- FLATLAND: A Romance of Many Dimensions
- A Square (Edwin A. Abbott) Basil Blackwell, Oxford. 1932.
Introduction by William Garnett.
Preface to the Second & Revised Edition,1884, by the Editor.
Part 1 THIS WORLD - Of the nature of Flatland / Climate & Houses in Flatland / Inhabitants of Flatland / The Women / Methods of recognizing one another / Concerning Irregular Figures / The Ancient Practice of Painting / Universal Colour Bill / Suppression of the Chromatic Sedition / Our Priests / Doctrine of our Priests.
Part II OTHER WORLDS - A vision of Lineland / Explaining the nature of Flatland / A Stranger from Spaceland / Its Mysteries / The Sphere / What I saw in Spaceland / The Sphere encourages me in a Vision / Trying to teach the Theory of Three Dimensions to my grandson/ How I tried to diffuse the Theory by other means, and the result. 102 pp.
- FLATLAND: A Romance of Many Dimensions
- A. Square (Edwin A. Abbott) Dover Publications, sixth ed. 1952.
Introduction by Banesh Hoffmann. 103pp.
- BEYOND THE THIRD DIMENSION: Geometry, Computer Graphics, & Higher Dimensions
- Thomas F. Banchof. Scientific American Library/W.H.Freeman, 1990.
The concept of dimensions is a theme that threads its way through mathematics and into the world beyond. Dimension does not have to be spatial, but could represent time, temperature, weight, energy, or other variables, and is of practical significance not only in mathematics, but also in physics, geology, medicine, and modern art. Ranging from Egyptian pyramids to the 19th century satire Flatland and the paintings of Salvador Dali, Banchoff recounts our long fascination with extra-dimensional spaces, shapes and structures. He explains how geometers, scientists, philosophers, and artists have explored higher dimensions through metaphor, analogy, and more formal methods of coordinate geometry, and shows how computer graphics enables us to grasp concepts previously beyond our reach. Further readings; illustration sources; index. 210 pages.
- THE FOURTH DIMENSION - and how to get there
- Rudy Rucker. Rider & Co, 1985. Penguin 1986.
Explores the whimsical world of Edwin Abbott's Flatland and then plunges into spaces above three, reaching the infinite dimensions of Hilbert space. Sets puzzles en route, with Answers; Bibliography and index. 228 pp.
- MIND TOOLS: The Mathematics of Information
- Rudy Rucker. Houghton Mifflin, 1987; Penguin Books 1988.
Mathematics is the study of pure pattern, and everything in the cosmos is a kind of pattern. The patterns of mathematics can be grouped into five archetypes: Number, Space, Logic, Infinity and Information. Information is the master concept of the computer age, which throws a completely new light on the other age-old concepts. The Five Modes of Thought: Maths as information / Number & Space / Logic & Infinity / Psychological Roots of Mathematical Concepts / Information & Communication / A History of Ideas.
1. Number: Zero & One / Numbers & Logs / Number patterns / Numerology, Numberskulls and Crowds / The Mind Reckoner / Words as Numbers / Limits of Knowledge.
2. Space: Math Space & Real Space / Tiles, Cells, Pixels & Grids / Algebraic Curves/ Wiggles and Whorls / Endless Complexity / Fractals / Life is a fractal in Hilbert Space / Hilbert Space.
3. Logic: Laws of Thought / Syllogism / Symbolic Logic / Exploring Logical Space / Godel's Theorem / Turing Machines / Unsolvable Problems / The Ocean of truth.
4. Infinity & Information: How Big is Infinity? / Info & the Continuum Problem / Infinitesimals in Perspective / Algorithmic Complexity / Inconceivability / Runtime / Everything is Information.
Index. 328 pp.
- THE FOURTH DIMENSION & NON-EUCLIDEAN GEOMETRY IN MODERN ART
- Linda Dalrymple Henderson. Princeton University Press, 1983.
During the first three decades of this century, the concept of the fourth dimension had a liberating effect on artists, from Cubism to Surrealism. Challenging art historians who attribute these concerns with new concepts of space to the influence of relativity theory, Henderson finds their source in the widespread contemporary interest in spaces beyond immediate sensory perception, and provides a history of the popularisation on the n-dimensional and non-Euclidean geometries during the 19th century.
Appendices include A Chronology of American articles popularizing the New Geometries, 1877-1920.
Bibliography. Index. 453 pp.
- FOURFIELD: Computers, Art & the 4th Dimension.
- Tony Robbin. Bullfinch Press, 1992.
Foreword by Rudy Rucker; Introduction by Linda Dalrymple Henderson.
The understanding and appreciation of space as presented in works of art have changed over the centuries, and it the time is ripe for our perception of space to undergo another radical shift.
Einstein's Cave / Emotional Equations: Space in Maths & Art / Chinese Boxes & Shadows from Heaven/ Patterns in the Ether / Curvature: Lobofour & Nonclid / Time in Space: Three Enigmas of Space / Exebar Speaks! Appendix 1 Vincent van Gogh's Letter & 2 Hardware & Software Notes. Glossary, Bibliography, Index. 199 pp.
- THE MATHEMATICAL TOURIST: Snapshots of modern mathematics
- Ivars Peterson. W.H.Freeman, 1988.
Maps - the 4-colour problem / Primes / Doughnuts and knots / Flatland & Beyond / Monsters and Fractal Excursions / Dragons of Chaos / Game of Life / The Burden of Proof. Further reading: index 240pp.
- ISLANDS OF TRUTH: A Mathematical Mystery Cruise
- Ivars Peterson. W.H.Freeman, 1990.
Inside out & Knot Physics / Paving the Plane / Fractal Forgeries / Number Play / Shadows
of Chaos / Math of the Spheres. Further reading: index. 325 pp.
- NEWTON'S CLOCK - Chaos in the Solar System
- Ivars Peterson. W.H.Freeman, 1993.
Are the orbits of planets and other bodies stable and predictable, or are there elements affecting the dynamics of the solar system that defy calculation? Illustrates progress in celestial mechanics and places in historical context contemporary research on dynamical sytems and chaos, highlighting important individuals in the field and their accomplishments. Bibliography, index. 317pp.
- A HISTORY OF MATHEMATICS
- Carl B. Boyer, rev. by Uta C. Merzbach. John Wiley, second ed. 1991.
Origins / Egypt / Mesopotamia / Ionia & the Pythagoreans / The Heroic Age / Plato & Aristotle / Euclid of Alexandria / Archimedes of Syracuse / Apollonius of Perga / Greek Trigonometry & Mensuration / Revival & Decline of Greek Mathematics / China & India / The Arab Hegemony / Europe in the Middle Ages / The Renaissance / Prelude to Modern Mathematics / Fermat & Descartes / Newton & Leibniz / The Bernoulli Era / Euler / French Revolution / Gauss & Cauchy / Geometry / Analysis / Algebra / Poincare & Hilbert / Aspects of the 20th Century. References; bibliography. Appendix: Chronological Table. Index. / 715 pp.
- THE MATHEMATICAL EXPERIENCE
- Philip J Davis & Reuben Hersh. Birkhauser, 1980. Penguin Books 1990.
The mathematical landscape--Brief chronological table to 1910, and Classification of
Mathematics: 1868 to 1979 compared. Varieties of mathematical experience--A
physicist looks at maths; Shafarevitch and the new Neoplatonism; Unorthodoxies. Outer
Issues--why mathematics works; mathematical models; utility: variety of mathematical
uses. Inner issues--symbols, abstraction, generalisation, formalisation, existence, proof,
infinity: the stretched string, the coin of Tyche, aesthetic component, pattern, order &
chaos; algorithmic vs dialectic mathematics; the drive to abstraction; mathematics as
enigma; unity within diversity. Selected topics--Group theory & classification of finite
simple groups; Prime number theorem; non-Euclidean geometry & non-Cantorian set
theory; Nonstandard analysis; Fourier analysis. Teaching & learning; pedagogy; Polya's
craft of discovery; New maths - an application of the Lakatos Heuristic; comparative
aesthetics; nonanalystic aspects. From certainty to fallibility -- Platonism, Formalism,
Constructivism; the Euclid myth; Foundations, found and lost; Formalist philosophy;
Lakatos & the philosophy of Dubitability. Mathematical Reality--Riemann Hypothesis;
and ; models, computers, and Platonism; intuition; 4-dimensional intuition; true facts
about Imaginary Objects.
Glossary, bibliography, index. 440 pp.
- DESCARTES' DREAM: The World According to Mathematics
- Philip J. Davis & Reuben Hersh. Harcourt Brace Jovanovich, 1986,
Penguin Books 1990.
An incisive look at how mathematics is applied in today's world.
I. This Mathematized World: Descartes' Dream / Where the dream stands today / Limits of mathematics/ Drowning in Digits? / Stochastized World / Feedback & Control: Equilibrium Machine / Computer Graphics & Possibility of High Art.
II. Social Tyranny of Numbers: Maths & Rhetoric / Criterion Makers: Maths & Social Policy / Computerization of Love / Testing / Maths as a Social Filler / "Marxian" analysis of the role of computing in organizations.
III. Cognition & Computation: Functions of Applied Maths / Intellectual Components of Technology, Maths & Computation / Metathinking / Scientific Computation / Why should I believe a computer / Whorfian Hypothesis / The Programming Milieu.
IV. Perspectives Through Time: Time & Maths / Non-Euclidean Geometry & Ethical Relativism / Unreasonable effectiveness of computers - are we hooked?
V. Maths & Ethics: Platonic Mathematics meets Platonic Philosophy of Religion / The Computer Thinks: An interpretation in Medieval mode / Maths & the end of the World.
VI. Personal Meanings Maths & Imposed Reality/ Loss of Meaning through Intellectual Processes: Mathematical Abstraction.
VII. Envoi. Bibliography. Index. 321 pp.
- THE BIRTH OF MATHEMATICS IN THE AGE OF PLATO
- François Lasserre. Hutchinson, 1964.
Professor Lasserre chooses the most important moment on the long history of discoveries
in mathematics--the first point at which mathematics was converted from a science of
concrete things to one based on abstractions. A change extending over several
generations, but which passed through a crucial phase in the years following its
introduction to the curriculum of Plato's Academy. It was then that many fundamental
ideas were defined in which a modern mathematician can still recognise his own modes
of thought. By examining the progress of mathematical thought in the time of Plato he
seeks to explain the beginnings of modern mathematics. Index. 191pp.
- MATHEMATICS FOR THE MILLION
- Lancelot Hogben. George Allen & Unwin, 1936. Reprinted 1943.
Mathematics, the Mirror of Civilization / Maths in Prehistory / Translating Number
Language / What you can do with Geometry / Beginnings of Arithmetic / Trigonometry
/ Algebra / Spherical Triangles/ Graphs / How Logarithms were discovered / Calculus /
Statistics, or the Arithmetic of Human Welfare.
Epilogue on Science; Appendices; Tables; Answers; Index. 686 pp.
- MATHEMATICS: A HUMAN ENDEAVOUR
- Harold R. Jacobs. W.H. Freeman, 1970.
The mathematical way of thinking / Number sequences / Functions and their graphs /
Large numbers and logarithms / Regular Polygons / Mathematical curves / Some methods
of counting / Mathematics of chance / Introduction to statistics / Topics in topology /
Some fundamental ideas. Index. 529pp.
- GEOMETRY
- Harold R. Jacobs. W.H. Freeman, 1974.
