Cogwheels of the Mind: The Story of Venn Diagrams

by Edwards, A. W. F.

• ISBN13: 9780801874345
• ISBN10: 0801874343
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-04-01
• Publisher: JOHNS HOPKINS UNIV PRESS

Summary

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.

Author Biography

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.

Table of Contents

Foreword, by Ian Stewart	ix
Preface	xv
1. John Venn and His Logic Diagram	1	(16)
John Venn, 1834-1923.
Logic diagrams.
Euler's circles.
Venn diagrams.
Boolean algebra.
Binary labeling.
Churchill's Venn diagram.
Earlier diagrams.
W. Stanley Jevons.
H.J.S. Smith.
2. Rings, Flags, and Balls	17	(12)
Precursors of the pattern.
The Trinity.
Borromean rings.
Lewis Carroll's diagram.
The Game of Logic.
Two-colorable property.
The flag of Greenland and the arms of Maryland.
The diagram on a sphere.
The globe.
Projections.
Tennis balls and basketballs.
3. Five and More Sets	29	(18)
Adding sets.
C.S. Peirce.
Venn's attempt.
Lewis Carroll's attempt.
Edwards's solution.
Other solutions: Grünbaum's, Humphries's, Fisher and Koh's.
4. The Gray Code, Binomial Coefficients, and the Revolving-Door Algorithm	47	(14)
Edwards's form maps a Gray code.
Alternative forms.
The binomial coefficients represented by necklaces of regions.
The revolving-door algorithm.
Jevons again.
5. Cosine Curves and Sine Curves	61	(16)
The linear diagram.
C.A.B. Smith's variant maps the binary code.
Martin Gardner's "proof" that π = 2.
Hybrid diagrams.
The rotatable diagram.
6. Ironing the Hypertube	77	(7)
The dual diagram maps the hypertube.
The maximal planar subgraph.
The Karnaugh map.
The Hamming code.
Hamilton circuits.
7. Diagrams with Rotational Symmetry	84	(11)
Leibniz's divisibility theorem and Henderson symmetry.
Grunbaum's symmetrical five-set diagram.
Polar symmetry.
Edwards's family of seven-set diagrams with rotational and polar symmetry.
More diagrams.
Appendix 1. Metrical Venn Diagrams	95	(6)
Appendix 2. A Rotatable Edwards-Venn Diagram	101	(4)
References	105	(4)
Index	109