\begin{theindex}

  \item abstract group, 208--209
  \item affine
    \subitem geometry, 79--89, 91, 119
    \subitem group $\Aff(n)$, 127, 197, 210
    \subitem linear
      \subsubitem dependence, 86, 89
      \subsubitem map, 9--10, 34, 86--87
      \subsubitem subspace, 36, 38, 79--86, 88, 91, 92
    \subitem space $\aff^n$, 79, 81, 86, 119, 210
      \subsubitem in projective space, 103
    \subitem span, 79, 83--84
    \subitem transformation, viii, 9--10, 86--88
  \item algebraic topology, vii, 140, 161
  \item algebraically closed field, 167--168
  \item angle, 1, 6--7, 35, 79, 87, 119
    \subitem bisector, 27, 31
    \subitem of rotation, 17--21
    \subitem signed, 6
    \subitem sum, 22--23, 44, 51, 63--69
  \item angular
    \subitem defect, excess, \see{angle sum}{iv}
    \subitem momentum, 116, 190
  \item area, 51--52, 63--69
  \item associative law, 35, 39, 40, 117, 209
  \item axiomatic projective geometry, 108--110, 207, 220, 221

  \indexspace

  \item ball, 72, 77, 135, 169, 180
  \item based loop, 161, 163, 167--168
  \item basis for a topology, 153--155
  \item bilinear form, 
		\see{Euclidean inner product, Lorentz dot product}{iv}, 
		199, 227--229

  \indexspace

  \item centre of rotation, 17
  \item centroid, 25, 87--89
  \item circumcentre, 25--27
  \item closed, \see{compact versus closed}{iv}, 72, 77, 95, 134, 
		139, 140, 170, 182
    \subitem and bounded, 143--158, 180
    \subitem diagonal, 157
    \subitem map, 158--160
  \item cofinite topology, 135, 137, 156
  \item commutative law, 17, 20, 36, 40
  \item compact, \see{maximal --, sequentially --}{iv}, vii, 94, 95, 
		142--145, 150, 164--170, 176, 180, 188
    \subitem Lie group, 180--182, 196
    \subitem surface, 147, 212--213
    \subitem versus closed, 157--158
  \item compactification, 95
  \item complex number, 14, 34, 167, 232
  \item composite
    \subitem of maps, 33--41
    \subitem of reflections, 17, 37--39, 41, 72, 77
    \subitem of rotation and glide, 41
    \subitem of rotation and reflection, 39
    \subitem of rotations, 34, 41
    \subitem of translations, 34
  \item congruent triangles, 22, 30, 31, 68
  \item connected, \see{path --, simply --}{iv}, 141--142, 144, 169, 
		182, 183, 186, 188
    \subitem component, 142, 150, 178, 182, 183, 188, 196, 198
    \subitem Lie group, 197
  \item continuous, vii, 5, 86, 124, 175--177, 182, 183
    \subitem family of paths, 161--162
  \item contractible loop, 161--163, 167, 173
  \item coordinate
    \subitem changes, vi
    \subitem frame, vi, 1
    \subitem geometry, v, viii, ix, 207
    \subitem system, 4
  \item Coventry market, 115--116
  \item cross-ratio, 99--102, 112, 131
  \item curvature, 44, 50, 61, 116, 206, 212--214, 225

  \indexspace

  \item Desargues' theorem, 103--105, 109, 112
  \item dimension, 84, 85, 88, 96, 178, 196
    \subitem of a Lie group, 178--180, 182, 197
    \subitem of intersection, 85, 88, 91--92, 97, 101, 104, 111
  \item direct motion, 11, 17, 20, 182, 186
  \item disc, 139, 150, 160, 163, 171
  \item discrete topology, 134, 137, 156, 177
  \item distance, 
		\see{Euclidean --, hyper\-bolic --, metric, shortest --, spherical --, ratio of --}{iv}
    \subitem function, 1, 2, 5, 7, 8, 45, 79, 119, 223, 224, 227
  \item duality, 107--108, 112