Introduction: Euclid, the Surfer, and the Spotter / Nature of deductive reasoning /
Fundamental ideas: lines and angles / Basic postulates & theorems / Congruent triangles
/ Transformations / Inequalities / Parallel lines / Quadrilaterals / Area / Similarity / Right
triangle / Circles / Concurrence Theorems / Regular polygons & the circle / Geometric
solids / Non-Euclidean geometries. Index. 701 pp.
- THE LANGUAGE OF MATHEMATICS
- Frank Land. John Murray, 1960.
Numbers / Systems of Units / Multiplication & Division / Extending the Number System
/ Time and the Calendar / Simple & Compound Units / Algebra / Number Sequences &
their Graphs / Logs, Pianos & Spirals / Squares, Parabolas & Telescopes / Geometry /
Shapes & Sizes / Fibonacci Sequence & Golden Section / Statistics. Index. 264 pp.
- MATHEMATICS AND THE UNEXPECTED
- Ivar Ekeland. U.of Chicago Press, 1988.
Poincaré in his famous book on celestial mechanics showed that such relatively simple
systems as three bodies moving under the control of gravity as given by Newton's
gravitational law could generate behaviour of incalculable complexity and even
disorder. The mathematics of long-time or asymptotic behaviour only describes what
happens in the long run, but in important examples in physics, the long run is what
actually is observed.
The Music of the Spheres: Marvel of Kepler's Laws; Celestial Mechanics;Classical
Determinism / The Shattered Crystal: Impossible calculations; Poincaré's contribution;
Deterministic but random; Unstable but stable / The Comeback of Geometry: Word of
warning; Dissipative systems; Catastrophes;Theory; A critique. Back to the Beginning.
Appendices: Prelude & Fugue on a Theme by Poincaré / The Feigenbaum Bifurcation.
Bibliography, index. 146pp.
- BEYOND NUMERACY: an uncommon dictionary of mathematics
- John Allen Paulos. Viking 1991.
Mathematical Accent / Algebra--Some Basic Principles / Analytic Geometry / Arabic
Numerals / Areas & Volumes / Binary numbers & Codes / Calculus / Chaos Theory /
Coincidences / Combinatorics, Graphs, and Maps / Complexity of Programs /
Computation & Rote / Correlation. Intervals, and Testing/ Differential Equations / E /
Mathematics in Ethics / Exponential Growth / Fermat's Last Theorem / Mathematical
Folklore / Fractals / Functions / Game Theory / Gödel & His Theorem / Golden
Rectangle, Fibonacci Sequences / Groups and Abstract Algebra / Human Consciousness,
its Fractal Nature / Humour & Mathematics / Imaginary & Negative numbers /
Impossibilities--three old, three new / Mathematical Induction / Infinite Sets / Limits /
Linear Programming / Matrices & Vectors / Mean, Median, & Mode / Möbius Strips
&Orientability / Monte Carlo Method of Simulation / Multiplication Principle /
Music, Art & Digitalization / Non-Euclidean Geometry / Notation /
OULIPO--Mathematics in Literature / Partial Orderings & Comparisons / Pascal's
Triangle / Philosophy / Pi / Platonic Solids / Prime Numbers / Probability / Pythagorean
Theorem / QED, Proofs & Theorems / Quadratic & other Formulas / Quantifiers in
Logic / Rational & Irrational Numbers / Recursion--from Definitions to Life/ Russell's
Paradox / Scientific Notation / Series--Convergence & Divergence / Sorting & Retrieving
/ Statistics--Two Theorems / Substitutability and More on Rote / Symmetry & Invariance
/ Tautologies & Truth Tables / Time, Space & Immensity / Topology / Trigonometry
/ Turing's Test, Expert Systems / Variables & Pronouns / Voting Systems / Zeno & Motion.
Chronological listing of the 'Top Forty'
Mathematicians, Suggested Readings. Index. 285 pp.
- MATHEMATICS & MATHEMATICIANS - 1
- D. Dedron & J. Itard. Open University, 1978.
The Lawgivers of Geometry: Euclid, Archimedes, Apollonius / The Greeks, Hindus,
Arabs, Byzantines, and the West / Italian triumphs: Cardan, Tartaglia / Father of Modern
Mathematics: François Viète / Napier and Logarithms / The Golden Age: Fermat,
Roberval, Descartes, Desargues, Pascal / Classical Work: Newton, Leibniz, Clairault,
D'Alembert, Lagrange, Laplace, Monge. Biographical references, Index of names.
Index. 325pp.
- MATHEMATICS & MATHEMATICIANS - 2
- D. Dedron & J. Itard. Open University, 1978.
Written numbers & Numerical calculations: Egypt, Mesopotamia, Greece. Latins,
Hindus, Arabs, amd Western Europeans. Algebraic Notations & First Degree Problems:
Notations, First Degree, Arithmetical & combinatorial problems. Second Degree
Problems: Square roots, Quadratic equations, Euclid, Applications to the conic sections,
Book X of Euclid's Elements. Pythagoras's Theorem. Trigonometry. Duplication of
the cube and trisection of the angle. Squaring the Circle. Bibliographical references,
index. 222pp.
- MATHEMATICIAN'S DELIGHT
- W. W. Sawyer. Penguin, 1943. Reprinted 1971.
Attempts to convince the general reader that mathematics is not a forbidding science but
an attractive mental exercise. Index. 238 pp.
- VISION IN ELEMENTARY MATHEMATICS :
Introducing Mathematics - 1
- W.W. Sawyer. Penguin, 1964, reprinted 1970.
Even and Odd / Divisibility / An unorthodox point of entry / Tricks, bags, & Machines
/ Words, Signs & Pictures / Sudden appearance of a Practical Result / Miniature problem
in Design / Investigations / Routines of Algebra I & II /Graphs / Negative numbers /
Fractions. Answers. 346 pp.
- THE SEARCH FOR PATTERN: Introducing Mathematics - 3
- W.W. Sawyer. Penguin, 1970.
Maths through the Hand / Maths through the Eye / Making the curved straight / Some
notes on Algebra/ Questions of Notation / Resistances, Condensers, Springs and Baths
/ Factors & the remainder theorem/ Method of discovery / Discovering the Binomial
Theorem / Algebra as a key to geometry / on Quadratic Equations / Transformations /
Algebra & Statistics / Square roots, Irrationals and all that. Symbols; Jargon; Answers to
exercises. 349 pp.
- A PATH TO MODERN MATHEMATICS: Introducing Mathematics - 4
- W.W. Sawyer. Penguin, 1966, reprinted 1971.
The Arithmetic of Space / Geometrical Dictionary / Maps & matrices / Hidden Simplicity
/ Benefits from Equations / Towards Applications / Towards Systematic Classification /
On Linearity / What is a Rotation? Metric & Banach Spaces. 224 pp.
- THE PSYCHOLOGY OF LEARNING MATHEMATICS
- Richard R. Skemp. Penguin, 1971.
In Part A, Dr Skemp first looks at the thought processes people adopt when they do
mathematics, and analyses it psychologically. In Part B, he applies the ideas of the first
part to some of the basic topics of mathematics, and includes an overview in the form
of schematic diagrams in which conceptual hierarchies are represented spatially,
covering chapters 8 to 15. Bibliography, index.319 pp.
- RIDDLES IN MATHEMATICS: A Book of Paradoxes
- Eugene P. Northrop. English UP 1944, rev. 1961. Pelican, 1964.
What is a paradox? / Paradoxes for everyone / Paradoxes in arithmetic -- largest known
prime; Fermat numbers; Division of circle; Perfect numbers; Positional notation; Binary
system / Paradoxes in Geometry -- Optical illusions; Cutting a square; Fibonacci series;
Leaf arrangements; Golden Section; Logarithmic spirals; Sunflower head; Dynamic
symmetry; Rolling discs, slabs and rollers; Cycloid family; Königsberg problem; Klein's
bottle; Möbius strips; Knots; Four-colour problem / Algebraic fallacies -- Misuses of
axioms; Illegal cancellation; Division by zero; Any two unequal numbers are equal;
Peculiar proportions; Contradictions in equations; Inequalities; Imaginary numbers /
Geometrical fallacies -- Any triangle is isosceles; Two perpendiculars from a point to
a line; A rightangle greater than itself; 45=60; Equality implies parallelism; A line
equal to part of itself; Two lengths whose sum is zero; Reasoning by analogy; Sum of
angles of a spherical triangle; Any number of perpendiculars from a point to a plane /
Paradoxes of the Infinite -- Achilles & the tortoise: is motion impossible?; Convergence
and divergence of infinite series; Oscillating sums; Summing a series to any desired
number; A point equal to a line; Proof of the parallel postulate; Pathological curves;
Deceptive limiting curves; Finite area and infinite length; Area-filling curve; Every point
a point of intersection; Comparing infinite classes; One-to-one correspondences; The
number of natural numbers, rational numbers, real numbers; A peculiar system of
arithmetic / Paradoxes in Probability--Measure of probability; D'Alembert's error;
Bertrand's box paradox; Picking a point at random; Random chord in a circle, planes in
space and points on a sphere; Life on Mars paradox; Weather forecasters; St Petersburg
paradox; Winning at roulette / Paradoxes in Logic -- Russell on maths & logic;
Epimenides and the liars; All rules have exceptions; Least integer problem; Autological
and heterological; Vicious circle; Theory of types; Greatest transfinite oe no greatest
transfinite? The class of all non-self-remembered classes; Richard paradox; Recent trends
in the foundations of mathematics; Paradoxes in Higher Mathematics -- Geometry and
trigonometry; Analytical geometry; Differential calculus; Integral calculus; Complex
numbers. Appendix on Chapter 2; Notes and references; index. 240pp.
- MATHEMATICS AND LOGIC: Retrospect & Prospects
- Mark Kac & Stanislaw Ulam. Praeger,1968, Pelican Books 1971.
The advance of technology has made it important that mathematics is seen as a way of
thinking, rather than a means of solving problems. The aim of this book is to show how
the mathematician thinks and what he is thinking about. Because of the simplicity and
elegance of mathematical symbolism, it would be difficult to show the depth of
mathematical thought without the use of some advanced techniques. This abridged
edition has been prepared by removing the more difficult mathematics.
1. Examples: The infinity of Primes / Irrationality of 2 / Approximation by Rational
Numbers / Transcendental Numbers: Cantor's argument / More proofs of Impossibility -
Sperner's Lemma / The Art & Science of Counting / Number System & Functions /
Elementary Probability / Measure / Probability revisited / Groups & Transformations /
Homology Groups / Vectors, Matrices & Geometry/ Special Theory of Relativity as
example of the geometric view in physics / Transformations, Flows & Ergodicity /
Iteration & Composition of Transformations / Proving the obvious.
2. Themes, Trends, and Syntheses.
3. Relations to other disciplines.
4. Summary and Outlook. Index. 204 pp.
- MATHEMATICS IN WESTERN CULTURE
- Morris Kline. OUP 1953, Pelican Books 1972.
A lively history of the cultural influence of mathematics -- ranging from Ancient Egypt,
where priests used their knowledge of the calendar to predict when the Nile would
flood, to Renaissance Italy where such artists as Uccello and Leonardo were the most
skilled mathematicians of their day, to the 20th C when Einstein's mathematical work
on relativity changed the whole course of science. Index. 543pp.
- MATHEMATICS AND THE IMAGINATION
- Edward Kasner & James Newman. USA pub.1940, Pelican Books 1968.
New names for old / Beyond the Googol / PIE (,i,e) - Transcendental & Imaginary /
Assorted Geometries: Plane & Fancy / Puzzles / Paradox Lost & Regained / Chance /
Rubber-sheet Geometry / Change & Changeability. Epilogue. Bibliography. Index. 332
pp.
