Polyhedra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1997
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIII, 451 S. Ill., zahlr. graph. Darst. |
| ISBN: | 0521554322 9780521664059 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804125908179615744 |
|---|---|
| adam_text | Contents
Preface xi
Acknowledgements xiii
Introduction 1
Polyhedra in architecture 1
Polyhedra in art 2
Polyhedra in ornament 3
Polyhedra in nature 4
Polyhedra in cartography 7
Polyhedra in philosophy and literature 9
About this book 9
The inclusion of proofs 10
Approaches to the book 11
Basic concepts 12
Making models 14
1. Indivisible, Inexpressible and Unavoidable 17
Castles of eternity 17
Egyptian geometry 19
Babylonian geometry 23
Chinese geometry 24
A common origin for oriental mathematics 28
Greek mathematics and the discovery of incommensurability 29
The nature of space 33
Democritus’ dilemma 35
Liu Hui on the volume of a pyramid 38
Eudoxus’ method of exhaustion 41
Hilbert’s third problem 44
2. Rules and Regularity 51
The Platonic solids 51
The mathematical paradigm 58
Abstraction 58
Primitive objects and unproved theorems 59
VI
Contents
The problem of existence 61
Constructing the Platonic solids 66
The discovery of the regular polyhedra 70
What is regularity? 74
Bending the rules 79
The Archimedean solids 79
Polyhedra with regular faces 86
3. Decline and Rebirth of Polyhedral Geometry 95
The Alexandrians 95
Mathematics and astronomy 97
Heron of Alexandria 98
Pappus of Alexandria 99
Plato’s heritage 100
The decline of geometry 101
The rise of Islam 102
Thabit ibn Qurra 103
Abu’l-Wafa IO4
Europe rediscovers the classics 104
Optics 105
Campanus’ sphere 106
Collecting and spreading the classics 107
The restoration of the Elements 108
A new way of seeing 109
Perspective HI
Early perspective artists H2
Leon Battista Alberti H4
Paolo Uccello H5
Polyhedra in woodcrafts ^g
Piero della Francesca ng
Luca Pacioli 122
Albrecht Dürer 12g
Wenzel Jamnitzer 12g
Perspective and astronomy ^22
Polyhedra revived 13g
4. Fantasy, Harmony and Uniformity 139
A biographical sketch 139
A mystery unravelled 142
The structure of the universe 148
vii
Fitting things together 149
Rhombic polyhedra 151
The Archimedean solids 156
Star polygons and star polyhedra 168
Semisolid polyhedra 173
Uniform polyhedra 175
5. Surfaces, Solids and Spheres 181
Plane angles, solid angles, and their measurement 183
Descartes’ theorem 187
The announcement of Euler’s formula 189
The naming of parts 191
Consequences of Euler’s formula 193
Euler’s proof 197
Legendre’s proof 198
Cauchy’s proof 200
Exceptions which prove the rule 202
What is a polyhedron? 205
Von Staudt’s proof 210
Complementary viewpoints 213
The Gauss-Bonnet theorem 215
6. Equality, Rigidity and Flexibility 219
Disputed foundations 220
Stereo-isomerism and congruence 225
Cauchy’s rigidity theorem 228
Cauchy’s early career 233
Steinitz’ lemma 235
Rotating rings and flexible frameworks 237
Are all polyhedra rigid? 240
The Connelly sphere 243
Further developments 244
When are polyhedra equal? 247
7. Stars, Steilations and Skeletons 249
Generalised polygons 249
Poinsot’s star polyhedra 251
Poinsot’s conjecture 256
Cayley’s formula 257
Cauchy’s enumeration of star polyhedra 259
vm
Face-stellation
Stellations of the icosahedron
Bertrand’s enumeration of star polyhedra
Regular skeletons
8. Symmetry, Shape and Structure
What do we mean by symmetry?
Rotation symmetry
Systems of rotational symmetry
How many systems of rotational symmetry are there?
Reflection symmetry
Prismatic symmetry types
Compound symmetry and the S2n symmetry type
Cubic symmetry types
Icosahedral symmetry types
Determining the correct symmetry type
Groups of symmetries
Crystallography and the development of symmetry
9. Counting, Colouring and Computing
Colouring the Platonic solids
How many colourings are there?
A counting theorem
Applications of the counting theorem
Proper colourings
How many colours are necessary?
The four-colour problem
What is proof?
10. Combination, Transformation and Decoration
Making symmetrical compounds
Symmetry breaking and symmetry completion
Are there any regular compounds?
Regularity and symmetry
Transitivity
Polyhedral metamorphosis
The space of vertex-transitive convex polyhedra
Totally transitive polyhedra
Symmetrical colourings
Colour symmetries
Contents
263
267
281
282
289
289
291
292
297
300
301
305
308
311
312
314
318
327
328
330
331
334
337
347
348
354
359
359
361
365
366
367
373
376
385
394
397
IX
Perfect colourings 400
The solution of fifth degree equations 402
Appendix 1 406
Appendix II 408
Sources of Quotations 411
Bibliography 416
Name Index 439
Subject Index 443
|
| any_adam_object | 1 |
| author | Cromwell, Peter R. 1964- |
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| ctrlnum | (OCoLC)34912586 (DE-599)BVBBV011395163 |
| dewey-full | 516.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.15 |
| dewey-search | 516.15 |
| dewey-sort | 3516.15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| indexdate | 2024-07-09T18:09:02Z |
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| isbn | 0521554322 9780521664059 |
| language | English |
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| physical | XIII, 451 S. Ill., zahlr. graph. Darst. |
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| spelling | Cromwell, Peter R. 1964- Verfasser (DE-588)136947093 aut Polyhedra Peter R. Cromwell 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1997 XIII, 451 S. Ill., zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Geometria larpcal Polyhedra Polyeder Motiv (DE-588)4138353-9 gnd rswk-swf Kunst (DE-588)4114333-4 gnd rswk-swf Polyeder (DE-588)4132101-7 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Polyeder (DE-588)4132101-7 s Geschichte (DE-588)4020517-4 s DE-604 Polyeder Motiv (DE-588)4138353-9 s Kunst (DE-588)4114333-4 s Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007660064&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cromwell, Peter R. 1964- Polyhedra Geometria larpcal Polyhedra Polyeder Motiv (DE-588)4138353-9 gnd Kunst (DE-588)4114333-4 gnd Polyeder (DE-588)4132101-7 gnd Geschichte (DE-588)4020517-4 gnd |
| subject_GND | (DE-588)4138353-9 (DE-588)4114333-4 (DE-588)4132101-7 (DE-588)4020517-4 |
| title | Polyhedra |
| title_auth | Polyhedra |
| title_exact_search | Polyhedra |
| title_full | Polyhedra Peter R. Cromwell |
| title_fullStr | Polyhedra Peter R. Cromwell |
| title_full_unstemmed | Polyhedra Peter R. Cromwell |
| title_short | Polyhedra |
| title_sort | polyhedra |
| topic | Geometria larpcal Polyhedra Polyeder Motiv (DE-588)4138353-9 gnd Kunst (DE-588)4114333-4 gnd Polyeder (DE-588)4132101-7 gnd Geschichte (DE-588)4020517-4 gnd |
| topic_facet | Geometria Polyhedra Polyeder Motiv Kunst Polyeder Geschichte |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007660064&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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