A geometrical picture book:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
New York [u.a.]
Springer
1998
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 279 - 286 |
| Physical Description: | XX, 291 S. graph. Darst. |
| ISBN: | 0387984372 |
Staff View
MARC
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| 245 | 1 | 0 | |a A geometrical picture book |c Burkard Polster |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1998 | |
| 300 | |a XX, 291 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Universitext | |
| 500 | |a Literaturverz. S. 279 - 286 | ||
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Record in the Search Index
| _version_ | 1804126703788752896 |
|---|---|
| adam_text | Contents
I Finite Geometries 1
1 Introduction via the Fano Plane 3
1.1 Geometries—Basic Facts and Conventions 3
1.2 Projective Planes 5
1.3 Affine Planes 7
1.4 Automorphisms 9
1.5 Polarities 11
1.6 Ovals and Hyperovals 12
1.7 Blocking Sets 13
1.8 Difference Sets and Singer Diagrams 13
1.9 Incidence Graphs 15
1.10 Spatial Models 15
2 Designs 19
2.1 The Smallest Nontrivial 2 Design 20
2.2 Hadamard Designs 21
2.2.1 The One Point Extension of the Fano Plane 21
2.3 Steiner Triple Systems 23
2.3.1 Kirkman s Schoolgirl Problem 25
3 Configurations 27
3.1 Configurations with Three Points on a Line 28
xvi Contents
3.1.1 The Fano, Pappus, and Desargues Configurations . . 29
3.1.2 The Configurations with Parameters (73) and (83) . 30
3.1.3 The Configurations with Parameters (93) 31
3.1.4 The Configurations with Parameters (103) 32
3.2 Configurations with Four Points on a Line 36
3.3 Tree Planting Puzzles 37
4 Generalized Quadrangles 39
4.1 The Generalized Quadrangle of Order (2,2) 41
4.1.1 A Plane Model—The Doily 41
4.1.2 A Model on the Tetrahedron 43
4.1.3 A Model on the Icosahedron 48
4.1.4 A Model in Four Space 50
4.2 The Petersen Graph 50
4.3 How to Construct the Models 53
4.4 The Generalized Quadrangle of Order (2,4) 58
4.5 The Generalized Quadrangle of Order (4,2) 60
4.6 Symmetric Designs and Generalized Quadrangles 65
4.7 Incidence Graph of a Generalized Quadrangle 65
5 The Smallest Three Dimensional Projective Space 67
5.1 A Plane Model 69
5.1.1 Line Pencils 70
5.1.2 Subplanes 71
5.1.3 Spreads and Packings 71
5.1.4 Reguli 75
5.1.5 Ovoids 76
5.1.6 A Labelling with Fano Planes 77
5.1.7 Fake Generalized Quadrangles 80
5.1.8 The Hoffmann Singleton Graph 81
5.2 Spatial Models 82
5.2.1 A Model on the Tetrahedron 82
5.2.2 Other Substructures 86
5.2.3 Spreads 88
5.2.4 A Model on the Icosahedron 90
5.3 Symmetric Designs Associated with Our Space 91
6 The Projective Plane of Order 3 93
6.1 More Models of the Affine Plane of Order 3 93
6.1.1 Two Triangular Models 94
6.1.2 Maximizing the Number of Straight Lines 96
6.2 Projective Extension 98
6.2.1 Blocking Sets 100
6.2.2 Desargues Configuration 101
Contents xvii
6.3 Singer Diagram and Incidence Graph 102
6.4 A Spatial Model on the Cube 103
6.5 Extending the Affine Plane to a 5 Design 104
7 The Projective Plane of Order 4 107
7.1 A Plane Model 108
7.2 Constructing the Plane Around a Unital 110
7.3 A partition into Three Fano Planes 112
7.4 A Spatial Model Around a Generalized Quadrangle 113
7.5 Another Partition into Fano Planes 115
7.6 Singer Diagram 116
8 The Projective Plane of Order 5 117
8.1 Beutelspacher s Model 117
8.2 A Spatial Model on the Dodecahedron 120
8.3 The Desargues Configuration Revisited 121
9 Stargazing in Affine Planes up to Order 8 125
9.1 Star Diagrams of the Affine Planes of Orders 2 and 3 .... 126
9.2 Star Diagram of the Affine Plane of Order 4 128
9.3 Star Diagram of the Affine Plane of Order 5 129
9.4 Star Diagram of the Affine Plane of Order 7 130
9.4.1 The Pascal Configuration in a Conic 130
9.5 Star Diagram of the Affine Plane of Order 8 131