  \indexspace

  \item electron, viii, 176, 189--196, 219
  \item empty set, 85, 88, 92, 96, 134, 153
  \item Erlangen program, vi--vii, 119--120, 139, 210--211
  \item Euclidean
    \subitem angle, 56
    \subitem distance, 1, 2, 4, 143, 185, 186
    \subitem frame, 1, 15, 16, 30, 50, 179
    \subitem geometry, 1--31, 43, 44, 57, 58, 87, 119, 205
    \subitem group $\protect\Eucl(n)$, viii, 196, 197
    \subitem inner product, 2, 6, 11, 29, 54, 71, 228, 229, 232
    \subitem line, 5
    \subitem motion, \see{motion}{iv}, 10, 11, 15, 16, 29, 30, 58, 115, 
		177
    \subitem plane $\E^2$, 6, 41
    \subitem space $\E^n$, 1, 4--11, 36, 44, 223
    \subitem translation, 23
  \item Euler characteristic, 213

  \indexspace

  \item family of paths, 161
  \item Feuerbach circle, 27--28
  \item frame, vi
    \subitem affine, 86--88
    \subitem Euclidean, 1, 15--16, 30, 179
    \subitem projective, 98--100, 112, 131
  \item frame of reference, \see{frame}{iv}
  \item fundamental
    \subitem group, 140, 161, 195
    \subitem theorem of algebra, 167
  \item fundamental group, vii

  \indexspace

  \item generators, 37, 124--125, 128, 131
  \item genus, 148, 172, 213
  \item geodesic, \see{shortest distance}{iv}
  \item glide, 17--20, 29, 39--41, 50, 58, 122
    \subitem reflection, \see{glide}{iv}
  \item glueing, \see{quotient topology}{iv}
  \item great circle, \see{spherical line}{iv}
  \item group, 
		\see{abstract --, fundamental --, Lie --, topological --, transformation --}{iv}

  \indexspace

  \item half-turn, 14, 40
  \item Hausdorff, 136, 156--158, 160, 171, 188
  \item Heine--Borel theorem, 143
  \item Hermitian form, 188, 193, 196, 201, 232--233
  \item homeomorphism, 133, 137--140, 144, 148--150, 160, 163--165, 167, 
		169, 171, 181--183, 187, 188, 196, 212
    \subitem criterion, 139, 160, 175, 187
    \subitem problem, vii, 140
  \item homogeneous space, 209--210
  \item hyperbolic
    \subitem distance, 54, 57, 71, 77
    \subitem geometry, 5, 23, 43--45, 52--77, 205
    \subitem line, 53, 57, 59--63, 75
    \subitem motion, 57, 178
    \subitem plane $\sH^2$, 49, 58--60, 71--75, 223
    \subitem sine rule, 73
    \subitem space, 44, 53, 63, 129
    \subitem translation, 58, 72, 75
    \subitem triangle, 55, 63, 72, 73
    \subitem trig, 44, 55--56
  \item hyperplane, 36, 38, 84, 85, 96, 98, 101--103, 111, 119
    \subitem at infinity, 109

  \indexspace

  \item ideal point, \see{infinity, point at}{iv}
  \item ideal triangle, 63, 66--69
  \item incentre, 27
  \item incidence of lines, 43, 44, 50, 59, 87, 105
  \item indiscrete topology, 134, 137, 170
  \item infinity
    \subitem hyperplane at, 91, 92, 96, 103, 113
    \subitem point at, 60, 63, 64, 66, 67, 73, 92, 94, 96, 100
  \item intersection, \see{dimension of --}{iv}, 134
  \item intrinsic
    \subitem curvature, 44, 50, 220
    \subitem distance, 50
    \subitem unit, 44, 61
  \item isometry, \see{motion, preserves distances}{iv}, 4, 7, 139, 224

