Experiments in Topology

Stephen Barr

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486259338
• eBook ISBN: 9780486152745
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

"A mathematician named Klein
Thought the Moebius band was divine.
Said he: 'If you glue
The edges of two,
You'll get a weird bottle like mine.' " — Stephen Barr

In this lively book, the classic in its field, a master of recreational topology invites readers to venture into such tantalizing topological realms as continuity and connectedness via the Klein bottle and the Moebius strip. Beginning with a definition of topology and a discussion of Euler's theorem, Mr. Barr brings wit and clarity to these topics:
New Surfaces (Orientability, Dimension, The Klein Bottle, etc.)
The Shortest Moebius Strip
The Conical Moebius Strip
The Klein Bottle
The Projective Plane (Symmetry)
Map Coloring
Networks (Koenigsberg Bridges, Betti Numbers, Knots)
The Trial of the Punctured Torus
Continuity and Discreteness ("Next Number," Continuity, Neighborhoods, Limit Points)
Sets (Valid or Merely True? Venn Diagrams, Open and Closed Sets, Transformations, Mapping, Homotopy)

With this book and a square sheet of paper, the reader can make paper Klein bottles, step by step; then, by intersecting or cutting the bottle, make Moebius strips. Conical Moebius strips, projective planes, the principle of map coloring, the classic problem of the Koenigsberg bridges, and many more aspects of topology are carefully and concisely illuminated by the author's informal and entertaining approach.
Now in this inexpensive paperback edition, Experiments in Topology belongs in the library of any math enthusiast with a taste for brainteasing adventures in the byways of mathematics.

Index

Achilles and the Tortoise

Alexandroff, Paul

annulus

Moebius strip from

Betti, Enrico

Betti numbers

Bolzano-Weierstrass theorem

boundary points

closed sets

defined

closures of sets

compact sets

compactness

complement of set

connectedness

connectivity

continuity

and discreteness

of a line, defined

of transformations

continuum

cross-caps

cross-cuts

cruciform models

cylinder, paper model of

denumerability

dimensions

discreteness

disjoint sets

distortion(see also homeomorphism)

domains

duality:

of cross-cuts

of loop-cuts

duals of maps

epsilon

Euclidean geometry

Euclidean spaces

Euler, Leonhard

Euler’s theorem;

proof of

on a torus

four-color map problem

functions

Gardner, Martin, solution of shortest Moebius strip

Gardner model

glove, inversion of

graphs

discontinuous

homeomorphism

defined

in distortion of Klein bottle

homotopy

images

infinite sets

inner tube, inversion of

intersection of sets

intuition

invariants, topological

inversion :

of glove

of inner tube (punctured torus)

irrational numbers

Jordan curves

Jordan curve theorem

Klein bottle

non-orientability of

paper models of

Slipped-disk

symmetrical

knots

in cut Moebius strip

Koenigsberg Bridges, the

Lefschetz, S.

left vs right

limit points

linkedness

loop-cuts

map coloring

game

on a Moebius strip

on a torus

mappings

maps:

duals of

regular

maxima and minima

metric spaces

minima and maxima

Moebius strip

annular

conical

knot in edge of

map coloring on

non-orientability of

the shortest

square

symmetrical

number of twists in

wire-and-rubber model

neighborhoods

networks

“Next Number,” the

next point, the

null sets

open sets

defined

orientability

not in Klein bottle

paper models:

advantages of, and objections to

of cylinder

of Klein bottle

of Moebius strip

of projective plane (see also Gardner model)

of torus

pi

polyhedra

problems

Programing

projections

projective geometry

projective plane

Gardner model of

punctured torus, inversion of.

puzzles, see problems

rational numbers

real numbers

regular maps

right vs left

Russell, Bertrand

sense, rotational

series

sets

closed defined

closures of

compact

complement of

disjoint

infinite

intersection of

limit points of

null

open defined

subsets

sum of

union of

shortest Moebius strip

Gardner’s solution of

sides, how many

simply connected

Slipped-disk Klein bottle

spaces:

Euclidean

metric

topological

subsets

sum of sets

symmetry

three-color problem, the

tiling theorem

topological invariants

topological spaces

topology:

defined

modern uses of

torus

map coloring on

paper model of

punctured, inversion of

transformations

traverse, law of

trees

true vs valid

twists, number of, in Moebius strip

union, see sum of sets

valid vs true

Venn diagrams

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Table of contents

• DOVER BOOKS ON MATHEMATICAL & LOGICAL PUZZLES, CRYPTOGRAPHY, AND WORD RECREATIONS
• Title Page
• Copyright Page
• Dedication
• Illustrations drawn by Ava Morgan
• Table of Contents
• I - What Is Topology?
• 2 - New Surfaces
• 3 - The Shortest Moebius Strip
• 4 - The Conical Moebius Strip
• 5 - The Klein Bottle
• 6 - The Projective Plane
• 7 - Map Coloring
• 8 - Networks
• 9 - The Trial of the Punctures Torus
• 10 - Continuity and Discreteness
• 11 - Sets
• A PRIMER - PAGE ONE
• In Conclusion
• Appendix
• Index