9.5.1 The Fano Configuration 132
9.5.2 An Oval That Is Not a Conic 133
10 Biplanes 135
10.1 The Biplane of Order 2 136
10.2 The Biplane of Order 3 140
10.3 The Three Biplanes of Order 4 143
10.3.1 A First Biplane of Order 4 143
10.3.2 A Second Biplane of Order 4 149
10.3.3 A Third Biplane of Order 4 151
10.4 Two Biplanes of Order 7 152
10.4.1 A First Biplane of Order 7 152
10.4.2 A Second Biplane of Order 7 152
10.5 A Biplane of Order 9 153
10.6 Blocking Sets 153
11 Semibiplanes 1« 5
11.1 The Semibiplanes on Hypercubes 156
11.2 The Semibiplane of Order (12,5) on the Icosahedron .... 158
11.3 Folded Projective Planes 159
xviii Contents
12 The Smallest Benz Planes 161
12.1 The Smallest Inversive Planes 161
12.1.1 The Inversive Plane of Order 2 161
12.1.2 The Inversive Plane of Order 3 163
12.2 A Unifying Definition of Benz Planes 167
12.3 The Smallest Laguerre Planes 168
12.3.1 The Laguerre Plane of Order 2 168
12.3.2 The Laguerre Plane of Order 3 170
12.4 The Smallest Minkowski Planes 171
12.4.1 The Minkowski Plane of Order 2 171
12.4.2 The Minkowski Plane of Order 3 171
13 Generalized Polygons 173
13.1 The Generalized Hexagon of Order (1,2) 174
13.2 The Generalized Hexagon of Order (1,3) 175
13.3 The Two Generalized Hexagons of Order (2,2) 176
13.4 The Generalized Octagon of Order (1,2) 178
13.5 The Generalized 12 gon of Order (1,2) 179
13.6 Cages 180
14 Colour Pictures and Building the Models 185
15 Some Fun Games and Puzzles 197
15.1 The Game Set —Line Spotting in 4 D 197
15.2 Mill and Ticktacktoe on Geometries 199
15.3 Circular Walks on Geometries 200
15.4 Which Generalized Quadrangles Are Magical? 200
15.5 Question du Lapin 201
II Geometries on Surfaces 203
16 Introduction via Flat Affine Planes 205
16.1 Some More Basic Facts and Conventions 205
16.2 The Euclidean Plane—A Flat Affine Plane 206
16.2.1 Proper R2 Planes 206
16.2.2 Pencils, Parallel Classes, Generator Only Pictures . 206
16.2.3 The Group Dimension 207
16.2.4 Topological Geometries 208
16.2.5 The Way Lines Intersect 208
16.2.6 Ovals and Maximal Arcs 209
16.3 Nonclassical R2 Planes 210
16.3.1 Moulton Planes 210
16.3.2 Shift Planes 211
16.3.3 Arc Planes 213
Contents xix
16.3.4 Integrated Foliations 214
16.3.5 Gluing Constructions 214
16.4 Classification 217
16.5 Semibiplanes 218
17 Flat Circle Planes—An Overview 219
17.1 The Axiom of Joining and the Map 219
17.2 Nested Flat Circle Planes 221
17.3 Interpolation 222
18 Flat Projective Planes 223
18.1 Models of the Real Projective Plane 224
18.1.1 The Euclidean Plane Plus Its Line at Infinity .... 225
18.1.2 The Geometry of Great Circles 225
18.1.3 Mobius Strip—Antiperiodic 2 Unisolvent Set .... 226
18.1.4 A Disk Model 227
18.2 Recycled Nonclassical Projective Planes 228
18.3 Salzmann s Classification 229
19 Spherical Circle Planes 231
19.1 Intro via Ovoidal Spherical Circle Planes 231
19.2 The Miquel and Bundle Configurations 233
19.3 Nonclassical Flat Spherical Circle Planes 236
19.3.1 The Affine Part 236
19.3.2 Ewald s Affine Parts of Flat Mobius Planes 236
19.3.3 Steinke s Semiclassical Affine Parts 238
19.4 Classification 238
19.5 Subgeometries 238
19.5.1 Double Covers of Flat Projective Planes 239
19.5.2 Recycled Projective Planes 239
19.6 Lie Geometries Associated with Flat Mobius Planes .... 240
19.6.1 The Apollonius Problem 240
19.6.2 Semibiplanes 241
19.6.3 The Geometry of Oriented Lines and Circles .... 242
20 Cylindrical Circle Planes of Rank 3 245
20.1 Intro via Ovoidal Cylindrical Circle Planes 245
20.2 The Miquel and Bundle Configurations 247
20.3 Nonclassical Flat Cylindrical Circle Planes 249
20.3.1 Maurer s Construction 249
20.3.2 The Affine Part 250
20.3.3 The Artzy Groh Construction 250
20.3.4 The Lowen Pfuller Construction 251
