Regular polytopes and their symmetry have a long history stretching back two and a half millennia, to the classical regular polygons and polyhedra. Much of modern research focuses on abstract regular polytopes, but significant recent developments have been made on the geometric side, including the exploration of new topics such as realizations and rigidity, which offer a different way of understanding the geometric and combinatorial symmetry of polytopes. This is the first comprehensive account of the modern geometric theory, and includes a wide range of applications, along with new techniques. While the author explores the subject in depth, his elementary approach to traditional areas such as finite reflexion groups makes this book suitable for beginning graduate students as well as more experienced researchers.
- Provides the first comprehensive coverage of the modern geometric theory
- Uses an elementary approach to topics and collects basic theory in one place, making it suitable for graduate students
- Introduces new techniques for the use of researchers
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Reviews & endorsements
'McMullen (emer., University College London) begins the text with a rapid, cogent review of relevant topics in linear algebra and group theory and proceeds to a thorough discussion of the properties and structures of geometric and abstract regular polytopes. The majority of the text details the classification and properties of geometric regular polytopes … This is a careful, comprehensive survey of the topic and will likely become a classic reference.' C. A. Gorini, Choice
'This book, along with his previous book … is well-written and indispensable for every researcher or every student who wants to pursue research in this area.' Uma Kant Sahoo, Encyclopedia of Mathematics and its Applications
'The book is without a doubt a modern bible on the current state of polytopes. It is no exaggeration to say that 'all of it' can be found in this book …' Peter McMullen, Nieuw Archief voor Wiskunde
'Whether you are an experienced researcher in the subject, a mathematician seeking to expand their research interests to include this subject area, or a student beginning their polytopal journey, this book is a great source of knowledge on the geometric perspective of regular polytopes.' Andrés R. Vindas-Meléndez, MathSciNet
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Product details
- Date Published: February 2020
- format: Hardback
- isbn: 9781108489584
- length: 619 pages
- dimensions: 240 x 160 x 32 mm
- weight: 1.1kg
- contains: 43 b/w illus. 19 colour illus. 3 tables
- availability: In stock
Table of Contents
Foreword
Part I. Regular Polytopes:
1. Euclidean space
2. Abstract regular polytopes
3. Realizations of symmetric sets
4. Realizations of polytopes
5. Operations and constructions
6. Rigidity
Part II. Polytopes of Full Rank:
7. Classical regular polytopes
8. Non-classical polytopes
Part III. Polytopes of Nearly Full Rank:
9. General families
10. Three-dimensional apeirohedra
11. Four-dimensional polyhedra
12. Four-dimensional apeirotopes
13. Higher-dimensional cases
Part IV. Miscellaneous Polytopes:
14. Gosset–Elte polytopes
15. Locally toroidal polytopes
16. A family of 4-polytopes
17. Two families of 5-polytopes
Afterword
References
Symbol index
Author index
Subject index.
Look Inside
Author
Peter McMullen, University College London
Peter McMullen is Professor Emeritus of Mathematics at University College London. He was elected a foreign member of the Austrian Academy of Sciences in 2006 and is also a member of the London Mathematical Society and the European Mathematical Society. He was elected a Fellow of the American Mathematical Society in 2012. He has co-edited several books and co-authored Abstract Regular Polytopes (Cambridge, 2002). His work has been discussed in the Encyclopaedia Britannica and he was an invited speaker at the International Congress of Mathematicians in 1974.