Cogwheels of the Mind: The Story of Venn Diagrams [Hardcover]

A. W. F. Edwards (Author), Ian Stewart (Foreword)

Book Description

Publication Date: April 7, 2004 | ISBN-10: 0801874343 | ISBN-13: 978-0801874345 | Edition: 1

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.

Editorial Reviews

From 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 © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved.

From Scientific American

Three overlapping circles intersect to create eight distinct areas: for well over a century, the classic Venn diagram has helped delineate relationships in fields as diverse as theoretical physics and business strategy. Edwards, a statistician at the University of Cambridge, provides an insightful history of the diagrams, which were developed by English logician John Venn (1834–1923), discussing their presence in everything from Christian iconography to tennis balls and flags.

Editors of Scientific American

See all Editorial Reviews

Product Details

• Reading level: Ages 18 and up
  
• Hardcover: 128 pages
• Publisher: The Johns Hopkins University Press; 1 edition (April 7, 2004)
• Language: English
• ISBN-10: 0801874343
• ISBN-13: 978-0801874345
• Product Dimensions: 9.2 x 6.8 x 0.6 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #115,171 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

8 of 8 people found the following review helpful:

5.0 out of 5 stars Venn Adventures, March 2, 2005

By 

R. Hardy "Rob Hardy" (Columbus, Mississippi USA) - See all my reviews
(TOP 100 REVIEWER)    (HALL OF FAME REVIEWER)    (REAL NAME)   

This review is from: Cogwheels of the Mind: The Story of Venn Diagrams (Hardcover)

Some animals are birds. Some animals are black. Some birds are black. Draw the Venn diagram for these propositions. Most high school students have done Venn diagrams; if you can do this picture, you might have drawn a big rectangle, representing animals, containing two overlapping circles, each representing respectively birds and black, and the overlap representing birds that are black. You may be able to do the diagram, but probably you don't know who Venn was, and you probably don't know what strange reaches of mathematics can be achieved simply by the study of the diagrams. _Cogwheels of the Mind: The Story of Venn Diagrams_ (Johns Hopkins University Press) by A. W. F. Edwards was written to bring light on these aspects of the subject, and very well succeeds. A short book, with a fittingly large number of illustrations, it summarizes a wealth of logical and geometric ideas. Some of the math that springs from these initially simple diagrams will be daunting for those who want to read through quickly, but much is basic and well explained, and the lovely diagrams will repay anyone's study.

It is a book that leads in surprising directions, and as befits such a work, Edwards was drawn to the subject almost by chance. He wanted to put a memorial window up to Venn at the University of Cambridge, and the familiar diagram of three overlapping circles suggested itself. In doing research for the simple window, Edwards began to wonder about drawing a Venn Diagram for more than three sets. John Venn had thought about it, too, and found that circles would do for three, not four. Venn did draw a pretty solution using four ellipses, and realized that any number of sets could be diagrammed, but that the shapes would have to be increasingly convoluted and thus decreasingly explicatory. Much of the book covers Edwards's own research and discovery in producing symmetrical seven-set Venn diagrams. There are a total of six versions of such diagrams, and in the color illustrations they look rather like mandalas, with each of the seven regions of exactly the same curvy shape (one looks rather like a fat footprint), rotated around the circle in seven equidistant steps. Another chapter tells about the author's own method of using spherical surfaces on which to plot the diagrams, and then turning them into Mercator projections. The attractive seam of a tennis ball looms large in his investigations.

This is a very personal book about the subject. It is written in the first person, and tells of the author's attempts to hit different problems he had proposed to himself. It is not without humor; at one point in his account of hunting down a symmetrical seven-set diagram (an account which "conveys some of the excitement of the chase"), he says that in 1992 he had drawn a hopeful initial diagram but that "On Monday evening, 16 November, I was bitten by a dog, so on the Tuesday I stayed in [college] in the morning." Thus do worldly concerns impede intellectual pursuits. Those familiar with themes frequently found in recreational mathematics will find friends here, with connections between Venn diagrams and Boolean algebra and Grey codes and hypercubes. It will be no surprise to find Lewis Carroll doing diagrams in his own way, but it might be a surprise to find the diagram, reproduced here, that Winston Churchill drew in 1948 at Hever Castle to show the relations of the sets of the British Empire, United Europe, and the English-Speaking World. It is a vivid illustration of how universal Venn diagrams are, and Edwards's book is a lively description of one researcher's route of enthusiastic discovery.

7 of 7 people found the following review helpful:

4.0 out of 5 stars words & PICTURES, June 22, 2005

By 

Bruce Crocker "agnostictrickster" (Whittier, California United States) - See all my reviews
(REAL NAME)   

This review is from: Cogwheels of the Mind: The Story of Venn Diagrams (Hardcover)

Cogwheels Of The Mind by A.W. F. Edwards is a pretty and frustrating little book. The text starts out as a history of Venn diagrams, but veers into a trip through Edwards's recent thinking on Venn diagrams [which is the author's prerogative, but does make the subtitle of the book - The Story of Venn Diagrams - a little misleading]. It's all good reading, but given the brevity of the text, I found myself wishing that Edwards had stayed on task a little more. The pictures are stunning! If you're into diagrams like I am, then the pictures will be a feast for your mind. To me the book feels like two well illustrated journal articles or maybe a really small coffee table book. Cogwheels Of The Mind is cool, but flawed.

9 of 10 people found the following review helpful:

4.0 out of 5 stars Interesting, but who's the audience?, July 27, 2004

By 

Jonathan Badger (Washington, DC United States) - See all my reviews
(REAL NAME)   

This review is from: Cogwheels of the Mind: The Story of Venn Diagrams (Hardcover)

I was attracted to this book largely because I was amused to see what the well known bio-statistician A.W.F. Edwards was doing with Venn diagrams -- it turns out that it has little to do with the main thrust of his research -- Edwards simply enjoys as a hobby recreational mathematics similar to what used to be presented in Martin Gardner's Scientific American columns.

Well, fair enough. Edwards writes an interesting story dealing with the life of John Venn, various rival presentation schemes, and ends with Edwards' own (successful) quest to generalize Venn diagrams to an arbitrary number of sets. The only problem is it isn't clear for whom Edwards is writing the book. If it's for mathematicians, even amateur ones, some proofs would be in order (none are in the book), and if it is for the general public, more historical detail would be in order. Still, the book is an enjoyable (if short) read.