- FACTS FROM FIGURES
- M.J. Moroney. Pelican,1951. Rev. reprint 1969.
Statistics Undesirable / Laws of Chance / Magic Lantern Technique / On the Average /
Scatter / Speeding up calculations / Fault-finding - the Binomial Distribution / The
Poisson Distribution / The Normal Distribution / Samples / Control Charts / How to be
a Good Judge - tests of significance/ Precise though vague - Estimation & Confidence
Limits / Association, Contingency, & Goodness of Fit -- the ² Distribution / Correlation,
cause & effect / Time series & fortune telling / Ranking Methods / Analysis of Variation
& Co-variation / Statistics Desirable. Bibliography, Answers, index. 472 pp.
- USE AND ABUSE OF STATISTICS
- W.J. Reichmann. Methuen 1961. Pelican Books 1964.
Age of statistics / Scope / Seeing things / Reality / Overworked Average / Persuasive
Percentage / Sense of proportion / What's in a name? / General & Aproximate / Cause
& effect / Artful advertising Is there an Index? / Time Series & Arithmancy / Probability
/ Normal & Other Distributions / Sampling / Populations & Samples / Popping the
Question / Matter of Opinion / Under Control / Linear Programming & Games.
Appendices: Coefficient of Correlation / Standard Deviation / Least Squares Method /
Geometric Indices & the Time Reversal Test / Factorial Designs / Latin Squares / Standard
Error of the Difference / Analysis of Variance / Chi-Square (²) Test. Glossary, index. 345
pp.
- HOW TO LIE WITH STATISTICS
- Darrell Huff. Gollancz, 1954.
Averages and relationships and trends and graphs are not always what they seem; there
may be more in them than meets the eye, and there may be a good deal less... This book
is a sort of primer in ways to use statistics to deceive. The crooks already know these
tricks; honest men must learn them in self-defence. 142 pp.
- HOW TO TAKE A CHANCE: A Lighthearted Introduction to the Laws of Probability
- Darrell Huff. Norton, 1959. Pelican Books 1970.
The author shows us all how to calculate our risks. With Probability Problems & Puzzles.
141 pp.
- MATHEMATICAL MODELS
- H.M. Cundy & A.P. Rollett. Oxford UP, second ed.1961, reprinted 1976.
Use & construction of models / Models in plane geometry / Polyhedra / Other models
in solid geometry / Mechanical models / Models for logic and computing. Bibliography.
Index. 286 pp.
- HOW TO SOLVE PROBLEMS: Elements of a Theory of Problems & Problem Solving
- Wayne A. Wickelgren. Freeman, 1974.
Introduction / Problem theory / Inference / Classification of action sequences / State
evaluation & hill climbing / Subgoals / Contradiction / Working backward / Relations
between problems / Topics in mathematical representation / Problems from mathematics,
science, & engineering. References. Index. 262 pp.
- AHA! INSIGHT
- Martin Gardner. Scientific American/W.H.Freeman, 1978.
We are entering an age in which there will be increasing temptation to solve all
mathematical problems by writing computer programs. . . It would be a sad day if
human beings, adjusting to the computer revolution, became so intellectually lazy that
they lost their power of creative thinking. A collection of puzzles to exercise and
improve ability in the technique of problem solving.... 1. Combinatorial aha! 2.
Geometry aha! 3. Number aha! 4. Logic aha! 5. Procedural aha! 6. Word aha!
Selected references. Answers to Posed Problems. 179 pp.
- MATHEMATICAL PUZZLES & DIVERSIONS
- Martin Gardner. Simon & Schuster, 1959. Pelican Books reprint 1971.
Hexaflagons / Magic with a Matrix / Nine Problems / Ticktacktoe / Probability Paradoxes
/ The Icosian Game & the Tower of Hanoi / Curious topological models / Game of HEX
/ Sam Loyd: America's greatest puzzlist / Mathematical card tricks / Memorizing
numbers / Nine more problems / Polyominoes/ Fallacies / Nim & Tac Tix / Left or Right?
References for further reading. 155 pp.
- MORE MATHEMATICAL PUZZLES & DIVERSIONS
- Martin Gardner. S&S 1961. Penguin Books 1974.
The Five Platonic Solids / Tetraflexagons / Henry Ernest Dudeney: England's Greatest
Puzzlist / Digital roots / Nine Problems / Soma Cube / Recreational topology / Phi - the
Golden Ratio / The Monkey & the coconuts / Mazes / Recreational logic / Magic Squares
/ James Hugh Riley Shows Inc / Nine more problems / Eleusis: the Induction Game /
Origami / Squaring the square / Mechanical Puzzles / Probability & Ambiguity.
References for further reading. 187 pp.
- FURTHER MATHEMATICAL DIVERSIONS
- Martin Gardner. S&S 1969, Pelican Books1977.
Paradox of the Unexpected Hanging / Knots & Borromean Rings / Transcendental
Number e / Geometric dissections / Scarne on gambling / Church of the Fourth
Dimension / Eight problems / Matchbox game-learning machine / Spirals / Rotations &
Reflections / Peg Solitaire / Flatlands / Chicago Magic Convention / Tests of divisibility
/ Nine problems / 8 Queens & other chessboard diversions / A loop of string / Curves of
constant width / Rep-Tiles: Replicating figures on the plane / 37 Catch questions.
Bibliography. 255 pp.
- MATHEMATICAL CARNIVAL
- Martin Gardner. Knopf 1975. Pelican Books 1978.
Sprouts & Brussels Sprouts / Penny Puzzles / Aleph-null & Aleph-one /Hypercubes /
Magic Stars & Polyhedrons / Calculating Progidies / Lightning Calculators / Art of M.C.
Escher / Red-Faced Cube & other problems / Card Shuffles / Mrs Perkins' Quilt & Square-packing problems / Numerology of Dr. Fliess / Random numbers / Rising Hourglass &
other Physics problems / Pascal's Triangle / Jam, Hot & other games / Cooks & Quibble-Cooks / Piet Hein's Superellipse / How to trisect an angle.
Bibliography. 274 pp.
- MATHEMATICAL CIRCUS
- Martin Gardner. Knopf 1979. Allen Lane 1981.
Optical Illusions / Matches / Spheres & Hyperspheres / Patterns of induction / Elegant
Triangles / Random walks & gambling / Random walks on the plane and in space /
Boolean Algebra / Can machines think? / Cyclic numbers / Eccentric Chess / Dominoes
/ Fibonacci & Lucas Numbers / Simplicity /Rotating Round table / Solar System oddities
/ Mascheroni Constructions / The Abacus / Palindromes: Words & Numbers / Dollar Bills.
Bibliography. 272 pp.
- MATHEMATICAL MAGIC SHOW
- Martin Gardner. Knopf 1977. Viking 1984.
Introduction, with glossary /Nothing / More Ado about Nothing / Game Theory, Guess
it, Foxholes / Factorial Oddities / Cocktail Cherry & other problems / Double Acrostics
/ Playing cards / Finger arithmetic / Möbius Bands / Ridiculous questions / Polyhexes &
Polyaboloes / Perfect, Amicable, Sociable / Polyominoes & Rectification / Knights of the
Square Table / Dragon Curve & other problems/ Coloured Triangles & Cubes / Trees /
Dice / Everything. Bibliography. 284 pp.
- THE AMBIDEXTROUS UNIVERSE: Mirror Asymmetry & Time-Reversed Worlds
- Martin Gardner. USA 1964. Second edition, revised & updated, reprinted Pelican Books, 1986.
Mirrors / Lineland & Flatland / Solidland / Magic / Art, Music, Poetry & Numbers /
Galaxies, Suns & Planets / Plants & Animals / Asymmetry in animals / The human body
/ Sinistral Minority / Crystals / Molecules / Carbon / Living molecules / Origin of Life /
Origin of asymmetry / Fourth Dimension / Ozma Problem / Mach's Shock / Parity /
Neutrinos / Mr Split / Fall of Time Invariance / The Arrows of Time / Entropy / Time-reversed Worlds / Time-reversed Persons & Particles / Epilogue. Answers. Index. 293 pp.
- THE NEW AMBIDEXTROUS UNIVERSE: Symmetry & Asymmetry from Mirror Reflections to Superstrings
- Martin Gardner. W.H. Freeman, 1990. Third revised edition.
As above, plus Antiparticles / The Fall of Parity / Where's the Anti-matter? / What
happened to the Monopoles? / Early theories of matter / Spin / Superstrings. Further
reading. Answers. Index. 392 pp.
- WHEELS, LIFE & OTHER MATHEMATICAL AMUSEMENTS
- Martin Gardner. W.H. Freeman, 1983.
Wheels / Diophantine Analysis & Fermat's Last Theorem / Knotted Molecule / Alephs &
Supertasks / Nontransitive Dice & other probability paradoxes / Geometrical fallacies /
Combinatorics of paper folding / Set of Quickies / Ticktacktoe games / Plaiting
polyhedrons / Game of Halma / Advertising premiums / Salmon on Austin's dog / Nim
& Hackenbush / Golomb's Graceful Graphs / Charles Addams' Skier / Chess Tasks /
Slither, 3x+1, & other curious questions /Tricks with cards / Game of Life Parts I, II & III.
Name Index. 261 pp.
- KNOTTED DOUGHNUTS & OTHER MATHEMATICAL ENTERTAINMENTS
- Martin Gardner. W.H. Freeman, 1986.
Coincidence / Binary Gray code / Polycubes / Bacon's Cipher / Doughnuts: Linked &
Knotted / Tour of the Arrows / Napier's Bones / Napier's Abacus / Sim, Chomp &
Racetrack / Elevators / Crossing Numbers / Point Sets on a Sphere / Newcomb's Paradox
& Reflections / Reverse the Fish / Look-See Proofs / Worm Paths / Waring's Problems /
Cram, Bynum & Quadraphage / The I Ching / Laffer Curve/ Index of names. 278 pp.
- TIME TRAVEL & OTHER MATHEMATICAL BEWILDERMENTS
- Martin Gardner. W.H. Freeman, 1988.
Time Travel / Hexes & Stars / Tangrams / Nontransitive Paradoxes / Combinatorial Card
problems / Melody-making Machines / Anamorphic art / The rubber rope / Six
Sensational Discoveries / The Császár Polyhedron / Dodgem / Tiling with convex
polygons / Tiling with Polyominoes, Polyiamonds, & Polyhexes / Curious Maps / The
Sixth symbol / Magic Squares & Cubes / Block Packing / Induction & Probability /
Catalan Numbers / Fun with a pocket calculator / Tree-Plant problems. Index of names.
295 pp.
- PENROSE TILES TO TRAPDOOR CIPHERS
- Martin Gardner. W.H. Freeman, 1989.
Penrose Tiling / Mandelbrot's Fractals / Conway's Surreal Numbers / Back from the
Klondike / OULIPO/ Wythoff's Nim / Pool-Ball Triangles / Mathematical Induction &
Colored Hats / Negative Numbers / Cutting shapes into n congruent parts / Trapdoor
Ciphers / Hyperbolas / New Eleusis / Ramsey Theory/ From Burrs to Berrocal / Sicherman
Dice, the Kruskal Count & other curiosities / Raymond Smullyan's Logic Puzzles / The
Return of Dr. Matrix. Index. 311 pp.
- FRACTAL MUSIC, HYPERCARDS AND MORE...
- Martin Gardner. W.H. Freeman, 1992.
