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.

'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

• 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.