When straight lines curve: non-Euclidean geometry
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London
RBA
2012
|
| Schriftenreihe: | Everything is mathematical
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | 151 S. Ill., graph. Darst., Kt. |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804149634370633728 |
|---|---|
| adam_text | Contents
Preface
Chapter
1.
A Taxi Ride
................................................................................................................. 11
Enchanted streets
.................................................................................................................................. 13
The taxicab distance
........................................................................................................................... 15
An example with triangles
...................................................................................................... 18
Circles
................................................................................................................................................... 19
Ellipses
.................................................................................................................................................. 21
Union Street
................................................................................................................................... 22
Chapter
2.
Euclidean Geometry
............................................................................................. 25
Euclid, Elements and the fifth postulate
................................................................................... 25
Statements equivalent to the fifth postulate
......................................................................... 32
Geometry of Renaissance paintings
......................................................................................... 34
Challenging Euclid
.............................................................................................................................. 38
Chapter
3.
Competing with Euclid
....................................................................................... 43
The last Greek master
........................................................................................................................ 43
The Medieval guardians of Greek learning
......................................................................... 46
The modern period
............................................................................................................................ 46
Saccheri s quadrilateral
...................................................................................................................... 48
Towards non-Euclidean geometries
......................................................................................... 51
Chapter
4.
The Establishment of Non-Euclidean Geometry
.......................... 53
Nikolai Lobachevski: the Russian soul of hyperbolic geometry
.............................. 53
János Bolyai:
mathematician and cavalryman
..................................................................... 55
Gauss s contribution
.................................................................................................................... 57
Correspondence between Gauss and
Bolyai
............................................................... 58
The joint achievements of Lobachevski and
Bolyai
................................................. 60
Common models of hyperbolic geometry
........................................................................... 61
Riemann and elliptical geometry
............................................................................................... 67
CONTENTS
The same but different
..................................................................................................................... 70
An ant race
........................................................................................................................................ 73
Einstein versus Euclid
........................................................................................................................ 74
The theory of relativity
............................................................................................................. 74
The correct geometry
................................................................................................................ 76
Chapter
5.
The Surprising Results of Hyperbolic Geometry
........................ 79
Limiting angle of parallels and straight lines
........................................................................ 80
Equidistant curves
................................................................................................................................ 82
Pythagoras, triangles and lengths
................................................................................................ 82
Triangles
.............................................................................................................................................. 83
Circles
................................................................................................................................................... 84
Pythagoras
.......................................................................................................................................... 86
Hyperbolic trigonometry
................................................................................................................ 88
Classical and hyperbolic trigonometry
................................................................................... 93
Chapter
6.
Contributions to Elliptical Geometry
...................................................... 95
The third geometry
............................................................................................................................. 95
Terminology of spherical triangles
............................................................................................ 97
A world of spherical triangles
....................................................................................................... 103
The sum of angles and sides of a spherical triangle
................................................ 103
The area of a triangle
................................................................................................................. 103
Lengths of circumferences
....................................................................................................... 104
The sine and cosine rules
........................................................................................................ 105
Pythagoras
s
theorem
.................................................................................................................. 105
Chapter
7.
Geometry of the Earth
...................................................................................... 107
Parallels and meridians
...................................................................................................................... 108
From the
Mappa
Mundi
to Google™ Earth
........................................................................ 113
What s the shortest distance between Barcelona and Tokyo?
........................... 113
Chapter
8.
Geometry in the 21st Century
.................................................................... 117
Integral geometry
................................................................................................................................. 117
From compasses to
computen
..................................................................................................... 120
Artificial eyes for a robot
.......................................................................................................... 122
Magnetic resonance
..................................................................................................................... 125
CONTENTS
Digital images
.................................................................................................................................. 128
CAD:
Computer-Aided
Design
............................................................................................ 133
Remote sensing: geographic information systems
.......................................................... 135
Appendix. The Theory of Relativity and the New Geometries
.......................... 139
Bibliography
........................................................................................................................................... 147
Index
............................................................................................................................................................ 149
When Straight
Lines Curve
Non-Euclidian geometry
Since the time of Euclid, more than two thousand years
ago, there seemed to be only one kind of geometry. However,
new developments in mathematics have put forward alternative
geometries in which worlds curve in a dizzying fashion. It
may seem crazy, but these worlds exist and we live in them all
simultaneously.
|
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| spelling | Gómez Urgellés, Joan 1959- Verfasser (DE-588)1141694433 aut When straight lines curve non-Euclidean geometry Joan Gómez London RBA 2012 151 S. Ill., graph. Darst., Kt. txt rdacontent n rdamedia nc rdacarrier Everything is mathematical Nichteuklidische Geometrie (DE-588)4042073-5 gnd rswk-swf Nichteuklidische Geometrie (DE-588)4042073-5 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025388611&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025388611&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Gómez Urgellés, Joan 1959- When straight lines curve non-Euclidean geometry Nichteuklidische Geometrie (DE-588)4042073-5 gnd |
| subject_GND | (DE-588)4042073-5 |
| title | When straight lines curve non-Euclidean geometry |
| title_auth | When straight lines curve non-Euclidean geometry |
| title_exact_search | When straight lines curve non-Euclidean geometry |
| title_full | When straight lines curve non-Euclidean geometry Joan Gómez |
| title_fullStr | When straight lines curve non-Euclidean geometry Joan Gómez |
| title_full_unstemmed | When straight lines curve non-Euclidean geometry Joan Gómez |
| title_short | When straight lines curve |
| title_sort | when straight lines curve non euclidean geometry |
| title_sub | non-Euclidean geometry |
| topic | Nichteuklidische Geometrie (DE-588)4042073-5 gnd |
| topic_facet | Nichteuklidische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025388611&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025388611&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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