White, Brown & Fractal Music / Tinkly Temple Bells / Mathematical Zoo / Charles
Sanders Pierce / Twisted Prismatic Rings / Thirty Color Cubes / Egyptian Fractions /
Minimal Sculpture / Tangent Circles / Rotating Table & Other Problems / Does
Time Ever Stop? / Generalized Ticktacktoe / Psychic Wonders & Probability /
Mathematical Chess Problems / Gödel, Escher, Bach / Imaginary Numbers /
Pi & Poetry / More on poetry / Packing Squares / Chaitin's Omega.
Name Index. 327 pp.
- THE ARMCHAIR UNIVERSE: An Exploration of Computer Worlds
- A.K. Dewdney. W.H. Freeman, 1988.
1. Infinite Graphics: Mandelbrot Set / Wallpaper for the Mind. Mathemagadgets: Analog
gadgets / Gadgets revisited / Golomb Rulers / Hypercubes. 2. Artificial Intelligence &
Artificial Insanity: Conversations with RACTER / Facebender / Perceptron
Misconceptions / Checkers Program that never loses? / Automated Magic. 4.Life in
Automata: One-dimensional computers / 3-dimensional life / Busy beavers.
5.Puzzles & Wordplay: Bill's Baffling Burr, Coffin's Cornucopia & Engel's Enigma / Towers of Hanoi & Chinese Rings / Anagrams & Pangrams. 6. Stimulation through
Simulation: 5 Easy Pieces / A Cosmic Ballet / Sharks & Fish on the planet Wa-Tor /
Evolution of Flibs / Extinction of Tribolites & Survival of Smiths / 7. Core Wars: Core War
/ Core War Encore / First Core War Tournament. Bibliography. List of Suppliers. Index.
330 pp.
- THE MAGIC MACHINE: A Handbook of Computer Sorcery
- A.K. Dewdney. W.H. Freeman, 1990.
Prologue / A Word to Programmers & Apprentice Programmers.
SPELL 1: Conjuring up Chaos - Mandelbrot Magic / Visions of Julia / Mandelbrew &
Mandelbus / The STRANGE attractions of Chaos / CatchingBiomorphs. SPELL 2: Weird
Machines - Vehicles of Thought/ Apraphulian Wonder / Atomic Computers / Paradoxical
Gold. SPELL 3: Deus ex machina - Fractal Mountains & Graftal Plants / Hodgepodge
Reactions /Demons of Cyclic Space / Slow Growths Programmed Parties / Palmiter's
Protozoa. SPELL 4: Puzzling Landscapes - Mazes & Minotaurs People Puzzles / Panning
for Primes / Trains of Thought / Prosodic Programs / The Martian Dictionary. SPELL 5:
Mathemagical Movies - Special FX / Balls in Boxes / Invisible Professor. SPELL 6: Battles
of the Magi - Enigma & the Bombe / Computers in the Crypt / Core Wars / Attacks of the
Viruses. Further Readings, List of Suppliers, Name Index, Subject Index. 357 pp.
- THE TINKERTOY COMPUTER & OTHER MACHINATIONS
- A.K. Dewdney. W.H. Freeman, 1993.
Prologue: The hidden Agenda. Theme 1: Matter Computes. Tinkertoy Computer / Rope-and-pulley Wonder / The Infinite Brain / Invasion of the Insectoids / Building a Brain /
Dance of the Tur-mites. Theme 2: Matter Misbehaves. Microminiature Golf / Star Trek
Dynamics / Weather in a Jar / Portrait of Chaos / Designer Fractals / Fractal Workshop.
Theme 3: Mathematics Matters. Mathematical Morsels /Golygon City / Scanning the Cat
/ Rigid Thinking / Automated Math. Theme 4: Computers Create: Computer composer
/ Chaos in A Major / Mark V. Shaney / Face Space / Voltage Sculptures / Latticeworks
by Hand. Index. 238 pp.
- INTRODUCTORY COMPUTER SCIENCE: Bits of Theory, Bytes of Practice
- A.K. Dewdney. W.H. Freeman, 1996.
A Program called Chaos/ The Automated Teacher: Programs from Algorithms /
A Variable named Money / Electronic Visions / The Ground Floor in Computing /
Piling Up Data / Simulating the Information Highway / Public & Private Codes/
The Tortoise & the Hare / The Daily Planet / Turbo Pascal Advisory / Index. 368pp.
- FRACTALS: Form, Chance, & Dimension
- Benoit B. Mandelbrot. W.H. Freeman, 1977.
Fractals are a class of highly irregular shapes that have myriad counterparts in the real
world, such as islands, continents, coastlines and snowflakes. This unique geometrical
investigation brings several studies together into one integrated essay, and claims that
the geometry of fractals is necessary to describe geometric shapes in nature. The classic
fractals include Brownian paths, Cantor sets, and Koch curves. Hausdorff dimension is
the main parameter of a fractal. This dimension serves as an
excellent measure of irregularity and fragmentation, and gives meaning to ideas like curves of dimension greater than 1, and surfaces of dimension greater than 2.
Biographical & Historical Sketches. Mathematical Lexicon & Addenda. Bibliography.
Index. Index of Selected Dimensions. 365 pp.
- THE FRACTAL GEOMETRY OF NATURE: Updated and Augmented
- Benoit B. Mandelbrot, W.H. Freeman, 1982-83.
Replaces 1977 essay, involving new art, a few deletions, extensive rewriting, additions devoted to older work, and extensive additions devoted to new developments.
I. Introduction: Theme / The Irregular & Fragmented in Nature / Dimension, Symmetry, Divergence / Variations & Disclaimers.
II. Three Classical Fractals, Tamed: How long is the coast of Britain? / Snowflakes & Other Koch Curves / Harnessing the Peano Monster Curves / Fractal Events & Cantor Dusts.
III. Galaxies & Eddies: Fractal View of Galaxy Clusters / Geometry of Turbulence; Intermittency / Fractal Singularities of Differential Equations.
IV. Scaling Fractals: Length-Area-Volume relations / Islands, Clusters, and Percolation; Diameter-Number relations / Ramification & Fractal Lattices.
V. Non-Scaling Fractals: Surfaces with Positive Volume & Flesh / Trees; Scaling Residues; Nonuniform Fractals / Trees & the Diameter Exponent.
VI. Self-Mapping Fractals: Self-Inverse Fractals, Apollonian Nets & Soap / Cantor & Fatou Dusts; Self-Squared Dragons / Fractal Attractors & Fractal ("Chaotic") Evolutions.
VII. Randomness: Chance as a Tool in Model Making / Conditional Stationarity & Cosmographic Principles.
VIII. Stratified Random Fractals: Random Curds: Contact Clusters & Fractal Percolation / Random Chains & Squigs / Brownian Motion & Brown Fractals / Random Midpoint Displacement Curves.
IX. Fractional Brown Fractals: River Discharges; Scaling Nets & Noises / Relief & Coastlines / The Areas of Islands, Lakes & Cups. A Book-within-the-Book, in colour.
X. Random Tremas; Texture: Interval Tremas; Linear Lévy Dusts; Ordered Galaxies / Disc & Sphere Tremas: Moon Craters & Galaxies / Texture: Gaps & Lacunarity; Cirri & Succolarity / General Tremas, & the Control of Texture.
XI. Miscellany: Logic of Fractals in Statistical Lattice Physics / Price Change & Scaling in Economics / Scaling & Power Laws without Geometry / Mathematical Backup & Addenda.
XII. Of Men & Ideas: Biographical Sketches / Historical Sketches / Epilog: The Path to Fractals.
List of References / Index of Selected Dimensions / Index of Names & Subjects / Update Added in the Second Printing. 468 pages.
- FRACTALS, CHAOS, POWER LAWS: Minutes from an Infinite Paradise
- Manfred Schroeder. W.H. Freeman & Co, 1991.
Introduction: Einstein, Pythagoras & Simple Similarity / A Self-Similar Array of Self-Preserving Queens / A Self-Similar Snowflake / A New Dimension for Fractals / A Self-Similar Tiling & 'Non-Euclidean' Paradox / At the Gates of Cantor's Paradise / The Sierpinski Gasket / Sir Pinski's Game & Deterministic Chaos / Three Bodies Cause Chaos / Strange Attractors, Their Basins & a Chaos Game / Percolating Random Fractals / Power Laws from Alvarez to Zipf / Newton's iteration and how to Abolish Two-Nation Boundaries / Could Minkowski hear the Shape of a Drum? / Discrete Self-Similarity: Creases & Centre Folds /Golden & Silver Means & Hyperbolic Chaos / Winning at Fibonacci Nim /Self-Similar Sequences from Square Lattices / John Horton Conway's "Death Bet".
2. Similarity & Dissimilarity: More Than One Scale / To scale or not to scale: A bit of Biology & Astrophysics / Similarity in Physics: Some Astounding Consequences / Similarity in Concert Halls, Microwaves & Hydrodynamics / Scaling in Psychology / Acousticians, Alchemy & Concert Halls / Preference & Dissimilarity: Concert Halls Revisited.
3. Self-Similarity--Discrete, Continuous, Strict & Otherwise: The Logarithmic Spiral, Cutting Knives & Wideband Antennas / Simple Cases of Self-Similarity / Weierstrass Functions & a Musical Paradox / More Self-Similarity in Music: the Tempered Scales of Bach The Excellent Relations between the Primes 3, 5, & 7.
4. Power Laws: Endless Sources of Self-Similarity: The Sizes of Cities & Meteorites / A Fifth Force of Attraction / Free of Natural Scales / Bach composing on All Scales / Birkhoff's Aesthetic Theory / Heisenberg's Hyperbolic Uncertainty Principle / Fractional Exponents / Peculiar Distribution of the First Digit / Diameter Exponents of Trees, Rivers, Arteries & Lungs.
5. Noises: White, Pink, Brown & Black: Pink Noise / Self-Similar Trends on the Stock Market / Black Noises & Nile Floods / Warning: World Warming / Fractional Integration: A Modern Tool / Brownian Mountains / Radon Transform & Computer Tomography / Fresh & Tired Mountains.
6. Brownian Motion, Gambling Losses & Intergalactic Voids: Random Fractals Par Excellence. Brownian Beast Tamed / Brownian Motion as a Fractal / How Many Molecules? / Spectrum of Brownian Motion / The Gambler's Ruin, Random Walks & Information Theory / Counterintuition Runs Rampant in Random Runs / More Food for Fair Thought / The St Petersburg Paradox / Shannon's Outguessing Machine / The Classical Mechanics of Roulette & Shannon's Channel Capacity / Clustering of Poverty & Galaxies / Levy Flights through the Universe / Paradoxes from Probalistic Power Laws / Invariant Distributions: Gauss, Gauchy & Beyond.
7. Cantor Sets: Self-Similarity & Arithmetic Dust: A Corner of Cantor's Paradise / Cantor Sets as Invariant Sets / Symbolic Dynamics & Deterministic Chaos / Devil's Staircases & a Pinball Machine / Mode Locking in Swings & Clocks / Frustrated Manhattan Pedestrian / Arnold Tongues.
8. Fractals in Higher Dimensions & a Digital Sundial: Cartesian Products of Cantor Sets / Leaky Gasket, Soft Sponges & Swiss Cheeses / Cantor-Set Sundial / Fat Fractals.
9. Multifractals: Intimately Intertwined Fractals: Distributions of People & Ore / Self-Affine Fractals without Holes / Multifractal Spectrum: Turbulence & Diffusion-Limited / Aggregation / Viscous Fingering / Multifractals on Fractals / Fractal Dimensions from Generalised Entropies / Relation between the Multifractal Spectrum f() and the Mass Exponents (q) / Strange Attractors as Multifractals / A Greedy Algorithm for Unfavorable Odds.
