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Geometric Regular Polytopes [Kietas viršelis]

5.00/5 (2 ratings by Goodreads)

Peter McMullen (University College London)

Formatas: Hardback, 619 pages, aukštis x plotis x storis: 240x160x32 mm, weight: 1100 g, Worked examples or Exercises; 3 Tables, black and white; 19 Line drawings, color; 43 Line drawings, black and white
Serija: Encyclopedia of Mathematics and its Applications
Išleidimo metai: 20-Feb-2020
Leidėjas: Cambridge University Press
ISBN-10: 1108489583
ISBN-13: 9781108489584

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Formatas: Hardback, 619 pages, aukštis x plotis x storis: 240x160x32 mm, weight: 1100 g, Worked examples or Exercises; 3 Tables, black and white; 19 Line drawings, color; 43 Line drawings, black and white
Serija: Encyclopedia of Mathematics and its Applications
Išleidimo metai: 20-Feb-2020
Leidėjas: Cambridge University Press
ISBN-10: 1108489583
ISBN-13: 9781108489584

Kitos knygos pagal šią temą:

Pastovi nuoroda: https://www.kriso.lt/db/9781108489584.html

Raktažodžiai:

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.

Regular polytopes and their symmetry have a long history. This book, the first to cover modern theory, explores the subject in depth, introducing new techniques and elementary approaches to familiar ideas. It caters for experienced researchers and also for graduate students in discrete and Euclidean geometry, combinatorics and group theory.

Recenzijos

'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

Daugiau informacijos

The first comprehensive account of modern geometric theory and its applications, written by a leader in the field.

Foreword

I Regular Polytopes

(206)

1 Euclidean Space

(60)

1A Algebraic Properties

(6)

1B Convexity

(6)

1C Euclidean Structure

(5)

1D Isometries

(8)

1E Reflexion Groups

(11)

1F Subgroup Relationships

(2)

1G Angle-Sum Relations

(1)

1H Group Orders

(12)

1J Ordinary Space

(3)

1K Quaternions

(5)

2 Abstract Regular Polytopes

(31)

2A Abstract Polytopes

(4)

2B Regularity

(5)

2C Regularity Criteria

(5)

2D Presentations

(7)

2E Regular Maps

(2)

2F Special Polytopes

(7)

3 Realizations of Symmetric Sets

(45)

3A Transitive Actions

(4)

3B Realization Cone

(3)

3C Cosine Vectors

101

(5)

3D Examples

106

(5)

3E Products of Realizations

111

(3)

3F Λ-Orthogonality

114

(4)

3G The Wythoff Space

118

(5)

3H Λ-Orthogonal Bases

123

(8)

3J Cosine Matrices

131

(3)

3K Cuts and Duality

134

(1)

3L Realizations over Subfields

135

(2)

3M Realizations and Representations

137

(2)

4 Realizations of Polytopes

139

(30)

4A Wythoff's Construction

139

(6)

4B Faithful Realizations

145

(3)

4C Degenerate Realizations

148

(1)

4D Induced Cosine Vectors

149

(4)

4E Alternating Products

153

(1)

4F Apeirotopes

154

(5)

4G Examples of Realizations

159

(10)

5 Operations and Constructions

169

(31)

5A Operations on Polyhedra

170

(9)

5B General Mixing

179

(6)

5C Twisting

185

(5)

5D Modifying Mirrors

190

(5)

5E Extensions

195

(2)

5F Vertex-Figures

197

(3)

6 Rigidity

200

(7)

6A Basic concept

200

(1)

6B Fine Schlafli symbols

201

(2)

6C Shapes

203

(1)

6D Rigidity Criteria

204

(3)

II Polytopes of Full Rank

207

(122)

7 Classical Regular Polytopes

209

(92)

7A Faces of Full Rank

209

(8)

7B Polytopes in All Dimensions

217

(9)

7C The 24-Cell

226

(8)

7D Pentagonal Polyhedra

234

(6)

7E The 600-Cell

240

(8)

7F The 120-Cell

248

(3)

7G Star-Polytopes

251

(11)

7H Honeycombs

262

(4)

7J Regular Compounds

266

(11)

