\addvspace {10\p@ }
\contentsline {figure}{\numberline {1.1}{\ignorespaces Triangle inequality}}{2}
\contentsline {figure}{\numberline {1.5}{\ignorespaces Angle with direction}}{6}
\contentsline {figure}{\numberline {1.6}{\ignorespaces Rigid body motion}}{7}
\contentsline {figure}{\numberline {1.9}{\ignorespaces Affine linear construction of $\la \@mathbf x+\mu \@mathbf y$}}{10}
\contentsline {figure}{\numberline {1.11a}{\ignorespaces A rotation in coordinates}}{12}
\contentsline {figure}{\numberline {1.11b}{\ignorespaces The rotation and the reflection}}{13}
\contentsline {figure}{\numberline {1.13}{\ignorespaces The Euclidean frames $P_0,P_1,P_2$ and $P'_0,P'_1,P'_2$}}{16}
\contentsline {figure}{\numberline {1.14a}{\ignorespaces $\Rot (O,\theta )$ and $\Glide (L,\@mathbf v)$}}{19}
\contentsline {figure}{\numberline {1.14b}{\ignorespaces Construction of glide}}{19}
\contentsline {figure}{\numberline {1.14c}{\ignorespaces Construction of rotation}}{19}
\contentsline {figure}{\numberline {1.15a}{\ignorespaces $\Twist (L,\theta ,\@mathbf v)$ and $\RotRefl (L,\theta ,\Pi )$}}{20}
\contentsline {figure}{\numberline {1.15b}{\ignorespaces A grid of parallel planes and their orthogonal lines}}{21}
\contentsline {figure}{\numberline {1.16a}{\ignorespaces Pons asinorum}}{22}
\contentsline {figure}{\numberline {1.16b}{\ignorespaces Sum of angles in a triangle $=\pi $}}{23}
\contentsline {figure}{\numberline {1.16c}{\ignorespaces Parallel lines fall on lines in the same ratio}}{24}
\contentsline {figure}{\numberline {1.16d}{\ignorespaces Similar triangles}}{24}
\contentsline {figure}{\numberline {1.16d}{\ignorespaces The centroid}}{25}
\contentsline {figure}{\numberline {1.16e}{\ignorespaces The circumcentre}}{26}
\contentsline {figure}{\numberline {1.16f}{\ignorespaces The orthocentre}}{26}
\contentsline {figure}{\numberline {1.16g}{\ignorespaces The Feuerbach 9-point circle}}{28}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {2.3}{\ignorespaces Composite of two reflections}}{35}
\contentsline {figure}{\numberline {2.7}{\ignorespaces Composite of a rotation and a reflection}}{39}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {3.0}{\ignorespaces Plane-like geometry}}{43}
\contentsline {figure}{\numberline {3.2}{\ignorespaces Spherical trig}}{47}
\contentsline {figure}{\numberline {3.6}{\ignorespaces Overlapping segments of $S^2$}}{51}
\contentsline {figure}{\numberline {3.7}{\ignorespaces The hyperbola $t^2=1+x^2$ and $t>0$}}{53}
\contentsline {figure}{\numberline {3.8}{\ignorespaces Hyperbolic space $\@mathcal H^2$}}{54}
\contentsline {figure}{\numberline {3.10}{\ignorespaces Hyperbolic trig}}{56}
\contentsline {figure}{\numberline {3.12}{\ignorespaces (a) Projection to the $(x,y)$-plane of the spherical lines $y=cz$. (b) Projection to the $(x,y)$-plane of the hyperbolic lines $y=ct$.}}{59}
\contentsline {figure}{\numberline {3.13}{\ignorespaces The failure of the parallel postulate in $\@mathcal H^2$}}{61}
\contentsline {figure}{\numberline {3.14a}{\ignorespaces The hyperbolic triangle $\triangle PQR$ with one ideal vertex}}{64}
\contentsline {figure}{\numberline {3.14b}{\ignorespaces Area and angle sum are ``additive''}}{65}
\contentsline {figure}{\numberline {3.14c}{\ignorespaces The subdivision of $\triangle PQR$}}{67}
\contentsline {figure}{\numberline {3.14d}{\ignorespaces The angular defect formula}}{68}
\contentsline {figure}{\numberline {3.14e}{\ignorespaces Area is an additive function}}{68}
\contentsline {figure}{\numberline {3.14f}{\ignorespaces Area is a monotonic function}}{69}
\contentsline {figure}{\numberline {3.15}{\ignorespaces $\@mathcal H$-lines}}{74}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {4.2}{\ignorespaces Points, vectors and addition}}{81}
\contentsline {figure}{\numberline {4.3a}{\ignorespaces The affine construction of the line segment $[\@mathbf p,\@mathbf q]$}}{83}
\contentsline {figure}{\numberline {4.3b}{\ignorespaces Parallel hyperplanes}}{84}
\contentsline {figure}{\numberline {4.7}{\ignorespaces The affine centroid}}{87}
\contentsline {figure}{\numberline {4.8}{\ignorespaces A weighted centroid}}{89}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {5.1a}{\ignorespaces A cube in perspective}}{93}
\contentsline {figure}{\numberline {5.1b}{\ignorespaces Perspective drawing}}{93}
\contentsline {figure}{\numberline {5.1c}{\ignorespaces Hyperbola and parabola}}{94}
\contentsline {figure}{\numberline {5.6a}{\ignorespaces The 3-transitive action of $\PGL (2)$ on $\PP ^1$}}{100}