10. Some Practical Fractals & Their Measurement: Dimensions from Box Counting / Mass Dimension / Correlation Dimension / Infinitely Many Dimensions / Determina tion of Fractal Dimensions from Time Series / Abstract Concrete / Fractal Interfaces Enforce Fractional Frequency Exponents / Fractal Dimensions of Fracture Surfaces / Fractal Shapes of Clouds & Rain Areas / Cluster Agglomeration / Diffraction from Fractals.
11. Iteration, Strange Mappings, & a Billion Digits for : Looking for Zeros & Encountering Chaos / The Strange sets of julia / Multifractal Julia Set / Beauty of Broken Linear relationships / Baker's Transformation & Digital Musical Chairs / Arnold's Cat Map / A Billion Digits for / Bushes & Flowers from Iterations.
12. A Self-Similar Sequence, the Logistic Parabola, & Symbolic Dynamics: Self-Similarity from the Integers / Logistic Parabola & Period Doubling / Self-Similarity in the Logistic Parabola / Scaling of the Growth Parameter / Self-Similar Symbolic Dynamics / Periodic Windows Embedded in Chaos / Parenting of New Orbits / Calculation of the Growth Parameters for Different Orbits / Tangent Bifurcations, Intermittency, & 1/f Noise /Case of Complete Chaos / Mandelbrot Set / Julia Sets of the Complex Quadratic Map.
13. A Forbidden Symmetry, Fibonacci's Rabbits & a New State of Matter: Forbidden Fivefold Symmetry / Long-range Order from Neighbourly Interactions / Generation of the Rabbit Sequence from the Fibonacci Number System / Self-Similar Spectrum of the Rabbit Sequence / Self-Similarity in the Rabbit Sequence / One-dimensional Quasiperiodic Lattice / Self-Similarity from Projections / More Forbidden Symmetries.
14. Periodic & Quasiperiodic Structures in Space --the Route to Spatial Chaos: Devil's Staircase for Ising Spins / Quasiperiodic Spatial Distributions / Beatty Sequence Spins / Scaling Laws for Quasiperiodic Spins / Self-Similar Winding Numbers / Circle Maps & Arnold Tongues / Mediants, Farey Sequences & the Farey Tree / Golden-Mean Route to Chaos.
15. Percolation: from Forest Fires to Epidemics: Critical Conflagration on a Square Lattice / Universality / Critical Density / Fractal Perimeters of Percolation / Finite-Size Scaling.
16. Phase Transitions & Renormalization: A First-Order Markov Process / Self-Similar & Non-Self-Similar Markov Processes / Scaling of Markov Outputs / Renormalization & Hierarchical Lattices / Percolation Threshold of the Bethe Lattice / Simple Renormalization.
17. Cellular Automata: The Game of Life / Cellular Growth & Decay / Biological Pattern Formation / Self-Similarity from a Cellular Automaton / Catalytic Converter as a Cellular Automaton / Pascal's Triangle Modlo N / Bak's Self-Organised Critical Sandpiles.
Appendix / References / Author Index / Subject Index. 429 pages.
- COMPUTERS & THE IMAGINATION: Visual Adventures Beyond The Edge.
- Clifford A. Pickover. Allan Sutton, 1991.
Introduction: Computers & the Unexpected / Simulation / Butterfly Curves / Cancer
Conundrum / Growing your own Font / Leaning Tower of Books / Artificial Webs /
Wiring Problem / Desktop Evolution / Research & Development / Speculation / Soda-Can-Sized Super-Super Computer / The Ten most influential scientists in history /
Mummy Project / PC placed in 1900 / Visualization / Pain inducing patterns / Ikeda
Attractor / Virtual Voltage Sculptures / World of Chaos / Fossil Sea Shells / Noise Spheres
/ Cantor Cheese Construction / Twisted Mirror Worlds / Polyhedral Paradise / Turning A
Universe Inside-Out / Monkey Curves & Spirals / Gleichniszahlen-Reihe Monster / Moire
Effect / Computer Exoskeletons / Exploration /Lute of Pythagoras / Very-Large-Number
Contest / Prime Plaid / Infinite Sequences in Centred Hexamorphic Numbers /
Cakemorphic Integers / Fractal Goose/ All known replicating Fibonacci-Digits less than
one billion / Juggler Sequence / Earthworm Algebra/ Friendly Technology /Palindromes
on Parade / Chimpanzee/Man Hybrid / Infinite Triangular Arrays / Undulating
Undecamorphic and Undulating Pseudofareymorphic Integers / Mandelbrot Set Pattern
/ 1/137 / Computer Smorgasbord / Million-Point Sculptures / Roots in Complex Plane /
Spherical Lissajous Figures / Fractal Faces / Invention / Self-Correcting Anti-Dyslexic Font
/ Speech Synthesis Grenade / Pictorial Password Systems / Evolving Computers /
Computer-Generated Poetry / Fiction / 24th Annual Meeting of the Chaos Society / Big
Black Bug / Soft Cattle / Conclusion. Appendices A-G. Glossary, Index. Gallery of
Computer Graphics. 423 pp.
- KEYS TO INFINITY
- Clifford A. Pickover. John Wiley & Sons Inc, 1995.
Preface / Too Many Threes: the search for Super-3 Numbers / Ladders to Heaven: Life on a Ladder World / Insight from Computer Programs / Infinite Ladder / Digressions / Infinity Machines: Infinity Anti-Lock Brakes / Infinity Keyboard / Infinity Program / Some Changing History of Infinity / Paradoxes Past the Speed of Light Infinity World: Fifth Avenue / Ancient Maps / Worlds without End / Vertical Infinity
Earths / Sociopolitical Impact / Physical Constraints to Infinity World / Alaska-Asia
Land Bridge / Peninsular Anistrophy / One Supercontinent / No Greece or Italy /
Reader Questions / Horizontal Repetition on Infinity World / Memicentrism /
Artificial Earths / Spiral Earths / Mecca Maps & Heart-shaped Maps / Grid of the
Gods: Digressions / Strahlkörper / To the Valley of the Sea Horses / The Million-
Dollar, Trillion-Digit, Pi Sequencing Initiative: Ants, God and Pi / The Grand Pi-
Sequencing Project / A Rupture in Geometry / Survey Results / Randomness /
Microbes, Hermann Schubert & Pi / Try it on yourself / Infinite Chess / Digressions /
The Loom of Creation / Slides in Hell / Alien Abduction Algebra / The Leviathan Number: Eternirty / Superfactorials, Super-Leviathans & Incomputability /
Welcome to Worm World: Internet Worm World Tournament / Abkhasian Areas /
Czech Logical Labyrinth / Hyperworms in Abkhasia / Fractal Milkshakes & Infinite
Archery: In the Great Beyond / Further Exploration / A Musical Tribute & PS /
Creating Life using the Cancer Game: Creation / The Game / Genetics, Programs, &
Evolution / A 40-Day Flood / No Zeros Allowed / Infinite Star Chambers /
Infinitely Exploding Circles: Not for the Meek / The Infinity Worms of Callisto /
The Undulation of the Monks: Other Undulants / Binary Undulants beyond
Imagination / The Fractal Golden Curlicue is Cool: Curlicues & the Infinite /
The Loneliness of the Factorions: Digression - Narcissistic Numbers / Escape
from Fractalia: Simple Equations make Interesting Patterns / Computer Art Critic /
Sample Artwork / Are Infinite Carotid-Kundalini Functions Fractal?: the Magical
Fractal Gaps / A Programming Tip for some BASIC Users / The Crying of Fractal
Batrachion 1,489: $10,000 Cash Award / Visualizing Infinity / Other Batrachions /
The Crying of Fractal Batrachions / Ramanujan, Infinity, and the Majesty of the
Quatuordecillion: Numerical Sumo Wrestlers / Absolute Reality / Recursive Worlds:
Recursion, Recursive Lattices / Beauty and the Bits / Higher Symmetry / Digressions /
Chaos in Ontario: Digressions and Contest / The Portuguese Maneuver / Cyclotron
Puzzles: Cyclotrons to Infinity / Vampire Numbers: Anne Rice / True Vampires v.
Pseudovampires / Vampires & the infinite / Computers, Randomness, Mind, &
Infinity: Humans cannot produce random Numbers / The Lure of Random Numbers/
Randomness / The Cliff RNG / Noise Spheres / Explore Extraterrestrial Lands / The
Bizarre Logistic Equation / The Logit Transformation / Pretty Good Random Numbers / Flipping a Quantum Mechanical Coin / Randomness & Infinity.
Appendix: 1-Program Code, 2-For Further Exploration. Notes. Further Reading.
Index. About the Author. 332 pages
- POETRY OF THE UNIVERSE: A Mathematical Exploration of the Cosmos
- Robert Osserman. Weidenfeld & Nicholson, 1995.
Sets out to provide a 'map' of the cosmos, as revealed by the latest theories of physics,
showing how mathematics has through the ages been the forerunner of scientific
advances. Measuring the unmeasurable / Encompassing the Earth / The Real World /
Imaginary Worlds / Curved Space / Invisible Universe / Looking back: the Observable
Universe / Another Dimension / A Galaxy of Shapes. Postlude; notes. Index. 210 pp.
- GAME, SET, & MATH -- Enigmas and Conundrums
- Ian Stewart. Basil Blackwell, 1989.
Selection of articles from Pour la Science, the French translation of Scientific American.
When Martin Gardner's 'Mathematical Games' column ceased, Stewart was invited to
write a replacement feature.
Mother Worm's Blanket / Drunken Tennis-Player / Infinormatics Laboratory / The
Autovoracious Ourotorus / Fallacy or Ycallaf? / Build your own virus / Parity Piece /
Close Encounters of the Fermat Kind / Pascal's Fractals / Worm Returns / All Parallels
lead to Rome / Twelve Games of Christmas.
Further reading after each section. 191 pp.
- A CATALOGUE OF SPECIAL PLANE CURVES
- J. Dennis Lawrence. Dover Publications 1972.
An illustrated study of plane algebraic and transcendental curves.
1. Properties of Curves. 2.Types of Derived Curves: Evolute, Involute, and Radial /
Parallel Curves / Inversion / Pedal Curves / Conchoid / Strophoid / Cissoid / Roulette /
Isoptic / Caustic. 3. Conics and Polynomials: Conics / Circle / Parabola / Ellipse /
Hyperbola / Power Function / Polynomial. 4. Cubic Curves: Semi-cubical Parabola /
Tschirnhausen's Cubic / Witch of Agnesi / Pedal of a Parabola / Cissoid of Diocles / Right
Striphoid / Trisectrix of Mclaurin / Folium of Descartes / Trident of Newton/ Serpentine.
5. Quartic Curves: Limacon of Pascal / Cardioid / Lemniscate of Bernoulli / Eight Curve
/ Bullet Nose / Cross curve / Deltoid / Conchoid of Nicomedes / Kappa Curve / Kampyle
of Eudoxus / Hippopede / Bicorn / Piriform / Devil's Curve / Folia / Cassinian Oval /
Cartesian Oval / Dürer's Conchoid. 6. Algebraic Curves of High Degree: Epitrochoid /
Hypotrochoid / Epicycloid / Nephroid/ Hypocycloid / Astroid / Rhodonea / Nephroid of
Freeth / Cayley's Sextic / Bowditch Curve. 7. Transcendental Curves: Sinusoidal Spiral
/ Logarithmic Spiral / Archimedean Spirals / Euler's Spiral / Involute of a Circle / Epi
Spiral / Poinsot's Spirals / Cochleoid / Cycloid / Quadratrix of Hippias / Catenary /
Tractrix. References. Appendix A: Tables of Derived Curves. Appendix B:
Further Reading. Index of Curve Names. 218 pp.
