Cogwheels of the Mind: The Story of Venn Diagrams

Overview

Used today in spheres of life as diverse as business strategy, creative writing, medicine, computer science, and theoretical physics, Venn diagrams possess fascinating properties. The basic Venn diagram is both elegantly simple—three overlapping circles that intersect to create eight distinct areas—and conceptually innovative. Devised by English logician John Venn (1834–1923) to visually represent complex logical propositions and algebraic statements, the diagrams drew the ...

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Overview

Used today in spheres of life as diverse as business strategy, creative writing, medicine, computer science, and theoretical physics, Venn diagrams possess fascinating properties. The basic Venn diagram is both elegantly simple—three overlapping circles that intersect to create eight distinct areas—and conceptually innovative. Devised by English logician John Venn (1834–1923) to visually represent complex logical propositions and algebraic statements, the diagrams drew the excited interest of both scholars and the general public.

In Cogwheels of the Mind, statistician and geneticist A. W. F. Edwards provides an accessible and engaging history of the Venn diagram, its reception and evolution, and its presence in such objects and images as Christian iconography, tennis balls, and flags which provide a rich source of Venn diagrams for Edwards, including those of Switzerland, Poland, and Japan (all one-set Venn diagrams), Greenland (a two-set Venn diagram), and Maryland (a three-set device).

Edwards begins with a sketch of Venn's life, his discovery of the three-circle design while developing a series of lectures on symbolic logic at Cambridge University, and the publication of his find in an 1880 paper, and, more influentially, his 1881 book, Symbolic Logic. Edwards discusses the rival diagrammatic scheme invented by Charles Dodgson, better known as Lewis Carroll, who also developed a board game based on his design. The author also recreates famous Venn diagrams from history, including Winston Churchill's of 1948 depicting the mutual interests of the British Empire, a united Europe, and the English-speaking world, with the United Kingdom located at the intersection.

Edwards goes on to show how different shapes can be linked together to form artistically beautiful and mathematically important, multi-set Venn diagrams, including the author's own influential Adelaide variation. And he delineates the possibilities for expanding the analytic power of these diagrams far beyond those first appreciated by Venn. Edwards even tells readers how to draw complex Venn diagrams on a spherical surface to create "Vennis balls." For anyone interested in mathematics or its history, Cogwheels of the Mind is invaluable and compelling reading.

Johns Hopkins University Press

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Editorial Reviews

Convergence - Jim Kiernan

A very short readable book on attempts to physically represent the intersection of any number of sets.

Review of Modern Logic - Amirouche Moktefi

This is a wonderful book which should be taken simply for what it is, the story of the Venn-Edwards diagrams.

School Science and Mathematics - Darlinda Cassel

An easy, friendly, and enlightening book to read... Would be of particular interest to college professors, especially those involved in teaching a History of Mathematics class and/or a Graph theory class.

Choice

Venn diagrams are familiar as pictorial representations of relationships among sets, and statistician Edwards discusses their development... He shows how to interlink shapes to form beautiful Venn diagrams.

Booklist

Edwards arrives with this pleasing little history about who John Venn was, why he conceived of the diagram, and the properties that lie secreted beneath such a seemingly simple mathematical object... A world-class authority, Edwards... proves himself wholly accessible to anyone interested in reading about mathematics.

Nature
Deserves to become a minor classic and may well go on to make many friends for mathematics.

— Jeremy Gray

Boston Sunday Globe

Entertaining.

New Scientist
Edwards is a charming if earnest guide, and the many illustrations of the beautiful cogwheels will fascinate and satisfy.

— Ben Longstaff

MAA Online
The kind of book that I can imagine giving to a wide range of readers: any junior high student would be able to follow the mathematics, and most professors would find it interesting... I found it to be a nice—if light—read, and it is well worth a look.

— Darren Glass

Significance
Give this book to any youngster with an enquiring mind, and watch delight develop.

— John Haigh

The Physicist
There is a nice balance between personal anecdotes, history, aesthetics and attention to detail... The overall result should become something of a classic.

— B. I. Henry

Choice

Venn diagrams are familiar as pictorial representations of relationships among sets, and statistician Edwards discusses their development... He shows how to interlink shapes to form beautiful Venn diagrams.

American Scientist
There is no better place to start than with Cogwheels of the Mind.

— Frank Ruskey

Scientific American

An insightful history of the diagrams.

The Times of Acadiana
A short book, with a fittingly large number of illustrations, it summarizes a wealth of logical and geometric ideas.

