A Smoother Pebble: Mathematical ExplorationsThis book takes a novel look at the topics of school mathematics--arithmetic, geometry, algebra, and calculus. In this stroll on the mathematical seashore we hope to find, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This book assembles a collection of mathematical pebbles that are important as well as beautiful. |
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... Tubeland . The cosmologists of Tubeland consider ques- tions that have analogies in our own universe . • The ascendancy of graphs in the twentieth century . · Algebra - arithmetic by other means . • Solving equations with passion ...
... Tubeland . The cosmologists of Tubeland consider ques- tions that have analogies in our own universe . • The ascendancy of graphs in the twentieth century . · Algebra - arithmetic by other means . • Solving equations with passion ...
Page vii
... Intervals , Scales , and Tuning Pythagorean tuning .. Approximating m octaves with n fifths Equal - tempered tuning . 5 6 8 13 15 16 18 20 22 24 26 33 35 36 39 44 45 46 49 49 51 54 II The Shape of Things 4 Tubeland Curvature of Smooth.
... Intervals , Scales , and Tuning Pythagorean tuning .. Approximating m octaves with n fifths Equal - tempered tuning . 5 6 8 13 15 16 18 20 22 24 26 33 35 36 39 44 45 46 49 49 51 54 II The Shape of Things 4 Tubeland Curvature of Smooth.
Page viii
... Tubeland - A fantasy 66 68 Triangular excess 71 Euclidean Geometry 73 The parallels axiom 75 Graphs Non - Euclidean Geometry Models of non - Euclidean geometries 5 The Calculating Eye The need for graphs . " Materials " for graphs ...
... Tubeland - A fantasy 66 68 Triangular excess 71 Euclidean Geometry 73 The parallels axiom 75 Graphs Non - Euclidean Geometry Models of non - Euclidean geometries 5 The Calculating Eye The need for graphs . " Materials " for graphs ...
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... Tubeland . The efforts of Tubelanders to understand their world is a re- flection of the efforts of our scientists to understand ours . Question : What geometric device was unknown in 1800 , a promising innovation in 1900 , and a ...
... Tubeland . The efforts of Tubelanders to understand their world is a re- flection of the efforts of our scientists to understand ours . Question : What geometric device was unknown in 1800 , a promising innovation in 1900 , and a ...
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Contents
Graphical Solutions | 129 |
Quadratic Equations | 130 |
Secrecy Jealousy Rivalry Pugnacity and Guile | 135 |
Solving a cubic equation | 138 |
Symmetry Without Fear | 142 |
Symmetries of a Square | 145 |
The Group Axioms | 148 |
Isometrics of the Plane | 150 |
| 22 | |
| 24 | |
| 26 | |
The Music of the Ratios | 33 |
Acoustics | 35 |
The rotating circle | 36 |
Waveforms and spectra | 39 |
Psychoacoustics | 44 |
Consonance versus dissonance | 45 |
Critical bandwidth | 46 |
Intervals Scales and Tuning | 49 |
Approximating m octaves with n fifths | 51 |
Equaltempered tuning | 54 |
Tubeland | 61 |
Curvature of Smooth Curves | 62 |
Curves embedded in two dimensions | 63 |
Curves embedded in three dimensions | 64 |
Curvature of Smooth Surfaces | 65 |
Gaussian curvature Extrinsic definition | 66 |
Tubeland A fantasy | 68 |
Triangular excess | 71 |
Euclidean Geometry | 73 |
The parallels axiom | 75 |
Models of nonEuclidean geometries | 77 |
The Calculating Eye | 82 |
Graphs | 84 |
The need for graphs | 85 |
Materials for graphs | 86 |
Clever people invented graphs | 89 |
Coordinate Geometry | 93 |
Synthetic versus analytic | 94 |
Synthetic and analytic proofs | 95 |
Straight lines | 99 |
Conic sections | 101 |
Algebra Rules | 111 |
Algebra Anxiety | 112 |
Arithmetic by Other Means | 115 |
Symbolic algebra | 116 |
Algebra and Geometry | 120 |
Aljabr | 121 |
Square root algorithms | 122 |
The Root of the Problem | 128 |
Patterns for Plane Ornaments | 151 |
Wallpaper watching | 158 |
The Magic Mirror | 160 |
The Magic Writing | 162 |
On the Shoulders of Giants | 167 |
Integration Before Newton and Leibniz | 168 |
Circular reasoning | 170 |
Completing the estimate of pi | 171 |
Differentiation Before Newton and Leibniz | 172 |
Descartess discriminant method | 173 |
Fermats difference quotient method | 176 |
Galileos Lute | 177 |
The inclined plane | 179 |
SixMinute Calculus | 184 |
Preliminaries | 185 |
Functions | 186 |
Limits | 188 |
Continuity | 189 |
The Damaged Dashboard | 191 |
The broken speedometer | 193 |
The derivative | 194 |
The broken odometer | 199 |
The definite integral | 201 |
Roller Coasters | 206 |
Time of descent | 208 |
RollerCoaster Science | 212 |
The Simplest Extremum Problems | 213 |
The lifeguards calculation | 215 |
A faster track | 217 |
A roadbuilding project for three towns | 219 |
Inequalities | 220 |
The inequality of the arithmetic and geometric means | 221 |
Cauchys inequality | 223 |
The geometry of the cycloid | 227 |
A differential equation | 228 |
The restricted brachistochrone | 230 |
The unrestricted brachistochrone | 235 |
Glossary | 243 |
Notes | 249 |
References | 257 |
Index | 261 |
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Common terms and phrases
achieves algebra algorithm ancient Greek angle antiderivatives approximation arbitrary arithmetic assertion Babylonian BAFS brachistochrone problem calculus called Cardano century chapter computation concept cone continued fraction critical bandwidth cubic equations curve cycloid defined definite integral derivative Descartes descent difference quotient discussion dissonance distance down-up ramp Egyptian equal Euclid's Euclidean geometry example fact fifths Flatland formula frequency function Galileo Gaussian curvature glide reflection graph hyperbolic inclined plane inequality instantaneous velocity intersecting interval inverse Leibniz line segment magnitudes mathematical mathematicians mean velocity method multiplication musical natural numbers Newton obtain octaves odometer osculating circle parallels axiom partials proof Proposition Pythagorean Pythagorean comma Pythagorean tuning quadratic ratio real numbers rectangle respectively rotation sexagesimal shown in Figure shows side simple tones slope smooth downward ramp solution solving speedometer square root square wave straight line surface symbolic symmetry Table tangent Tartaglia theorem tion triangle Tubeland unit fractions variable vertical
Popular passages
Page 83 - the words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. . . . The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily
Page 1 - I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
Page 142 - Tyger! Tyger! burning bright In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry? In what distant deeps or skies Burnt the fire of thine eyes? On what wings dare he aspire? What the hand dare seize the fire? And what shoulder, & what art, Could twist the sinews of thy heart?
Page 24 - Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Page 61 - Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows • — only hard and with luminous edges — and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said " my universe " : but now my mind has been opened to higher...
Page 23 - A ratio is a sort of relation in respect of size between two magnitudes of the same kind. 4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Page 112 - When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean — neither more nor less." "The question is," said Alice, "whether you can make words mean so many different things.
Page 33 - ... they saw that the modifications and the ratios of the musical scales were expressible in numbers; - since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.
Page 249 - Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.
Page 61 - I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers who are privileged to live in Space.