On Space and Time

Preface......Page 12
1. Euclidean Barycentric Coordinates and the Classic Triangle Centers......Page 17
1.1 Points, Lines, Distance and Isometries......Page 18
1.2 Vectors, Angles and Triangles......Page 21
1.3 Euclidean Barycentric Coordinates......Page 24
1.4 Analogies with Classical Mechanics......Page 27
1.5 Barycentric Representations are Covariant......Page 28
1.6 Vector Barycentric Representation......Page 30
1.7 Triangle Centroid......Page 33
1.8 Triangle Altitude......Page 35
1.9 Triangle Orthocenter......Page 40
1.10 Triangle Incenter......Page 43
1.11 Triangle Inradius......Page 49
1.12 Triangle Circumcenter......Page 52
1.13 Circumradius......Page 56
1.14 Triangle Incircle and Excircles......Page 58
1.15 Excircle Tangency Points......Page 63
1.16 From Triangle Tangency Points to Triangle Centers......Page 68
1.17 Triangle In-Exradii......Page 71
1.18 A Step Toward the Comparative Study......Page 73
1.19 Tetrahedron Altitude......Page 74
1.20 Tetrahedron Altitude Length......Page 78
1.21 Exercises......Page 79
2. Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry......Page 81
2.1 Einstein Addition......Page 82
2.2 Einstein Gyration......Page 86
2.3 From Einstein Velocity Addition to Gyrogroups......Page 89
2.4 First Gyrogroup Theorems......Page 93
2.5 The Two Basic Equations of Gyrogroups......Page 98
2.6 Einstein Gyrovector Spaces......Page 102
2.7 Gyrovector Spaces......Page 105
2.8 Einstein Points, Gyrolines and Gyrodistance......Page 111
2.9 Linking Einstein Addition to Hyperbolic Geometry......Page 115
2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles......Page 117
2.11 The Law of Gyrocosines......Page 122
2.12 The SSS to AAA Conversion Law......Page 124
2.13 Inequalities for Gyrotriangles......Page 125
2.14 The AAA to SSS Conversion Law......Page 127
2.16 The ASA to SAS Conversion Law......Page 131
2.17 Gyrotriangle Defect......Page 132
2.18 Right Gyrotriangles......Page 134
2.19 Einstein Gyrotrigonometry and Gyroarea......Page 136
2.20 Gyrotriangle Gyroarea Addition Law......Page 140
2.21 Gyrodistance Between a Point and a Gyroline......Page 143
2.22 The Gyroangle Bisector Theorem......Page 149
2.23 Mobius Addition and Mobius Gyrogroups......Page 151
2.24 Mobius Gyration......Page 152
2.25 Mobius Gyrovector Spaces......Page 154
2.26 Mobius Points, Gyrolines and Gyrodistance......Page 155
2.27 Linking Mobius Addition to Hyperbolic Geometry......Page 158
2.28 Mobius Gyrovectors, Gyroangles and Gyrotriangles......Page 159
2.29 Gyrovector Space Isomorphism......Page 164
2.30 Mobius Gyrotrigonometry......Page 169
2.31 Exercises......Page 171
3.1 Extension of R into Tn+1......Page 173
3.2 Scalar Multiplication and Addition in Tn+1......Page 178
3.3 Inner Product and Norm in Tn+1......Page 179
3.4 Unit Elements of Tn+1......Page 181
3.5 From Tn+1 back to R......Page 189
4.1 Gyrobarycentric Coordinates in Einstein Gyrovector Spaces......Page 195
4.2 Analogies with Relativistic Mechanics......Page 199
4.3 Gyrobarycentric Coordinates in Mobius Gyrovector Spaces......Page 200
4.4 Einstein Gyromidpoint......Page 203
4.5 Mobius Gyromidpoint......Page 205
4.6 Einstein Gyrotriangle Gyrocentroid......Page 206
4.7 Einstein Gyrotetrahedron Gyrocentroid......Page 213
4.8 Mobius Gyrotriangle Gyrocentroid......Page 215
4.9 Mobius Gyrotetrahedron Gyrocentroid......Page 216
4.10 Foot of a Gyrotriangle Gyroaltitude......Page 217
4.11 Einstein Point to Gyroline Gyrodistance......Page 221
4.12 Mobius Point to Gyroline Gyrodistance......Page 223
4.13 Einstein Gyrotriangle Orthogyrocenter......Page 225
4.14 Mobius Gyrotriangle Orthogyrocenter......Page 235
4.15 Foot of a Gyrotriangle Gyroangle Bisector......Page 240
4.16 Einstein Gyrotriangle Ingyrocenter......Page 245
4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance......Page 253
4.18 Mobius Gyrotriangle Ingyrocenter......Page 256
4.19 Einstein Gyrotriangle Circumgyrocenter......Page 260
4.20 Einstein Gyrotriangle Circumgyroradius......Page 265
4.21 Mobius Gyrotriangle Circumgyrocenter......Page 266
4.22 Comparative Study of Gyrotriangle Gyrocenters......Page 269
4.23 Exercises......Page 273
5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters......Page 275
5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points......Page 281
5.3 Useful Gyrotriangle Gyrotrigonometric Relations......Page 284
5.4 The Tangency Points Expressed Gyrotrigonometrically......Page 285
5.5 M obius Gyrotriangle Ingyrocenter and Exgyrocenters......Page 291
5.6 From Gyrotriangle Tangency Points to Gyrotriangle Gyrocenters......Page 296
5.7 Exercises......Page 299
6.1 Gyrotetrahedron Gyroaltitude......Page 301
6.2 Point Gyroplane Relations......Page 310
6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters......Page 312
6.4 In-Exgyrosphere Tangency Points......Page 321
6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the Gyrotetrahedron In-Exgyrocenters......Page 323
6.6 Gyrotetrahedron Circumgyrocenter......Page 332
6.7 Exercises......Page 336
7.1 Gyromidpoints and Gyrocentroids......Page 339
7.2 Two and Three Dimensional Ingyrocenters......Page 342
7.3 Two and Three Dimensional Circumgyrocenters......Page 344
7.4 Tetrahedron Incenter and Excenters......Page 345
7.5 Comparative study of the Pythagorean Theorem......Page 347
7.6 Hyperbolic Heron\'s Formula......Page 349
7.7 Exercises......Page 350
Notation And Special Symbols......Page 351
Bibliography......Page 353
Index......Page 357