- CHAOS AND FRACTALS: New Frontiers of Science
- H-O. Peitgen / H. Jürgens / D. Saupe. Springer-Verlag NY, 1992.
Foreword by M.J. Feigenbaum.
Provides a broad view of the underlying notions behind fractals, chaos and dynamics,
shows how fractals and chaos relate to each other and to many other aspects of
mathematics as well as natural phenomena. A third motif is the inherent visual and
imaginative beauty in the structures and shapes of fractals and chaos. Introduction:
Causality Principle, Deterministic Laws & Chaos. 1: Backbone of Fractals - Feedback
and the Iterator. 2: Classical Fractals & Self-Similarity. 3: Limits & Self-Similarity. 4:
Length, Area and Dimension - Measuring Complexity and Scaling Properties. 5:
Encoding Images by Simple Transformations. 6: The Chaos Game - How randomness
Creates Deterministic Shapes. 7: Recursive Structures - Growing of Fractals & Plants.
8: Pascal's Triangle - Cellular Automata & Attractors. 9: Irregular Shapes - Randomness
in Fractal Constructions. 10: Deterministic Chaos - Sensitivity, Mixing, and Periodic
Points. 11: Order & Chaos - Period-Doubling & its Chaotic Mirror. 12: Strange Attractors
- The Locus of Chaos. 13: Julia Sets - Fractal Basin Boundaries. 14: The Mandelbrot Set
- Ordering the Julia Sets. Appendix A: Yuval Fisher - A Discussion of Fractal Image
Compression. Appendix B: Carl Evertsz & Benoit Mandelbrot - Multifractal Measures.
Bibliography. Index. 984 pp.
- MATHEMATICS: An Introduction to its Spirit and Use
- Selected & introduced by Morris Kline. W.H. Freeman, 1979.
Articles from the Scientific American.
History: Newman - The Rhind Papyrus / Review of Cardano / Review of Blaise Pascal
/ Crombie - Descartes / Cohen - Review of Mathematical Papers of Isaac Newton /
Kreiling - Leibnitz / Boyer - The Invention of Analytic Geometry. Number & Algebra:
Gardner - Remarkable Lore of Prime Numbers / Hawkins - Mathematical Seives/ Gardner
- Useless Elegance of Perfect Numbers & Amicable Pairs / Diophantine Analysis & the
Problem of Fermat's Legendary "Last theorem" / Herwitz - Theory of Numbers / Gardner -
Hierarchy of Infinities and the problems it spawns / Davis - Number.
Geometry: Gardner - Geometric Constructions with Compass & Straightedge, & Compass
alone / Elegant Triangle Theorems not to be found in Euclid / Diversions that involve one
of the classic Conic Sections: the Ellipse / Curious Properties of a Cycloid Curve / Curves
of constant width, one of which makes it possible to drill square holes / Kline - Projective
Geometry / Gardner - Geometric Fallacies: Hidden Errors pave the road to absurd
conclusions / Euler - The Koenigsberg Bridges / Gardner - Various problems based on
Planar Graphs, or sets of "Vertices" connected by "Edges" / Topological Diversions,
including a bottle with no Inside or Outside / Kline - Geometry. Statistics & Probability:
Weaver - Statistics / Carnap - What is Probability? / Weaver - Probability /
Gardner - On the Fabric of Inductive Logic & some Probability Paradoxes. Symbolic Logic & Computers: Gardner - Boolean Algebra, Venn Diagrams & the Propositional
Calculus / Pfeiffer - Symbolic Logic / Evans - Computer Logic & Memory / Ridenour -
Role of the Computer. Applications: ineback - Musical Tones / Saunders - Physics &
Music / Dalton - The Practice of Quality Control / Hurwicz - Game Theory & Decisions
/ Morgenstern - Theory of Games / Cooper & Charnes - Linear Programming / Levinson
& Brown - Operations Research.
Bibliographies. Index. 249 pp.
- MATHEMATICS: The Science of Patterns
- Keith Devlin. Scientific American Library, NY. 1994. Pbk 1997.
Preface / Prologue / Counting / Reasoning & Communicating / Motion & Change /
Shape / Symmetry & Regularity / Position.
Postscript, Further Readings, Index. 216 pp.
- AMUSEMENTS IN MATHEMATICS
- H.E. Dudeney. Nelson, 1917. Dover Reprint 1970.
Edition dedicated to the memory of Henry E. Dudeney, 1847-1930, "with the hope that
it will keep alive for many years his remarkable contributions to intellectual and
mathematical recreations." With a note on British coins & stamps, and the game of
cricket, for American readers. Solutions, index. 258 pp.
- RECREATIONAL PROBLEMS IN GEOMETRIC DISSECTIONS & HOW TO SOLVE THEM
- Harry Lindgren. D.Van Nostrand, 1964. Dover revised reprint, 1972.
Revisions by Greg Frederickson. P-Strips and PP Dissections / T-Strips and PT Dissections
/ TT Dissections / Piecemeal Dissections / Strips from Tessellations I / Triangles &
Quadrilaterals / Dissections from Tessellations I / Classifications / Dissections from
Tessellations II / Completing the Tessellation / Strips from Tessellations II / Rational
Dissections / Heptagon / Nine-gon / Decagon / Rectilinear Letters / Stars / More Stars /
E Pluribus Unum / Three-Figure Dissections / Curvilinear Dissections / Solid Dissections.
Postscript. Appendices: A. Problems for Solution B. Drawings for Polygons etc. C.
Dimensions of Polygons. D. Solutions of Problems. E. List of Dissections. F. Sources &
Credits. G. Recent progress. H. Eight Years After. 184 pp.
- MAZES & LABYRINTHS: Their History & Development
- W.H. Matthews. Longmans, Green, 1922. Dover Reprint 1970.
Introduction / The Egyptian Labyrinth: Accounts of the Ancient Writers and of Later
Explorers / Theseus and the Minotaur: Caverns of Gortnya, Knossus / The Etruscan or
Italian Labyrinth / Labyrinth in Ancient Art / Church Labyrinths / Turf Labyrinths / Maze
Etymology / Labyrinth Design & the Solution of Mazes / The Labyrinth in Literature /
Miscellanea & Conclusion. Bibliographical Appendix, Index.
254 pp.
- GÖDEL, ESCHER, BACH: an Eternal Golden Braid
- Douglas R. Hofstadter. Harvester Press, 1979.
Introduction: A Musico-Logical Offering / The MU-puzzle / Meaning & Form in
Mathematics / Figure and Ground / Consistency, Completeness & Geometry / Recursive
Structures & Processes / Location of Meaning / Propositional Calculus / Typographical
Number Theory / Mumon & Gödel / Levels of Description & Computer Systems / Brains
& Thoughts / Minds & Thoughts / BlooP and FlooP and GlooP / On Formally
Undecidedable Propositions of TNT & Related Systems / Jumping Out of the System /
Self-Ref & Self-Rep / Church, Turing, Tarski & Others / Artificial Intelligence: Retrospects
/ Artificial Intelligence: Prospects / Strange Loops, or Tangled Hierarchies.
Overview, Notes, Bibliography, List of illustrations, Index. 777 pp.
- METAMAGICAL THEMAS: Questing for the Essence of Mind and Pattern
- Douglas R. Hofstadter. Basic Books,1985. Penguin Books 1986.
1: Snags & Snarls - Self-Referential Sentences & a Follow-Up / Viral Sentences & Slf-Replicating Structures / Nomic: a Self-modifying Game based on Reflexivity in Law. 2:
Sense & Society - World Views in Collision / Number Numbness / Changes in Default
Words & Images / Purity in Language.
3: Sparking & Slipping - Pattern, Poetry & Power in music of Chopin / Parquet
Deformations / Stuff & Nonsense / Variations on a Theme as the Crux of Creativity /
Metafont, Metamathematics, & Metaphysics.
4: Structure & Strangeness - Magic Cubology / Crossing the Rubicon / Mathematical
Chaos & Strange
Attractors / Lisp: Atoms & Lists / Lisp: Lists & Recursion / Lisp: Recursion & Generality
/ Heisenberg's Uncertainty Principle & the Many-Worlds Interpretation of Quantum
Mechanics.
5: Spirit & Substrate - Review of Alan Turing: The Enigma / The Turing Test / Seeming
Paradox of Mechanizing Creativity / Analogies & Roles in Human & Machine Thinking
/ Who shoves Whom around inside the Careenium? / Subcognition as Computation.
6: Selection & Stability - The Genetic Code: Arbitrary? / Psychological Games with
Numbers / The Prisoner's Dilemma Computer Tournaments & the Evolution of
Cooperation. 7: Sanity & Survival - Dilemmas for Superrational Thinkers / Irrationality
is the Square Root of All Evil / Tale of Happiton / The Tumult of Inner Voices or What
is the Meaning of the Word "I"? Long Contents, Notes on the Cover, Epilogue,
Bibliography, Index. 852 pp.
- THE EMPEROR'S NEW MIND: Concerning Computers, Minds, & the Laws of Physics
- Roger Penrose. Oxford UP 1989.
Foreword by Martin Gardner / Prologue.
1. Can a computer have a mind? The Turing test / Artificial intelligence / Hardware &
software.
2. Algorithms & Turing machines: Background to the algorithm concept / Turing's
concept / Binary coding of numerical data / Church-Turing Thesis / Numbers other than
natural numbers / Universal Turing machine / Insolubility of Hilbert's problem / How to
outdo an algorithm / Church's lambda calculus.
3. Maths & Reality: Land of Tor'Bled Nam / Real numbers & how many are there? /
'Reality' of real numbers / Complex numbers / Construction of the Mandelbrot set /
Platonic reality of maths concepts?
4. Truth, proof and insight: Hilbert's programme for maths / Formal maths systems /
Gödel's theorem / Mathematical insight / Platonism or intuitionism? / Gödel-type
theorems from Turing's result / Recursively enumerable sets / Is Mandelbrot set recursive?
/ Some examples of non-recursive maths / Is Mandelbrot set like non-recursive maths?
/ Complexity theory / Complexity and computability in physical things.
5. The classical world: Status of physical theory / Euclidean geometry / Dynamics of
Galileo & Newton / Mechanistic world of Newtonian dynamics / Is life in the billiard-ball world computable? / Hamiltonian mechanics / Phase space / Maxwell's
electromagnetic theory / Computability & the wave equation / Lorentz equation of
motion: runaway particles / Special realtivity of Einstein & Poincaré / Einstein's general
relativity / Relativistic causality & determinism / Computability in classical physics:
where do we stand? / Mass, matter, and reality.
6. Quantum Magic & Quantum Mystery: Do philosophers need quantum theory? /
Problems with classical theory / Beginnings of quantum theory / Two-slit experiment /
Probability amplitudes / Quantum state of a particle / Uncertainty principle / Evolution
procedures U and R / Particles in two places at once / Hilbert Space / Measurements /
Spin & the Riemann sphere of states / Objectivity & measurability of quantum states /
Copying a quantum state / Photon spin / Objects with large spin / Many-particle systems
/ 'Paradox' of Einstein, Podolsky, and Rosen / Experiments with photons: a problem for
relativity? / Schrödinger's equation; Dirac's equation / Quantum field theory /
Schrödinger's cat / Various attitudes in existing quantum theory / Where does all this
leave us?
