Stillwell J. — Yearning for the Impossible: The Surprising Truths of Mathematics :: Электронная библиотека попечительского совета мехмата МГУ

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Stillwell J. — Yearning for the Impossible: The Surprising Truths of Mathematics

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Название: Yearning for the Impossible: The Surprising Truths of Mathematics

Автор: Stillwell J.

Аннотация:

This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress:

* Irrational and Imaginary Numbers
* The Fourth Dimension
* Curved Space
* Infinity and others

The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. By imbedding mathematics into a broader cultural context and through his clever and enthusiastic explication of mathematical ideas the author broadens the horizon of students beyond the narrow confines of rote memorization and engages those who are curious about the place of mathematics in our intellectual landscape.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 250

Добавлена в каталог: 11.06.2008

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Предметный указатель

$\mathbb{N}$       197
$\mathbb{R}$       25 26 197
$\mathbb{R}$ and the limit concept      208
$\mathbb{R}$ as a line of points      201
$\mathbb{R}$ is not countable      197 202
$\mathbb{R}$ , stands for real      27
$\mathbb{R}^2$       197
$\mathbb{R}^3$       143 197
$\mathbb{R}^3$ and regular polyhedra      144
$\mathbb{R}^3$ , rotations of      146
$\mathbb{R}^4$       149
$\mathbb{R}^4$ and regular polytopes      151
$\mathbb{R}^4$ , tilings of      153
$\mathbb{R}^n$       135
$\pi$       84
$\pi$ is transcendental      207
$\pi$ , computation of      85
$\pi$ , evaluation by calculus      96
$\pi$ , Indian formula      85 198 208
$\pi$ , not constructible      208
$\sqrt{-1}$       25
$\sqrt{-1}$ , as a symbol      213
$\sqrt{-1}$ , obeys laws of algebra      33
$\sqrt{2}$       1 13
$\sqrt{2}$ and decimal fractions      13
$\sqrt{2}$ and Huclidean algorithm      17
$\sqrt{2}$ as a gap in the rationals      14 197
$\sqrt{2}$ as infinite decimal      15 209
$\sqrt{2}$ in geometry      7
$\sqrt{2}$ , continued fraction for      16 19
$\sqrt{2}$ , irrationality of      9 18 22 162
120-cell      151
2-gon      181
2-sphere      104 108
2-torus      187
2-torus as a doubly periodic plane      191
2-torus, constructed from square      187
2-torus, topological properties      188
24-cell      147 149
24-cell, shadow      151
24-cell, tiling of $\mathbb{R}^4$       153
3-cylinder      177 183 189
3-cylinder contains tribar      183
3-sphere      104 108 189
3-sphere, stereographic image      107
3-torus      188 189
3-torus as periodic space      188
3-torus, constructed from cube      188
4-cube      150
4-orthoplex      150
4-simplex      150
600-cell      151
Abel, Niel Henrik      195
Absolute value      35
Absolute value, multiplicative property      35 129
Absolute value, multiplicative property, and two-square identity      130
Absolute value, multiplicative property, for quaternions      139
Adams, John Couch      33
Adcquality      91
Addition in projective geometry      72
Addition of angles      38 186
Addition of equivalence classes      186
Addition, multiplication perceived as      4
Addition, vector      131 136
Additive identity      70
Additive, inverse      70 131
al-Khazin, Abu Ja'far      35
Al-Khwarizmi, Abu Ja'far Muhammed ibn Musa      9
Alberti, Leon Battista      55
Algebra      10
Algebra and projective coincidences      69
Algebra, laws of      69
Algebra, linear      127
Algebraic-curve      44
Algebraic-curve, geometry      87
Algebraic-curve, number      205
Algebraic-curve, topology      195
Angle sum theorem      110
Angle sum theorem and flatness      114
Angle sum theorem of Gauss      112
Angle sum theorem on 2-sphere      111 112
Angle sum theorem on 2-sphere, Harriot proof      112
Angles and the parallel axiom      50
Angles as equivalence classes      186
Angles of pentagon      124
Angles, addition of      186
Angles, additive property      36
Apollonius      88
Archimedes      84
Area and slope      93
Area as sum of infinitesimals      93
Area of disk      84
Area of parallelogram      79
Area of polygons      77
Area of rectangle      78
Area of square      5
Area of triangle      79
Area, equality of      10
Aristophanes      207
Arithmetization      126
Artmann, Benno      28
Associative law      70
Axiom, Desargues      65
Axiom, incidence      60
Axiom, Pappus      65 76
Axiom, parallel      49 71
Axiom, parallel, of Bolyai and Lobachevsky      126
Axiom, parallel, of Euclid      126
Axioms about lines      127
Axioms of Euclid      126
Axioms of geometry      76
Axioms of projective geometry      76
Axioms of set theory      209
Axioms, Euclid-style      76
Axioms, inner product      76
Axioms, vector space      76
Bach, Johann Sebastian      21
Beeckman, Isaac      3
Beeckman, Isaac, law      3
Beltrami, Eugenio      119 124
Beltrami, Eugenio, assumed Euclidean space      125
Berkeley, George      99
Bernoulli, Daniel      190
Bezout, Etienne      45
Black holes      125
Bolyai, Janos      115 125
Bombelli, Rafael      32 36
Bosse, Abraham      62
Brewer Dictionary      207
Broombridge      138
Bruegel, Pieter      179
Bryson, Bill      55
Burke, Robert      138
Calculus      96 99
Calculus and elliptic functions      191
Calculus, infinitesimal      100 200
Calculus, notation      99
Calcut, Jack      168
Cantor, Georg      201
Cantor, Georg, and diagonal argument      204
Cantor, Georg, and the continuum problem      209
Cantor, Georg, and transcendental numbers      206
Cantor, Georg, discovered uncountability      202
Cardano, Girolamo, Ars Magna      30 41
Cardano, Girolamo, formula      32 34
Carroll, Lewis      98
Cassiodorus      9
Cayley, Arthur      62 144
Circular functions      190
Claudius Ptolemy      107