  \indexspace

  \item Klein bottle, 172

  \indexspace

  \item length of path, 5
  \item Lie group, \see{compact --}{iv}, 175--201, 203, 208
  \item line, 4, 5, 82
    \subitem hyperbolic, 55
    \subitem segment, 3, 82
    \subitem spherical, 45
  \item loop, 134--168, 172, 195
  \item Lorentz
    \subitem basis, 54, 67, 77, 229--232, 234
    \subitem complement, 59, 230
    \subitem dot product $\cdot_L$, 54, 228, 230
    \subitem form $q_L$, 53, 58, 228, 230, 234
    \subitem group, 116, 196, 198
    \subitem matrix, 53, 57--58, 198, 232, 234
    \subitem norm, 55, 229, 231
    \subitem orthogonal, 55
      \subsubitem matrix, 232
    \subitem pseudometric, 52, 71, 217
    \subitem reflection, 58
    \subitem space, 52, 57, 232
    \subitem transformation, 58--59, 67, 115, 178
    \subitem translation, \see{hyperbolic --}{iv}

  \indexspace

  \item M\"obius strip, vi, 133, 134, 146--148, 150, 151, 171
  \item maximal compact subgroup, 196
  \item metric, 223--225
    \subitem geometry, 82, 212
    \subitem space, 1, 4, 48, 223--225
    \subitem topology, 135, 154, 177, 188
  \item minimum over paths, 6, 223
  \item motion, vi, 1, 7, 8, 10--12, 15--18, 20--23, 29--31, 33, 36--41, 
		43, 44, 48--50, 57--59, 71, 72, 75, 77, 116, 118, 119, 
		121, 122, 124, 128, 130, 131, 178, 183, 184, 186, 190, 
		194, 197
  \item mousetrap topology, 151
  \item Mus\'ee Gr\'evin, 128, 130

  \indexspace

  \item non-Euclidean geometry, 43--77, 205
  \item normal form of a matrix, 12--15, 20, 37, 122--123, 183, 234

  \indexspace

  \item open set, 134--138, 140--142, 145, 146, 150, 154, 176, 182
  \item opposite motion, 11, 17, 20, 182
  \item orthocentre, 25--27
  \item orthogonal, \see{Lorentz orthogonal}{iv}
    \subitem axes, 1
    \subitem complement $V^\perp$, 15, 59, 179, 211, 229
    \subitem direct sum, 185
    \subitem frame, 49
    \subitem group $\protect\SO(n)$, 177--187
    \subitem line, 194
    \subitem magnetic field, 190, 194
    \subitem matrix, 7, 11--15, 29, 37, 48, 49, 122, 177, 180--183, 196, 
		232
    \subitem plane, 36
    \subitem transformation, 11, 115, 122, 232
    \subitem vector, 6, 36, 47, 185, 199, 229

  \indexspace

  \item Pappus' theorem, 105--107, 109, 112
  \item parallel
    \subitem axes, 38
    \subitem hyperplanes, 21, 81, 84, 85
    \subitem lines, 17, 18, 23--27, 35, 44, 50, 61, 79, 85, 88, 92, 103, 
		204
    \subitem mirrors, 128
    \subitem postulate, 23, 60, 62, 74, 204
    \subitem sides, 39
    \subitem vector, 18, 119
  \item path, \see{length of path, minimum over paths}{iv}, 141, 161, 
		195
    \subitem connected, 141, 142, 149, 163, 173, 183
  \item perpendicular bisector, 18, 25, 27, 29, 36--38, 71, 76
  \item perspective, 92, 93, 102--105, 110, 112, 113
  \item physics, vii, viii, 116, 117, 197, 214--220
  \item point at infinity, \see{infinity, point at}{iv}
  \item preserves distances, 7--8, 29, 49, 224
  \item principal homogeneous space, \see{torsor}{iv}
  \item Pringle's potato chip, 72, 212
  \item product topology, 155, 157, 171, 177
  \item profinite topology, 154, 155
  \item projective
    \subitem geometry, 91--113
    \subitem linear subspace, 92--97
  \item punctured disc $D^*$, 149, 160, 163, 167

