Abstract chiral polytopes / Daniel Pellicer.

Format:

Book

Author/Creator:

Pellicer, Daniel, author.

Series:

New mathematical monographs ; 49.

New mathematical monographs ; 49

Language:

English

Subjects (All):

Polytopes.

Physical Description:

1 online resource (viii, 484 pages) : digital, PDF file(s).

Edition:

First edition.

Place of Publication:

Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2025.

Summary:

Abstract polytopes are partially ordered sets that satisfy some key aspects of the face lattices of convex polytopes. They are chiral if they have maximal symmetry by combinatorial rotations, but none by combinatorial reflections. Aimed at graduate students and researchers in combinatorics, group theory or Euclidean geometry, this text gives a self-contained introduction to abstract polytopes and specialises in chiral abstract polytopes. The first three chapters are introductory and mostly contain basic concepts and results. The fourth chapter talks about ways to obtain chiral abstract polytopes from other abstract polytopes, while the fifth discusses families of chiral polytopes grouped by common properties such as their rank, their small size or their geometric origin. Finally, the last chapter relates chiral polytopes with geometric objects in Euclidean spaces. This material is complemented by a number of examples, exercises and figures, and a list of 75 open problems to inspire further research.

Contents:

Cover

Half-title

Series information

Title page

Imprints page

Contents

Preface

1 Introduction

2 Abstract Regular and Chiral Polytopes

2.1 Abstract Polytopes

2.2 Regular and Two-Orbit Polytopes

2.3 Chiral Polytopes

2.4 Toroidal Rotary Polytopes

3 Groups Related to Chiral Polytopes

3.1 Automorphism Group

3.2 Connection Group

3.3 Chirality Group

4 Polytopes Constructed from Other Polytopes

4.1 Operations

4.2 Embeddings

4.3 Quotients and Covers

4.4 Mixing

5 Families of Chiral Polytopes

5.1 Rank 3

5.2 Tight Chiral Polytopes

5.3 Chiral Polytopes from Geometric Constructions

5.4 Chiral Amalgamations

5.5 Chiral Polytopes of High Ranks

6 Skeletal Polytopes

6.1 Regular and Chiral Skeletal Polytopes

6.2 Chiral Polyhedra in E[sup(3)]

6.3 Skeletal Chiral Polytopes

Appendix A A Few Treats on Euclidean Geometry

A.1 Euclidean Isometries

A.2 Tessellations by Triangles

A.3 Tessellations of E[sup(3)]

A.4 An Interesting Arrangement of Lines in E[sup(3)]

Appendix B A Few Words about Numbers

Appendix C Open Problems

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 24 Mar 2025).

Includes bibliographical references and index.

ISBN:

9781108759021

1108759025

9781108695046

1108695043

OCLC:

1478244797