— Rob Hardy

Mathematics Teacher
If you have an interest in the history of mathematical ideas and the creative process of mathematicians, then I recommend this book.

— John Wilkins

Convergence
A very short readable book on attempts to physically represent the intersection of any number of sets.

— Jim Kiernan

Review of Modern Logic
This is a wonderful book which should be taken simply for what it is, the story of the Venn-Edwards diagrams.

— Amirouche Moktefi

Historia Mathematica

An engaging, very readable, and profusely illustrated account.

School Science and Mathematics
An easy, friendly, and enlightening book to read... Would be of particular interest to college professors, especially those involved in teaching a History of Mathematics class and/or a Graph theory class.

— Darlinda Cassel

Nature - Jeremy Gray

Deserves to become a minor classic and may well go on to make many friends for mathematics.

New Scientist - Ben Longstaff

Edwards is a charming if earnest guide, and the many illustrations of the beautiful cogwheels will fascinate and satisfy.

MAA Online - Darren Glass

The kind of book that I can imagine giving to a wide range of readers: any junior high student would be able to follow the mathematics, and most professors would find it interesting... I found it to be a nice—if light—read, and it is well worth a look.

Significance - John Haigh

Give this book to any youngster with an enquiring mind, and watch delight develop.

The Physicist - B. I. Henry

There is a nice balance between personal anecdotes, history, aesthetics and attention to detail... The overall result should become something of a classic.

American Scientist - Frank Ruskey

There is no better place to start than with Cogwheels of the Mind.

British Society for the History of Mathematics Bulletin - Peter M. Neumann

Lovely little book... which should not be summarized. It should be bought and enjoyed first-hand. It is a book in which mathematics and its history are combined in a lovely autobiographical account of a voyage of discovery. The mathematics is interesting, the history is interesting, the personal account is interesting, and the book, with its elegant full-colour pictures and diagrams is beautifully produced. The author and his publisher are to be warmly thanked and congratulated.

The Times of Acadiana - Rob Hardy

A short book, with a fittingly large number of illustrations, it summarizes a wealth of logical and geometric ideas.

Mathematics Teacher - John Wilkins

If you have an interest in the history of mathematical ideas and the creative process of mathematicians, then I recommend this book.

Jeremy Gray
[Cogwheels of the Mind] deserves to become a minor classic and may well go on to make many friends for mathematics.
"Nature", 22 July 2004
School Library Journal
Adult/High School-While it appears to be a simple construct with no underlying theory, the diagram invented by English logician John Venn (1843-1923) has provided new foundations for the study of logic and new insights into the structure and function of organizations. Those three partially overlapping circles can show many things, including the interrelationships between the key components of a business plan or the ways that different groups can support one another. This fascinating book traces the diagram's development and makes interesting research contributions to the subject. A key feature is Edwards's description of the ways in which mathematicians and logicians undertake their research and advance knowledge that can be used by others. Particularly interesting is the author's account of his own research, which has produced new concepts in logic and led to the construction of new diagrams of outstanding visual beauty and complexity. Many excellent and graphically exciting illustrations of Venn diagrams transform what might have been a simple math book into one that shows that mathematics can generate pictures that could be considered to be in the forefront of modern art. This title will appeal to readers studying mathematics and logic, to those who would like to know how scientific and mathematical research is carried out, and to those who are involved in graphic design and the study of the history of art as it relates to math.-Ted Woodcock, George Mason University, Arlington, VA Copyright 2004 Reed Business Information.
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Product Details

  • ISBN-13: 9780801874345
  • Publisher: Johns Hopkins University Press
  • Publication date: 4/28/2004
  • Pages: 128
  • Product dimensions: 7.00 (w) x 9.25 (h) x 0.56 (d)

Meet the Author

A. W. F. Edwards is, like John Venn before him, a fellow of Gonville and Caius College, Cambridge. He is the author of Likelihood and Pascal's Arithmetical Triangle, both available from Johns Hopkins.

Johns Hopkins University Press

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Table of Contents

Foreword
Preface
1 John Venn and his logic diagram 1
2 Rings, flags, and balls 17
3 Five and more sets 29
4 The gray code, binomial coefficients, and the revolving-door algorithm 47
5 Cosine curves and sine curves 61
6 Ironing the hypercube 77
7 Diagrams with rotational symmetry 84
App. 1 Metrical Venn diagrams 95
App. 2 A Rotatable Edwards-Venn diagram 101
References 105
Index 109
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