  \indexspace

  \item quadratic form, 6, 11, 53, 152, 184, 186, 227
  \item quaternions, 183--187
  \item quotient topology, 136, 145--148, 150--152, 154, 171--173, 177, 
		187, 188

  \indexspace

  \item reflection, 1, 13, 17, 18, 29, 35--38, 41, 44, 50, 72, 76, 77, 
		128, 130
    \subitem group, 128--130
    \subitem matrix, 7, 12, 29, 52, 178
  \item rigid body motion, \see{motion}{iv}
  \item rotary reflection, 41, 50
  \item rotation, 1, 12, 17--19, 21, 29, 30, 34, 36--38, 40, 41, 44, 
		49, 50, 58, 121, 124, 128, 175, 176, 183--187, 190, 
		194, 198
    \subitem group, 187
    \subitem matrix, 7, 12, 52, 178
  \item rubber-sheet geometry, vi, 133

  \indexspace

  \item sequentially compact, 143, 169
  \item shortest distance, \see{minimum over paths}{iv}, 5, 6, 50, 57, 
		72
  \item similar triangles, 24--27
  \item simplex of reference, \see{frame}{iv}
  \item simply connected, 161, 163, 180, 197
  \item special
    \subitem linear group $\SL(n)$, 196
    \subitem orthogonal group $\SO(n)$, 183, 187
    \subitem relativity, vii, 116, 178, 214, 216--217
    \subitem unitary group $\SU(n)$, 188, 220
  \item sphere $S^2$, 45, 46, 49--51, 53, 69, 72, 140, 223, 225
  \item sphere $S^n$, 71, 143, 150, 179, 186
  \item spherical
    \subitem disc, 69
    \subitem distance, 45, 46, 48, 50, 69, 143
    \subitem frame, 44, 50
    \subitem geometry, 5, 23, 43--52, 57, 70, 71, 206, 220, 221, 225
    \subitem line, 48--50
    \subitem motion, 48, 49
    \subitem triangle, 46, 48, 51, 70, 225
    \subitem trig, 46, 47, 77, 207
  \item spin, 176, 189, 191
  \item spinor group $\Spin(n)$, 188, 196
  \item Standard Model, 219--220
  \item subspace topology, 145, 150, 157, 177, 181, 188
  \item symmetry, 115--118, 197, 208, 216--221

  \indexspace

  \item topological
    \subitem group, 176--178, 196
    \subitem property, vii, 140, 156, 162, 167, 207
  \item topology, 118, 133--173, 176
    \subitem of $\PP^n$, 113, 150, 171
    \subitem of $\SO(3)$, 175, 176, 183
    \subitem of $S^3$, 187
  \item torsor, 209--210
  \item torus, 147, 148, 171
  \item transformation group, 33--41, 115, 117, 119, 120, 125, 129, 139, 
		175--201
  \item translation, 1, 17, 18, 21, 23, 30, 37, 38, 40, 41, 49, 85, 121, 
		124--128, 131, 194, 197
    \subitem map, 154
    \subitem subgroup, 125, 130
    \subitem vector, 17, 29, 34, 39
  \item triangle inequality, 1--6, 48, 223
  \item trichotomy, 212--214

  \indexspace

  \item ultraparallel lines, 60--63, 73, 76
  \item UMP, \see{universal mapping property}{iv}
  \item unitary
    \subitem group, 188, 220
    \subitem matrix, 188, 195, 232--234
    \subitem representation, 218
  \item universal mapping property, 146, 171, 187

  \indexspace

  \item winding number, vii, 133--134, 160--168

\end{theindex}