\contentsline {figure}{\numberline {5.6b}{\ignorespaces The cross-ratio $\{P,Q;R,S\}$}}{101}
\contentsline {figure}{\numberline {5.8}{\ignorespaces The inclusion $\aff ^n\subset \PP ^n$}}{103}
\contentsline {figure}{\numberline {5.9a}{\ignorespaces The Desargues configuration in $\PP ^2$ or $\PP ^3$}}{104}
\contentsline {figure}{\numberline {5.9b}{\ignorespaces Lifting the Desargues configuration to $\@mathbb P^3$}}{105}
\contentsline {figure}{\numberline {5.10}{\ignorespaces The Pappus configuration}}{106}
\contentsline {figure}{\numberline {5.12a}{\ignorespaces Axiomatic projective plane}}{109}
\contentsline {figure}{\numberline {5.12b}{\ignorespaces Geometric construction of addition}}{110}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {6.0}{\ignorespaces The plan of Coventry market}}{116}
\contentsline {figure}{\numberline {6.4a}{\ignorespaces The conjugate rotation $g\Rot (P,\theta )g\1=\Rot (g(P),g(\theta ))$}}{121}
\contentsline {figure}{\numberline {6.4b}{\ignorespaces Action of $\Aff (n)$ on vectors of $\aff ^n$}}{122}
\contentsline {figure}{\numberline {6.6a}{\ignorespaces Kaleidoscope}}{128}
\contentsline {figure}{\numberline {6.6b}{\ignorespaces ``Mus\'ee Gr\'evin''}}{129}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {7.2a}{\ignorespaces Hausdorff property}}{136}
\contentsline {figure}{\numberline {7.2b}{\ignorespaces $S^1=[0,1]$ with the ends identified}}{137}
\contentsline {figure}{\numberline {7.3a}{\ignorespaces $(0,1)\simeq \@mathbb R$}}{138}
\contentsline {figure}{\numberline {7.3b}{\ignorespaces Squaring the circle}}{139}
\contentsline {figure}{\numberline {7.4b}{\ignorespaces Path connected set}}{141}
\contentsline {figure}{\numberline {7.6a}{\ignorespaces The M\"obius strip $M$}}{147}
\contentsline {figure}{\numberline {7.6b}{\ignorespaces The cylinder $S^1\times [0,1]$}}{147}
\contentsline {figure}{\numberline {7.6c}{\ignorespaces The torus}}{148}
\contentsline {figure}{\numberline {7.6d}{\ignorespaces Surface with $g$ handles}}{148}
\contentsline {figure}{\numberline {7.6e}{\ignorespaces Boundary and interior points}}{149}
\contentsline {figure}{\numberline {7.7}{\ignorespaces Topology of $\@mathbb P^2_\@mathbb R$: M\"obius strip with a disc glued in}}{151}
\contentsline {figure}{\numberline {7.8a}{\ignorespaces The mousetrap topology}}{152}
\contentsline {figure}{\numberline {7.8b}{\ignorespaces Equivalence classes of quadratic forms $ax^2+2bxy+cy^2$}}{153}
\contentsline {figure}{\numberline {7.10}{\ignorespaces Balls for product metrics}}{156}
\contentsline {figure}{\numberline {7.12}{\ignorespaces Separating a point from a compact subset}}{158}
\contentsline {figure}{\numberline {7.13a}{\ignorespaces Closed map}}{159}
\contentsline {figure}{\numberline {7.13b}{\ignorespaces Nonclosed map}}{159}
\contentsline {figure}{\numberline {7.15a}{\ignorespaces Continuous family of paths}}{162}
\contentsline {figure}{\numberline {7.15b}{\ignorespaces $D^*$ covered by overlapping open radial sectors}}{165}
\contentsline {figure}{\numberline {7.15c}{\ignorespaces Overlapping intervals}}{165}
\contentsline {figure}{\numberline {7.15a}{\ignorespaces Glueing patterns on the square}}{172}
\contentsline {figure}{\numberline {7.15b}{\ignorespaces The surface with two handles and the octagon}}{173}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {8.0}{\ignorespaces The geometry of the group of planar rotations}}{176}
\contentsline {figure}{\numberline {8.7a}{\ignorespaces The Stern--Gerlach experiment}}{190}
\contentsline {figure}{\numberline {8.7b}{\ignorespaces The modified Stern--Gerlach device}}{191}
\contentsline {figure}{\numberline {8.7c}{\ignorespaces Two identical SG devices}}{192}
\contentsline {figure}{\numberline {8.7d}{\ignorespaces Two different SG devices}}{192}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {9.1a}{\ignorespaces The parallel postulate. To meet or not to meet?, that is the question.}}{205}
\contentsline {figure}{\numberline {9.1b}{\ignorespaces The parallel postulate in the Euclidean plane}}{206}
\contentsline {figure}{\numberline {9.1c}{\ignorespaces The ``parallel postulate'' in spherical geometry}}{207}
\contentsline {figure}{\numberline {9.3}{\ignorespaces The cap, flat plane and Pringle's chip }}{212}
\contentsline {figure}{\numberline {9.3}{\ignorespaces The famous trichotomy}}{213}
\addvspace {10\p@ }
\contentsline {figure}{\numberline {A.1}{\ignorespaces The bear}}{225}
\addvspace {10\p@ }