Excursions into mathematics:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
New York, NY
Worth
1982
|
| Edition: | 5. print. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 489 S. Ill., graph. Darst. |
| ISBN: | 0879010045 |
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MARC
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| 245 | 1 | 0 | |a Excursions into mathematics |c Mitarb.: Anatole Beck* |
| 250 | |a 5. print. | ||
| 264 | 1 | |a New York, NY |b Worth |c 1982 | |
| 300 | |a XXI, 489 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathématiques | |
| 700 | 1 | |a Beck, Anatole |e Sonstige |4 oth | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-005695373 | ||
Record in the Search Index
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| adam_text | CONTENTS
NOTE TO THE INSTRUCTOR xvii
CHAPTER 1 EULER S FORMULA FOR POLYHEDRA, AND RELATED TOPICS
Section 1 Introduction 3
1A Prelude and Fantasy. Being an account of Kepler s first law of
astrology.
IB Euler s Formula. Proof of Euler s theorem, v — e •{¦ f = 2, for
convex polyhedra.
Section 2 Regular Polyhedra 12
Proof of the existence of at most five types of regular polyhedra. Regular
tessellations. Occurrence in nature.
Section 3 Deltahedra 21
Derivation of all convex polyhedra with equilateral triangles as faces.
Construction patterns.
Section 4 Polyhedra without Diagonals 31
A generalization of Euler s formula. Solids with tunnels through them.
The only known such solid without diagonals. Do it yourself kit.
Section 5 The n dimensional Cube and the Tower of Hanoi 40
5A The n cube. The n dimensional cube and n dimensional
tetrahedron. How to visualize them. A round trip tour in n dimensions
(Hamilton circuits on the n cube).
5B The Tower of Hanoi. A solution to the Tower finds a
Hamilton circuit on the n cube.
5C Chinese Rings.
5D An Application. How to use a Hamilton circuit to make
your radar work better.
Section 6 The Four color Problem and the Five color Theorem 55
Every map has some region with fewer than six neighbors. No map
can have five regions each bordering the other four. Every map can
be colored with five colors. Some special maps which can be colored
with four colors. Will four always work?
CONTENTS
Section 7 The Conquest of Saturn, the Scramble for Africa,
and Other Problems 64
7A The Conquest of Saturn. Euler s formula for a ring of Saturn
(torus). Every map on the torus can be colored with seven colors.
A coloring theorem for generalized pretzels (Heawood s theorem).
7B The Scramble for Africa. Twelve colors are required if every
country has a colony. What about colonies on the moon?
7C Some Problems Equivalent to the Four color Problem.
Also, a list of problems—some easy, some hard, some still
unsolved—about cigarettes, maps, and ping pong balls. Bibliography
for further reading.
CHAPTER 2 THE SEARCH FOR PERFECT NUMBERS
Section 1 Introduction 81
Definition and history of perfect numbers.
Section 2 Prime Numbers and Factorization 84
Unique factorization into primes.
Section 3 Euclidean Perfect Numbers 93
Mersenne primes, final digits of Euclidean perfect numbers,
congruence modulo m.
Section 4 Primes and Their Distribution 102
The infinitude of primes, the number of primes less than n,
construction and history of tables of primes.
Section 5 Factorization Techniques 110
Fermat s method of difference of squares, last digits of squares,
casting out 9s and similar tests, Fermat s theorem, Euler s
theorem on factors of Mersenne primes, known Euclidean
perfect numbers.
Contents
Section 6 Non Euclidean Perfect Numbers 124
Sums of divisors of numbers. Odd perfect numbers; the number
of divisors of an odd perfect number.
Section 7 Extensions and Generalizations 137
Multiply perfect numbers, amicable numbers, Hadrian s headache.
CHAPTER 3 WHAT IS AREA?
Section 1 Introduction 147
The problem of area. Greek ideas. Properties which area ought
to have.
Section 2 Rectangles and Grid Figures 153
Areas for rectangles oriented in the same direction (the class (R).
Areas for combinations of such rectangles (the class g).
Section 3 Triangles 161
Preliminary results for special triangles. Every triangle which
has an area satisfying Al to A4 must have area bh.
Section 4 • Polygons 166
Decomposition of triangles into triangles. Every convex polygon
can be decomposed into triangles ( triangulated ) Every reflex
polygon can be triangulated. Proof by induction.
Section 5 Polygonal Regions 177
Triangulation of polygons with holes in them (polygonal regions).
The area of a triangle is the sum of the areas of subtriangles.
The area of a polygonal region does not depend on the
particular triangulation.
Section 6 Area in General 185
Area of a circle (the classical proof ) Upper numbers and lower
numbers. The figures of class Qi. Figures of class Q. Proofs that various
xii CONTENTS
combinations of figures in Q are also in Q (E (~ F, E F, E*U F).
Every figure of class Q has an area. Examples of figures of class Q.
Proof that every bounded convex figure is of class Q.
Section 7 Pathology 201
Are there figures which are not of class Q? The psychedelic
puff ball. A curve whose interior is not of class Q.
CHAPTER 4 SOME EXOTIC GEOMETRIES
Section 1 Historical Background 213
Euclid s contribution. Objections to Euclid.
Section 2 Spherical Geometry 217
A well known non Euclidean geometry, where two lines always meet.