7K Realizations of {5, 3, 3}

277

(24)

8 Non-Classical Polytopes

301

(28)

8A Polytopes in All Dimensions

301

(8)

8B Apeirotopes in All Dimensions

309

(8)

8C Apeirohedra and Polyhedra

317

(3)

8D Higher-Dimensional Exceptions

320

(9)

III Polytopes of Nearly Full Rank

329

(146)

9 General Families

331

(30)

9A Blends

331

(8)

9B Twisting Small Diagrams

339

(2)

9C Families of Polytopes

341

(9)

9D Families of Apeirotopes

350

(11)

10 Three-Dimensional Apeirohedra

361

(22)

10A The Classification

361

(4)

10B Groups of the Apeirohedra

365

(8)

10C Rigidity of the Apeirohedra

373

(10)

11 Four-Dimensional Polyhedra

383

(48)

11A Mirror Vectors

383

(3)

11B Mirror Vector (3, 2, 3) and Its Relatives

386

(9)

11C A Family of Petrials

395

(16)

11D Mirror Vector (2, 3, 2)

411

(3)

11E Mirror Vector (2, 2, 2)

414

(12)

11F Further Connexions

426

(5)

12 Four-Dimensional Apeirotopes

431

(19)

12A Imprimitive Groups

431

(6)

12B Group U5 and Relatives

437

(5)

12C Twisting P5

442

(8)

13 Higher-Dimensional Cases

450

(25)

13A The Gateway

451

(1)

13B Rotational Symmetry Groups

452

(8)

13C The Gosset-Elte Polytopes

460

(1)

13D The First Gosset Class

461

(3)

13E The Second Gosset Class

464

(6)

13F The Third Gosset Class

470

(2)

13G A Degenerate Gosset Class

472

(3)

IV Miscellaneous Polytopes

475

(107)

14 Gosset-Elte Polytopes

477

(29)

14A Rank 6: {32, 32,1}

477

(4)

14B Rank 6: {3, 32,2}

481

(2)

14C Rank 6: {3, 32,2}*

483

(8)

14D Rank 7: {33, 32,1}

491

(2)

14E Rank 7: {32, 33,1}

493

(2)

14F Rank 8: {34, 32,1}

495

(3)

14G Rank 8: {32, 34,1}

498

(8)

15 Locally Toroidal Polytopes

506

(24)

15A {{4. 4 : 4}, {4, 3}} and its Dual

506

(2)

15B {{3, 4}, {4, 4 | 3}} and {{4, 4 | 3}, {4, 4 | 3}}

508

(2)

15C {{4, 4 : 4}, {4, 4 | 3}} and its Dual

510

(4)

15D {{4, 4 : 4}, {4, 4 : 6}} and its Dual

514

(3)

15E {{4, 4 | 4}, {4, 4 | 3}} and its Dual

517

(3)

15F Polytopes of Type {3m-2, 6}

520

(6)

15G Polytopes of Type {3m-1, 6, 3n-1}

526

(4)

16 A Family of 4-Polytopes

530

(16)

16A The Polyhedron {5, 5 : 4}

530

(7)

16B A Permutation Representation

537

(1)

16C The Polytope {{5, 5 : 4}, {5, 3}}

538

(2)

16D Layers and Strata of {{5, 5 : 4}, {5, 3}}

540

(1)

16E The Dual Polytope {{3, 5}, {5, 5 : 4}}

541

(1)

16F The Extended Family

541

(3)

16G {3, 5, 3 :: 4} and {5,
3. 5 :: 4}

544

(2)

17 Two Families of 5-Polytopes

546

(36)

17A An Intuitive Approach

547

(2)

17B Group and Geometry

549

(10)

17C A Quotient of {3, 5, 3, 5}

559

(3)

17D The Dual Polytope

562

(4)

17E Double Covers

566

(5)

17F An Extended Family

571

(5)

17G Another Symmetric Set

576

(4)

17H Other Combinations

580

(2)

Afterword

582

(1)

Bibliography

583

(8)

Notation Index

591

(3)

Author Index

594

(2)

Subject Index

596

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.

Geometric Regular Polytopes [Kietas viršelis]

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