spheres with centers in $\Lambda _2$ adjacent to one, two, three and four adjacent spheres with centers in $\Lambda _2$

Appendix 4. The Mathieu group $M_{24}$ and the order of $M_{22}$

Appendix 5. The proof of lemma 3.3

Appendix 6. The sporadic simple groups