7. Cosmology & the arrow of time: Flow of time / Entropy / Cosmology & big bang /
Black holes / Structure ofspace-time singularities / How special was the big bang?
8. In search of quantum gravity: Weyl curvature hypothesis / Time-asymmetry in state-vector reduction / Hawking's box / When does the state-vector reduce?
9. Real brains & model brains: Seat of consciousness /Split-brain experiments / Blindsight
/ Information processing in the visual cortex /Nerve signals / Computer models / Brain
plasticity / Parallel computers and 'oneness' of consciousness / Quantum mechanics in
brain activity? / Quantum computers / Beyond quantum theory?
10. Where lies the physics of mind? What are minds for? / What does consciousness
actually do? / Natural selection of algorithms? / Non-algorithmic nature of mathematical
insight / Inspiration, insight and originality / Non-verbality of thought / Animal
consciousness? / Contact with Plato's world / View of physical reality / Determinism &
strong determinism /Anthropic principle / Tilings & quasi-crystals / Relevance to brain
plasticity / Time-delays of consciousness / Strange role of time in conscious perception
/ Conclusion: a child's view.
Epilogue; References. Index. 466 pp.
- SHADOWS OF THE MIND: a search for the missing science of consciousness
- Roger Penrose. Oxford UP, 1994. Reprinted 1995, with corrections.
Prologue / Part I: Why We Need Physics to Understand the Mind -The non-computability
of conscious thought.
1. Consciousness & computation. Mind & science / Can robots save this troubled world?
/ The A, B, C, D of computation & conscious thinking / Physicalism vs mentalism /
Computation; top-down & bottom-up procedures / Does viewpoint C violate the Church-Turing thesis? /Chaos / Analogue computation / What kind of action would be non-computational? / Can computers have rights or responsibilities? / 'Awareness',
'understanding', 'consciousness', 'intelligence' / John Searle's argument / Difficulties
with the computational model / Do limitations of present-day AI provide a case for C ?
/Argument from Gödel's theorem / Platonism or mysticism? / Relevance of mathematical
understanding? / Gödel's theorem & commonsense behaviour / Mental visualization &
virtual reality / Is mathematical imagination non-computational?
2. The Gödelian case
Gödel's theorem & Turing machines / Computations / Non-stopping computations /
Families of computations: the Gödel-Turing conclusion G / Possible technical objections
/ Deeper mathematical considerations / Condition of -consistency / Formal systems &
algorithmic proof / Further possible technical objections to G .
3. The case for non-computability in mathematical thought
Could an unsound algorithm knowably simulate mathematical understanding? / Could a knowable algorithm unknowably simulate mathematical understanding? / Do mathematicians unwittingly use an unsound algorithm? / Can an algorithm be unknowable? / Natural selection or act of God? / One algorithm or many? / Natural selection of unworldly esoteric mathematicians / Learning algorithms / May the environment provide a non-algorithmic external factor? / How can a robot learn? / Or attain 'firm mathematical beliefs' ? / Mechanisms underlying robot mathematics / Basic contradiction / Averting the contradiction / Does the robot need to believe in M? / Robot errors & robot 'meanings' / How to incorporate randomness - ensembles of robot activity
/ Removal of erroneous IM-assertions need to be considered / Adequacy of safeguards? / Can chaos save the computational model of mind? Reductio ad absurdum--a fantasy dialogue / Using paradoxical reasoning? / Complication in mathematical proofs / Computational breaking of loops / Top-down or bottom-up computational mathematics? / Conclusions.
Part II: What New Physics We Need to Understand the Mind - The quest for a non-computational physics of mind.
4. Does mind have a place in classical physics? Mind & physical laws / Computability
& chaos in the physics of today / Consciousness: new physics or 'emergent
phenomenon'? / The Einstein tilt / Computation & physics.
5. Structure of the quantum world - Quantum theory: puzzle & paradox / The Elitzur-Vaidman bomb-testing problem / Magic dodecahedra / Experimental status of EPR-type
Z-mysteries / Quantum theory's bedrock: a history extraordinary / Basic rules of quantum
theory / Unitary evolution U / State-vector reduction R / Solution of the Elitzur-Vaidman
bomb-testing problem / Quantum theory of spin: the Riemann sphere /Position &
momentum of a particle / Hilbert space & description of R / Commuting measurements
/ The quantum-mechanical 'and' / Orthogonality of product states / Quantum
entanglement / The magic dodecahedra explained. Appendices: Non-colourability of the
dodecahedron / Orthogonality between general spin states.
6.. Quantum theory & reality - Is R a real process? / Many-worlds-type viewpoints / Not
taking > seriously / Density matrix / For EPR pairs / A FAPP explanation of R? / Does
FAPP explain the squared modulus rule? Is it consciousness that reduces the state vector?
/ Taking > really seriously / Gravitationally induced state-vector reduction? / Absolute
units / The new criterion.
7.Quantum theory & the brain - Large-scale quantum action in brain function? /
Neurons, synapses, and computers / Quantum computation / Cytoskeletons &
microtubules / Quantum coherence within microtubules? / Microtubules &
consciousness / A model for a mind?/ Non-computability in quantum gravity / Oracle
machines & physical laws / Time & conscious perceptions / EPR & time.
8. Implications? - Intelligent artificial 'devices' / Things that computers do well--or badly
/ Aesthetics/ Some dangers inherent in computer technology / The puzzling election /
The physical phenomenon of consciousness ? / Three worlds & three mysteries.
Epilogue; Bibliography. Index. 457 pp.
- THE INFINITE IN THE FINITE
- Alistair Macintosh Wilson. Oxford UP, 1995.
Combining historical fact with a retelling of ancient myths and legends, each chapter
contains a mathematical case study where mathematics is applied to the problems of the
era.
1.Symphonies of stone 2.The pyramid builders 3. Theban mysteries 4.Babylonian
maths 5. Chinese maths 6. The Achaeans 7. A world made of numbers-- Pythagoras,
the shapes of numbers, regular polyhedra, Euler's number, polyhedra in the world 8.
The thoughts of Zeus - symmetries of polygons; symmetry groups of regular polyhedra
9. The Philosopher's criticism - geometry; Peloponnesian War; Socrates Plato; Aristotle's
logic; Stoics construct the truth 10.The Elements of Euclid - Euclid's dream; Similar
triangles; angles of triangles; area of triangles; Pythagoras' theorem; triangles in circles
11. An island interlude 12. Proportion - geometrical solution of aha problems; theory
of proportion; construction of regular polygons; uses of proportion; problem of maxima
13. Divine Archimedes - measurement of a circle; method of exhaustion; surface area
& volume of a sphere; volume of a cone; quadrature of a spiral, and a parabola;
Archimedes' principle; Rancher's Dilemma 14. Apollonius the great geometer: Conics;
lowers from three conic sections; tangents to conic sections; property of the parabola
used by Archimedes; centres of conics; foci of a conic; reflection properties of conic
sections; focal construction of conics. 15.Science of numbers - Pythagorean
numerology; prime numbers; irrational numbers; Pythagorean triples; patterns of primes
16. School of Alexandria - Alexandria; Heron; Diophantus; Pappus; last of the Greeks;
Eudemian summary 17.The dark subcontinent of India - The Aryans; Sanskrit and the
Hindu numerals; Hindu astronomy; maths of Brahmagupta & Mahavira; a pearl of
number theory; 18.The contribution of Islam - Conquests of the Arabs; Trigonometry;
geometry of the sphere; gnomon curve; algebra; summation of powers of integers; Spain
under Islam. Bibliography; index. 524 pp.
- THREE-TOED SLOTHS & SEVEN-LEAGUE BOOTS: A Dictionary of Numerical Expressions
- Laurence Urdang. Barnes & Noble, NY, 1986.
Foreword / Organization of Information / Dictionary / Appendix / Index. 324 pages.
- TILINGS AND PATTERNS
- Branko Grünbaum & G.C. Shephard. W.H. Freeman, NY, 1987.
While many books have been published on the aesthetic aspects of tilings and patterns, most consist of little more than collections of examples with no attempt
at presenting a general theory. This is the first truly authoritive, comprehensive,
and systematic treatment of the subject. Focusing on classification and enumeration
of tilings, the authors provide detailed surveys of coloring problems, tilings by
polygons, and tilings by topologically unusual tiles, and present the first complete coverage of aperiodic tilings.
Introduction / Basic Notions / Tilings by Regular Polygons & Star Polygons / Well-behaved Tilings / The Topology of Tilings / Patterns / Classification of Tilings with Transitivity Properties / Classification with Respect to Symmetries / Coloured Patterns & Tilings / Tilings by Polygons / Aperiodic Tilings / Wang Tiles / Tilings with Unusual Kinds of Tiles. References. Index. 700 pages.
- TILINGS AND PATTERNS: An Introduction
- Branko Grünbaum & G.C. Shephard. W.H.Freeman, NY, 1989.
Introduction / Basic Notions / Tilings by Regular Polygons & Star Polygons / Well-behaved Tilings / The Topology of Tilings / Patterns / Classification of Tilings with Transitivity Properties / Classification with Respect to Symmetries.
This volume is a "brief" paperback edition, comprising the first seven chapters of the earlier book, with corrections. References. Index. 446 pages.
- GEOMETRIC CONCEPTS IN ISLAMIC ART
- Issam El-Said & Aye Parman. World of Islam Festival, London. 1976.
Introduction / Man & Measure: Historical background, Measuring & Dividing
Dimensions, Organisation of Space, Proportion.
Geometric Patterns in Islamic Design: Concept of the Repeat Unit, Square & root-2
System of Proportion, Hexagon & 3 System of Proportion, Pentagon & Golden Ratio, Patterns based on the Double Hexagon, Artistic Creativity & Geometric Method of Design.
Architecture: Tracing Islamic Methods, Analyses of Ancient Egyptian Monuments, Divinbe Order of the Universe, Islamic Concepts in Architecture, Planning.
Arabic Calligraphy, Poetry, Music. Analyses of Patterns in the Applied Arts.
Conclusion. Bibliography. 154pp.
- PATTERN IN ISLAMIC ART
- David Wade. Studio Vista, London, 1976.
Introduction / Historical Summary / The Islamic ethos / The influence of Classical
Greece / Underlying geometry / The Drawings.
The Patterns: Square grid / Interlock / Isometric grid / Curvilinear variant / Hexagon /
Octagon / Star & Cross variations / Dodecagon / Overlay / Compound patterns /
Interlacing patterns / Borders / Openwork lattices. 144 pp.
- THE ISLAMIC COLOURING BOOK
- David Wade. Wildwood House, London. 1976.
The lazo or interlacing form of geometric ornament has been a consistent part of
Islamic art for over a millennium.
- GEOMETRIC PATTERNS & BORDERS
- David Wade. Wildwood House, London. 1982.
Over 500 patterns and 200 borders in 23 categories, drawn from over forty cultures.
Introduction / The Basic Grids / Coloring Schemes / Notes on the patterns.
Square-based / Square + diagonal / Square + compound diagonal / Special arrange ments of squares / Swastika / Counterchange through 90 / Other shapes on grids / The interlock / Key patterns / Knots / Spirals / Woven / Triangular-based / Islamic triangular-based / Hexagonal / Comb-cell / Octagonal / Dodecagonal / Stratiform / More star patterns / Star & Cross / Mouj / Overlaid circles / Circles & Arcs / Ecalé /
Borneo / Petal / Borders. Appendix.