Cohn-Vossen, Stefan      151
Coincidence      61
Coincidence and laws of algebra      72
Coincidence, projective explanation      67
Coincidence, projective explanation, via Pappus and Desargues      72
Columbus, Christopher      101 156
Common divisors      12
Common measure      1
Common measure by Euclidean algorithm      16
Commutative law      70
Commutative law, abandoned by Hamilton      136
Commutative law, fails for quaternions      138
Complex conjugate      141
Complex curves      45
Complex, factors      44
Complex, functions      45 191
Complex, numbers      see numbers complex
Complex, origin of word      44
Complex, period      191
Complex, points      45
Congruent triangles      73
Conic sections      62 88
Conic sections, equations of      88
Conjugation map      144
Continued fraction      19
Continued fraction, for $\sqrt{2}$       19
Continuum problem      209
coordinates      51 127
Coordinates and the parallel axiom      51 52
Coordinates, origin of      51 87
Cosine      42 190
Cosine, hyperbolic      116
Costrtizione legillima      55
Countability      200
Countability of algebraic numbers      200
Countability of rational numbers      200
Countability, implies zero length      203
Coxeter, Harold Scott MacUonald      152
cube      144
Cube and 3-torus      188
Cube n-dimensional      153
Cube, symmetry      145
Cubic curves      62
Cubic equations      30 41
Cubic equations, Cardano formula      32
Curvature      101
Curvature and angle sum      112
Curvature and infinitesimals      126
Curvature of a circle      114
Curvature of a smooth curve      116
Curvature of a sphere      114
Curvature of a surface      116
Curvature of latitude circles      105
Curvature, constant negative      118
Curvature, detection      108
Curvature, Gaussian      114 117
Curvature, Gaussian, negative      117
Curvature, Gaussian, of cylinder      117
Curvature, Gaussian, positive      117
Curvature, Gaussian, zero      117
Curvature, negative      116
Curvature, principal      117
Curvature, variable      125
Curved space      xi 108
Cutting and pasting      11 78
Cylinder      109
Cylinder and the plane      180
Cylinder as a periodic plane      181 182
Cylinder, curvature of      117
Cylinder, flatness of      109 182
Cylinder, geodesies on      109 177 182
Cylinder, locally like a plane      109 180
Cylinder, non-Euclidean      119
Cylinder, seal      181
Cylinder, three-dimensional      177 183
Cylinder, two-sided polygon on      177
Dante Alighieri      101 103
De la Hire, Phillipe      62
Decimal fractions      13
Dedekind, Richard      172
Dedekind, Richard, and $\mathbb{R}$       201
Dedekind, Richard, and ideals      173
Dedekind, Richard, Continuity and Irrational Numbers      201
Dedekind, Richard, cuts      202
Definition of line      202
Degree      44
Dehn, Max      11
del Ferro, Scipione      30
Desargues, Girard      62
Desargues, Girard, theorem      63
Desargues, Girard, theorem, as an axiom      65
Desargues, Girard, theorem, spatial version      63
Descartes, Rene      41
Descartes, Rene, and conic sections      88
Descartes, Rene, and length of curves      77 208
Descartes, Rene, coordinate geometry      51 71
Descartes, Rene, factor theorem      41
Descartes, Rene, Geometry      41 77
Diagonal argument      203
Diagonal argument for uncountability      204
Diagonalization      see diagonal argument
DIMENSION      127 135
Diophantus      34
Diophantus, "product" of triangles      35
Diophantus, Arithmetica      34 170
Diophantus, two-square identity      34 130 140
Diophantus, two-square identity, and multiplicative norm      167
Discriminant      30
Distance and absolute value      38
Distance in the hyperbolic plane      123
Distance in the plane      38
Distance on the pseudosphere and disk      121
Distributive law      26 70
Distributive law for areas      28
Distributive law for quaternions      137
Distributive law in Euclid      28
Distributive law in Greek geometry      27
Distributive law, used by Stevin      27
Divine Comedy      101
Divisibility      12 158
Division of Gaussian integers      164
Division of positive integers      158
Division property of Gaussian integers      165
Division property of integers      159
Division with remainder      158
Division with remainder in Gaussian integers      164
Division with remainder in positive integers      159
dodecahedron      144
Dodecahedron and dodecahedral space      189
Dodecahedron, symmetry      146
Duerer, Albrecht      48
e      43
e is transcendental      207
Eight-square identity      140
Einstein, Albert      127
Elements      12
Elements and regular polyhedra      144
Elements, Book II      28
Elements, Book IX      159
Elements, Book V      12 13
Elements, Book VII      12 16 161
Elements, Book X      18
Elements, Book XII      81
Elements, unique prime factorization      155
Elements, unstated assumptions      126
ellipse      191
Elliptic functions      191
Elliptic functions as functions on 2-torus      191
Elliptic functions, periodicity of      191
Elliptic integrals      191

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