20.3.5 Steinke s Two Types of Semiclassical Planes 252
20.3.6 Integrated Flat Affine Planes 252
xx Contents
20.4 Classification 252
20.5 Subgeometries 252
20.5.1 Recycled Projective Planes 253
20.5.2 Semibiplanes 255
20.6 Lie Geometries Associated with Flat Laguerre Planes . . . 255
20.6.1 Biaffine Planes 255
20.6.2 More Semibiplanes 257
20.6.3 Generalized Quadrangles 258
21 Toroidal Circle Planes 259
21.1 Intro via the Classical Minkowski Plane 259
21.2 The Miquel, Bundle, and Rectangle Configurations 261
21.3 Nonclassical Toroidal Circle Planes 262
21.3.1 The Affine Part 262
21.3.2 The Two Parts of a Toroidal Circle Plane 263
21.3.3 The Artzy Groh Construction 264
21.3.4 Semiclassical Planes 264
21.3.5 Proper Toroidal Circle Planes 265
21.4 Classification 265
21.5 Subgeometries 266
21.5.1 Flat Projective Planes Minus Convex Disks 266
A Models on Regular Solids 267
B Mirror Technique Stereograms 269
B.I The Generalized Quadrangle of Order (4,2) 269
B.2 Two PCs with Parameters 103 272
B.3 The Smallest Projective Space 273
References 279
Index 287
|
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| callnumber-label | QA447 |
| callnumber-raw | QA447 |
| callnumber-search | QA447 |
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| classification_tum | MAT 500f |
| ctrlnum | (OCoLC)611361469 (DE-599)BVBBV012097897 |
| dewey-full | 516/.12/0222 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.12/0222 |
| dewey-search | 516/.12/0222 |
| dewey-sort | 3516 212 3222 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012097897 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:21:40Z |
| institution | BVB |
| isbn | 0387984372 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008192866 |
| oclc_num | 611361469 |
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| physical | XX, 291 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
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| series2 | Universitext |
| spelling | Polster, Burkard 1965- Verfasser (DE-588)113535384 aut A geometrical picture book Burkard Polster New York [u.a.] Springer 1998 XX, 291 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Literaturverz. S. 279 - 286 Geometry Pictorial works Kombinatorische Geometrie (DE-588)4140733-7 gnd rswk-swf Topologische Geometrie (DE-588)4330788-7 gnd rswk-swf Endliche Geometrie (DE-588)4014650-9 gnd rswk-swf Endliche Geometrie (DE-588)4014650-9 s DE-604 Topologische Geometrie (DE-588)4330788-7 s Kombinatorische Geometrie (DE-588)4140733-7 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008192866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Polster, Burkard 1965- A geometrical picture book Geometry Pictorial works Kombinatorische Geometrie (DE-588)4140733-7 gnd Topologische Geometrie (DE-588)4330788-7 gnd Endliche Geometrie (DE-588)4014650-9 gnd |
| subject_GND | (DE-588)4140733-7 (DE-588)4330788-7 (DE-588)4014650-9 |
| title | A geometrical picture book |
| title_auth | A geometrical picture book |
| title_exact_search | A geometrical picture book |
| title_full | A geometrical picture book Burkard Polster |
| title_fullStr | A geometrical picture book Burkard Polster |
| title_full_unstemmed | A geometrical picture book Burkard Polster |
| title_short | A geometrical picture book |
| title_sort | a geometrical picture book |
| topic | Geometry Pictorial works Kombinatorische Geometrie (DE-588)4140733-7 gnd Topologische Geometrie (DE-588)4330788-7 gnd Endliche Geometrie (DE-588)4014650-9 gnd |
| topic_facet | Geometry Pictorial works Kombinatorische Geometrie Topologische Geometrie Endliche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008192866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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