- CRYSTAL & DRAGON: The Cosmic Two-Step
- David Wade. Green Books, Bideford. 1991.
World views of successive ages: from Plato's philosophy of the Ideal Form, and the
ancient Chinese philosophy of change, through to the modern scientific view of
structure and indeterminacy as embodied in the laws of science. How these
perceptions about the nature of the universe inform the art of their times: the form
and fluidity of primitive art, the crystalline order of Islamic patterns, and the subtlety
and vitality of Chinese landscapes and calligraphy.
Introduction / 1:The Individual & Society - Things & processes; the 'Two Forces of
Nature'; Snakes & Ladders. 2: Science & Nature - Crystalline Imperative; Symmetry
& Beyond; Flux & complication. 3: Art & Aesthetics - Formalism & Vitalism in art;
'Crystalline form in the art of Islam; Vitality & Fluidity in Taoist art.
Afterword, Glossary, Select Bibliography, Index. 287pp.
- THE LANGUAGE OF PATTERN: an enquiry inspired by Islamic decoration.
- Keith Albarn, Jenny Miall Smith, Stanford Steele & Dinah Walker.
Thames & Hudson, London, 1974, reprinted 1976.
Inspired by Islamic decorative pattern, the authors (who are all designers) explore
pattern step by step, beginning with simple numerical and geometrical relationships
and progressing through the dimensions, drawing the reader into a visual and conceptual game of increasing complexity.
Introduction / Number & pattern / Islamic pattern / Process & concept / Islamic
constructions / Structural developments / A way of thinking.
Further reading. 112 pp.
- DIAGRAM: The Instrument of Thought
- Keith Albarn & Jenny Miall Smith. Thames & Hudson, 1977.
Maps, charts, graphs - all diagrams - are used to organise our experience of the
world outside; the authors suggest a method for applying the diagram to our internal
world of thoughts, ideas, feelings and emotions.
Introduction / Context / Games / Duality / Coincidence / Pattern / Perception / Image
/ Thought / Reflections / Analogue / Model / Personality / Diagram / Cosmology /
Conclusion. Notes; Bibliography; Index. 144pp.
- HANDBOOK OF REGULAR PATTERNS: An Introduction to Symmetry in 2 Dimensions
- Peter S. Stevens. MIT Press, Cambridge, Mass. 1980.
Reveals the structural anatomy of patterns, demonstrates how the artist can play
limitless variations upon the few permitted fundamental patterns and structural
arrangements. Preface / Symmetry Groups / Point Groups / The Seven Line Groups /
The Seventeen Plane Groups. Appendix, Bibliography, Index. 400pp.
- MATHOGRAPHICS
- Robert Dixon. Basil Blackwell, Oxford. 1987.
Explores the possibilities of mathematical drawing through compass constructions
and computer graphics. Preface / How to use this book / 1. Compass drawings
2. String drawings 3. Perspective drawings 4. The Story of Trigonometry
5. Computer drawings. Bibliography; Index. 214 pp.
- ORDER IN SPACE: a design source book
- Keith Critchlow. Thames & Hudson, London, 1969.
A fresh and imaginative insight into the area where mathematics and art meet: a
systematic analysis of those space-defining geometries that are relevant to solving
design problems and understanding the ordered division of space.
Appendices: 1 A Periodic Arrangement of the Elements of Spatial Order / 2.Of the
Multiple Single, Dual & Triple All-space Filling Solids / 3. Aspects & Definitions /
4. Aspects of the 12 Degrees of Freedom. Bibliography. 120 pp.
- ISLAMIC PATTERNS: an analytical and cosmological approach.
- Keith Critchlow. Thames & Hudson, London, 1976.
Foreword by Seyyed Hossein Nasr. Introduction / The Point of Departure / The
Manifestation of Shape / Magic Squares / Pattern & Cosmology / The Pentagon /
The Tetractys / Mathematics of 2-dimensional space filling / The Circle & Cosmic
Rhythms / Specimen Islamic Patterns. 192 pp.
- GEOMETRIC SYMMETRY
- E.H. Lockwood & R.H. Macmillan. Cambridge UP, 1978.
Historical note / Introduction. Part I - Descriptive: Reflexions & rotations / Finite
patterns in the plane / Frieze patterns / Wallpaper patterns / Finite objects in three
dimensions / Rod patterns / Layer patterns / Space patterns / Patterns allowing
continuous movement / Dilation symmetry / Colour symmetry / Classifying and
identifying plane patterns / Making patterns.
Part II - The mathematical structure.
Movements in the plane / Symmetry groups; point groups / Line groups in two
dimensions / Nets / Plane groups in two dimensions / Movements in three dimensions / Point groups in 3D / Line groups in 3D / Plane groups in 3D / Lattices / Space groups / Limiting groups / Colour symmetry.
Appendices; Books; Summary tables; Notation & axes; Index of groups, nets and lattices; Index. 228 pp.
- OPTICAL & GEOMETRICAL PATTERNS & DESIGNS
- Spyros Horemis. Dover Publications, NY. 1970.
500 new and original designs with introduction by Janet Koch. 85pp.
- CHINESE LATTICE DESIGNS
- Daniel Sheets Dye. Dover Publications, NY. 1974.
Republication of A Grammar of Chinese Lattice, Harvard UP, 1937. Shows more
than 1200 designs arranged in a clear system of classification. 469 pp.
- GEOMETRICAL DESIGNS & OPTICAL ART
- Jean Larcher. Dover Publications, NY. 1974.
70 drawings by Jean Larcher, leading French graphic artist, with preface, and biography of the artist.
- OP ART COLORING BOOK
- Jean Larcher. Dover Publications, NY. 1975. 30 original designs.
- GRAPHIC & OP-ART MAZES
- Dave Phillips. Dover Publications, NY. 1976.
40 new mazes - includes several 3D mazes, an eccentric circles maze, an 'impossible' perspective construction etc. With solutions. 56pp.
- ARABIC GEOMETRICAL PATTERN & DESIGN
- J. Bourgoin. Dover Publications, NY. 1973.
Reprints the illustrations from Les Elements de l'art arabe: le trait des entrelacs. Paris, 1879. The classification used by Bourgoin is a trifle obscure, and the original text has not been translated. 200pp.
- DESIGN DISCOVERY COLORING BOOK: 46 Endless Patterns
- J. Bourgoin. Dover Publications, NY. 1976.
Based on patterns in earlier volume. 48pp.
- ISLAMIC PATTERNS: An Infinite Design Coloring Book
- J. Bourgoin. Dover Publications, NY. 1977.
A further 45 designs adapted from earlier volume. 48pp.
- INFINITE DESIGN COLORING BOOK: 46 Original Designs.
- Muncie Hendler. Dover Publications, NY. 1976.
- TRIAD OPTICAL ILLUSIONS: How to design them.
- Harry Turner. Dover Publications, NY. 1978.
How to create Triad designs: drawing 'impossible' figures that extend to form infinite patterns. Text and 32 plates for coloring. 64pp.
- ISOMETRIC PERSPECTIVE DESIGNS: How to create them
- John Locke. Dover Publications, 1981.
Isometric perspective is the picture of an object adrift in imaginary space; it eliminates normal perspective to create a new visual vocabulary of geometric design. 31 plates for coloring. 48pp.
- MOSAIC & TESSELLATED PATTERNS: How to create them
- John Willson. Dover Publications, NY. 1983.
History of tessellated designs / Basic kinds and how they are formed / Creating designs from common geometrical shapes. Text illustrated with 179 figures, and 32 pages of tessellations for coloring. 64pp.
- MODULE, SYMMETRY, PROPORTION
- Gyorgy Kepes, editor. Studio Vista, London. 1966.
Philip Morrison: The Modularity of Knowing / C.H. Waddington: Modular Principle
& Biological Form / Arthur L. Loeb: Architecture of Crystals / Stanislaw Ulam: Patterns of Growth: Mathematical Aspects / Modular Ideas in Science & Art: Visual
Documents / Lawrence B. Anderson: Module: Measure, Structure, Growth &
Function / Ezra D. Ehrenkrantz: Modular materials & Design Fleibility / Richard P. Lohse: Standard, Series, Module: New Problems and Tasks of Painting / Anthony Hill: Structural Syndrome in Constructive Art / Ernö Lendvai: Duality & Synthesis in Music of Béla Bartók / John Cage: Rhythm etc / François Molnar: The Unit & the Whole / Rudolf Arnheim: A Review of Proportion Biographical Notes on the Authors. 233pp.
- STRUCTURE IN ART & SCIENCE
- Gyorgy Kepes, Editor. Studio Vista, London. 1965.
Essays by Richard Held, Jacob Bronowski, R. Buckminster Fuller, Max Bill etc.
Biographical Notes. 190pp.
- THE MAN WHO KNEW INFINITY: a life of the Genius Ramanujan
- Robert Kanigel. Abacus, London, 1992. Reprint 1995.
Biography of the brilliant self-taught Indian mathematician Srinivasa Ramanujan
(1887-1920).See also 'Ramanujan and Pi', p.647, World Treasury of Physics, Astronomy & Mathematics. Index. 438 pp.
Books on Mathematics
Supplement
June 1998
TARQUIN BOOKS
- MAKE SHAPES: MATHEMATICAL MODELS / Gerald JENKINS, Anne WILD.
- Series No 1 19 simpler models -- Tetrahedron, pentagonal prism and pyramid,
octahedron, truncated tetrahedron, step pyramid, double hexagonal pyramid,
trunc. square pyramid, cuboctohedron, icosahedron, dodecahedron, rhomboid,
trunc. octahedron, rhombicuboctahedron, rotable ring of tetrahedra, tetrahemi-
hexahedron, octahemioctahedron, stella octangula, small stell. dodecahedron.
Series No 2 8 larger models -- Facetted cube, icosidodecahedron, cross of
octahedron, rhombicosidodecahedron, great stell. dodecahedron, third stellation
of icosahedron, great dodecahedron, compound of 5 tetrahedra.
Series No 3 3 intricate models -- Ninth stell. of icosahedron, great icosahedron, compound of 10 tetrahedra.
- MATHEMATICAL CURIOSITIES
- Gerald JENKINS, Anne WILD, 1980-1-2.
Book 1 Polyhedra flower, diabolic frame, double-sided magic square, möbius strips, Klein cube, double helix, shapes of constant width, family of hexaflexagons, set folding/unfolding cubes.
Book 2 Hexagonal rotating ring, hypercube, spherical icosahedron, rainbow squares, cat-and-mouse tetraflexagon, square rotating ring, Dudenay's dissection, nesting pyramids, magic cube.
Book 3 7-colour torus & rotating ring, cube&octahedron, flexitube, geodesic sphere, pop-up parallel pyramids, triangular dissection, pentagram and pyramid, flip-flop parallelepipeds.
- TARQUIN POLYHEDRA
/ Gerald JENKINS, Magdalen BEAR, 1985.
- No.1 Sixth Stellation of the Icosahedron
No.2 Compound of five Cubes
No.3 Final Stellation of the Icosahedron
- PENTOMINOES
- Jon MILLINGTON, 1987.
- POLYSYMETRICS: the art of making geometric patterns
- June OLIVER, 1979.
- KALEIDOMETRICS: the art of making beautiful patterns from circles
- Sheilah SHAW, 1981
- ALTAIR DESIGN BOOKS 1 - 3 1970, 1975,1979.
- Designs derived by Ensor HOLIDAY from a study of the geometrical construction of 10thC Arab design.
- ALTAIR DESIGN PAD 2 , 1975
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