Section 3 Absolute Geometry 221
The theorems of Euclid which do not depend on the parallel
postulate. The angle sum in a triangle.
Section 4 More History—Saccheri, Bolyai, and Lobachevsky 226
An accidental discovery of non Euclidean geometry, and two
intentional ones.
Section 5 Hyperbolic Geometry 230
The geometry of Bolyai and Lobachevsky, where there are two kinds
of parallels to a given line. Angle of parallelism. The angle sum in a
triangle is less than 180°. Relevance to the real world.
Section 6 New Beginnings 242
Hilbert s axioms for Euclidean geometry.
Contents xiii
Section 7 Analytic Geometry—A Reminder 248
Coordinate system, straight lines, and conies briefly reviewed.
Section 8 Finite Arithmetics 255
The integers modulo/) (/ prime), Ip, as a special case of a finite field.
Section 9 Finite Geometries 262
The points and lines of the analytic geometry coordinatized by a
finite field. The special case of 75.
Section 10 Application 273
How to use finite geometries to grow tomato plants. (Design of
experiments.)
Section 11 Circles and Quadratic Equations 279
Second degree equations over 75. Interior and exterior points of a
circle.
Section 12 Finite Affine Planes 289
Section 9 redone axiomatically.
Section 13 Finite Projective Planes 292
The symmetrized geometry, where two points always determine a
line and two lines always determine a point.
Section 14 Ovals in a Finite Plane 296
An axiomatic treatment of circles. Tangents, secants, interior
points, and exterior points. The strange behavior of circles in the
even order case.
Section 15 A Finite Version of Poincare s Universe 303
The classical model of non Euclidean geometry. A finite version
of this model.
Appendix Excerpts from Euclid 308
xiv CONTENTS
CHAPTER 5 GAMES
Section 1 Introduction 317
Section 2 Some Tree Games 318
The game of Nim and related games. Natural outcomes for
these games.
Section 3 The Game of Hex 327
Analysis of the game. White should always win, but no one knows
how. Beck s Hex, and the theorem which wrecks Beck s Hex.
Section 4 The Game of Nim 340
Complete analysis of Nim, including a recipe for winning. The
Marienbad Game.
Section 5 Games of Chance 359
Coin tossing, dice, roulette. Mr. Mark vs. Mr. House. Analysis of
some simple examples.
Section 6 Matrix Games 365
The value of a matrix game. Proof of the minimax theorem
for 2X2 games.
Section 7 Applications of Matrix Games 378
Airplanes vs. submarines (the North Atlantic Game). Similar war games.
Section 8 Positive sum Games 381
Analysis of a simple positive sum game.
Section 9 Sharing 383
Further remarks about positive sum games.
Section 10 Cooperative Games 385
Formation of coalitions to increase payoff. (The game called
Production.)
Contents xv
CHAPTER 6 WHAT S IN A NAME?
Section 1 Introduction 39
General comments on what we do and don t know about our
number system.
Section 2 Historical Background 392
How number systems developed. Examples from different cultures.
Section 3 Place Notation for the Base b 396
Development of the decimal system of writing integers for
bases other than 10.
Section 4 Some Properties of Natural Numbers Related
to Notation 402
Divisibility criterion; casting out 9s. Perfect squares. Distribution of
digits and density.
Section 5 Fractions: First Comments 414
Egyptian fractions and Babylonian fractions and some questions
about them.
Section 6 Farey Fractions 416
Definition and fundamental properties. Mediation.
Section 7 Egyptian Fractions 421
Basic and not so basic properties. Fibonacci and Farey series
algorithms; the splitting method.
Section 8 The Euclidean Algorithm 435
Greatest common divisors and least common multiples. Number of
steps involved.
Section 9 Continued Fractions 440
Connection with Euclidean algorithm and elementary properties.
xvi CONTENTS
Section 10 Decimal Fractions 444
Finite, repeating, and nonrepeating decimals. What is a decimal?
The decimal algorithm. Completeness. Arithmetic properties of
decimal digits: theorems of Leibniz, Gauss, Midy, and others.
Section 11 Concluding Remarks 470
What s in a name? A comparison of the different ways of naming
fractions.
GLOSSARY OF SYMBOLS 477
INDEX 479
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| discipline | Mathematik |
| edition | 5. print. |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:22:34Z |
| institution | BVB |
| isbn | 0879010045 |
| language | English |
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| physical | XXI, 489 S. Ill., graph. Darst. |
| publishDate | 1982 |
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| publishDateSort | 1982 |
| publisher | Worth |
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| spelling | Excursions into mathematics Mitarb.: Anatole Beck* 5. print. New York, NY Worth 1982 XXI, 489 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathématiques Beck, Anatole Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005695373&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Excursions into mathematics Mathématiques |
| title | Excursions into mathematics |
| title_auth | Excursions into mathematics |
| title_exact_search | Excursions into mathematics |
| title_full | Excursions into mathematics Mitarb.: Anatole Beck* |
| title_fullStr | Excursions into mathematics Mitarb.: Anatole Beck* |
| title_full_unstemmed | Excursions into mathematics Mitarb.: Anatole Beck* |
| title_short | Excursions into mathematics |
| title_sort | excursions into mathematics |
| topic | Mathématiques |
| topic_facet | Mathématiques |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005695373&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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