目錄
Part Ⅰ
1 TRIANGLES
1.1 Euclid
1.2 Primitive concepts and axioms
1.3 Pons asinorum
1.4 The medians and the centroid
1.5 The incircle and the circumcircle
1.6 The Euler line and the orthocenter
1.7 The nine-point circle
1.8 Two extremum problems
1.9 Morley's theorem
2 REGULAR POLYGONS
2.1 Cyclotomy
2.2 Angle trisection
2.3 Isometry
2.4 Symmetry
2.5 Groups
2.6 The product of two reflections
2.7 The kaleidoscope
2.8 Star polygons
3 ISOMETRY IN THE EUCLIDEAN PLANE
3.1 Direct and opposite isometries
3.2 Translation
3.3 Glide reflection
3.4 Reflections and half-turns
3.5 Summary of results on isometries
3.6 Hjelmslev's theorem
3.7 Patterns on a strip
4 TWO-DIMENSIONAL CRYSTALLOGRAPHY
4.1 Lattices and their Dirichlet regions
4.2 The symmetry group of the general lattice
4.3 The art of M. C. Escher
4.4 Six patterns of bricks
4.5 The crystallographic restriction
4.6 Regular tessellations
4.7 Sylvester's problem of collinear points
5 SIMILARITY IN THE EUCLIDEAN PLANE
5.1 Dilatation
5.2 Centers of similitude
5.3 The nine-point center
5.4 The invariant point of a similarity
5.5 Direct similarity
5.6 Opposite similarity
6 CIRCLES AND SPHERES
6.1 Inversion in a circle
6.2 Orthogonal circles
6.3 Inversion of lines and circles
6.4 The inversive plane
6.5 Coaxal circles
6.6 The circle of Apollonius
6.7 Circle-preserving transformations
6.8 Inversion in a sphere
6.9 The elliptic plane
7 ISOMETRY AND SIMILARITY IN EUCLIDEAN SPACE
7.1 Direct and opposite isometries
7.2 The central inversion
7.3 Rotation and translation
7.4 The product of three reflections
7.5 Twist
7.6 Dilative rotation
7.7 Sphere-preserving transformations
Part Ⅱ
8 COORDINATES
8.1 Cartesian coordinates
8.2 Polar coordinates
8.3 The circle
8.4 Conics
8.5 Tangent, arc length, and area
8.6 Hyperbolic functions
8.7 The equiangular spiral
8.8 Three dimensions
9 COMPLEX NUMBERS
9.1 Rational numbers
9.2 Real numbers
9.3 The Argand diagram
9.4 Modulus and amplitude
9.5 The formula eπi + 1 = 0
9.6 Roots of equations
9.7 Conformal transformations
10 THE FIVE PLATONIC SOLIDS
10.1 Pyramids, prisms, and antiprisms
10.2 Drawings and models
10.3 Euler's formula
10.4 Radii and angles
10.5 Reciprocal polyhedra
11 THE GOLDEN SECTION AND PHYLLOTAXIS
11.1 Extreme and mean ratio
11.2 De divina proportione
11.3 The golden spiral
11.4 The Fibonacci numbers
11.5 Phyllotaxis
Part Ⅲ
12 ORDERED GEOMETRY
12.1 The extraction of two distinct geometries from Euclid
12.2 Intermediacy
12.3 Sylvester's problem of collinear points
12.4 Planes and hyperplanes
12.5 Continuity
12.6 Parallelism
13 AFFINE GEOMETRY
13.1 The axiom of parallelism and the "Desargues" axiom
13.2 Dilatations
13.3 Affinities
13.4 Equiaffinities
13.5 Two-dimensional lattices
13.6 Vectors and centroids
13.7 Barycentric coordinates
13.8 Affine space
13.9 Three-dimensional lattices
14 PROJECTIVE GEOMETRY
14.1 Axioms for the general projective plane
14.2 Projective coordinates
14.3 Desargues's theorem
14.4 Quadrangular and harmonic sets
14.5 Projectivities
14.6 Collineations and correlations
14.7 The conic
14.8 Projective space
14.9 Euclidean space
15 ABSOLUTE GEOMETRY
15.1 Congruence
15.2 Parallelism
15.3 Isometry
15.4 Finite groups of rotations
15.5 Finite groups of isometries
15.6 Geometrical crystallography
15.7 The polyhedral kaleidoscope
15.8 Discrete groups generated by inversions
16 HYPERBOLIC GEOMETRY
16.1 The Euclidean and hyperbolic axioms of parallelism
16.2 The question of consistency
16.3 The angle of parallelism
16.4 The finiteness of triangles
16.5 Area and angular defect
16.6 Circles, horocycles, and equidistant curves
16.7 Poincar?'s "half-plane" model
16.8 The horosphere and the Euclidean plane
Part Ⅳ
17 DIFFERENTIAL GEOMETRY OF CURVES
17.1 Vectors in Euclidean space
17.2 Vector functions and their derivatives
17.3 Curvature, evolutes, and involutes
17.4 The catenary
17.5 The tractrix
17.6 Twisted curves
17.7 The circular helix
17.8 The general helix
17.9 The concho-spiral
18 THE TENSOR NOTATION
18.1 Dual bases
18.2 The fundamental tensor
18.3 Reciprocal lattices
18.4 The critical lattice of a sphere
18.5 General coordinates
18.6 The alternating symbol
19 DIFFERENTIAL GEOMETRY OF SURFACES
19.1 The use of two parameters on a surface
19.2 Directions on a surface
19.3 Normal curvature
19.4 Principal curvatures
19.5 Principal directions and lines of curvature
19.6 Umbilics
19.7 Dupin's theorem and Liouville's theorem
19.8 Dupin's indicatrix
20 GEODESICS
20.1 Theorema egregium
20.2 The differential equations for geodesics
20.3 The integral curvature of a geodesic triangle
20.4 The Euler-Poincar? characteristic
20.5 Surfaces of constant curvature
20.6 The angle of parallelism
20.7 The pseudosphere
21 TOPOLOGY OF SURFACES
21.1 Orientable surfaces
21.2 Nonorientable surfaces
21.3 Regular maps
21.4 The four-color problem
21.5 The six-color theorem
21.6 A sufficient number of colors for any surface
21.7 Surfaces that need the full number of colors
22 FOUR-DIMENSIONAL GEOMETRY
22.1 The simplest four-dimensional figures
22.2 A necessary condition for the existence of(p, q, r)
22.3 Constructions for regular polytopes
22.4 Close packing of equal spheres
22.5 A statistical honeycomb
TABLES
REFERENCES
ANSWERS To EXERCISES
INDEX
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