From: Dave Rusin [SMTP:rusin@math.niu.edu] Date: Thursday, April 16, 1998 12:21 AM To: lbpt@aol.com Subject: Re: Impossible constructions I assume you know about the True Accounts of these. Reviews below. dave (Ask Dave Barry; he probably submitted a few impossible constructions to his geometry teacher...) 94m:51028 51M15 01A05 51-01 Dudley, Underwood(1-DPAW) The trisectors. (English. English summary) Revised edition. MAA Spectrum. Mathematical Association of America, Washington, DC, 1994. xviii+184 pp. $27.50. ISBN 0-88385-514-3 _________________________________________________________________ \{The first edition has been reviewed [MR 89a:51045].\} From the preface: "I have not added any new constructions to this edition since the new ones are much like the old. There is a small amount of new material and the old material has been revised in small ways. Many of the errors in the first edition have been corrected and, unfortunately but inevitably, brand-new errors have been added to this edition, though not on purpose. The error-free book, like the trisection-free world, is an unattainable ideal." _________________________________________________________________ Next Review 93m:00003 00A08 01A80 Dudley, Underwood(1-DPAW) Mathematical cranks. (English) MAA Spectrum. Mathematical Association of America, Washington, DC, 1992. x+372 pp. $25.00. ISBN 0-88385-507-0 _________________________________________________________________ Mathematics seems to attract an extraordinary number of cranks. Already in 414 BC, Aristophanes' The birds mentions circle-squarers, but most cranks, like most mathematicians, are comparatively recent, say from about 1700. There have only been two other books on this topic. Augustus De Morgan's A budget of paradoxes [Longmans, Green, London, 1872; second edition, Open Court, Chicago, IL, 1915; reprint, Dover, New York, 1954] is a posthumous collection of his essays in the Athenaeum. It covers cranks in all fields and much else besides. Being a collection of articles, it is not a systematic exposition and certainly cannot be considered an organized survey of mathematical crankery. More closely related is a previous book by the author [A budget of trisections, Springer, New York, 1987; MR 89a:51045], but this covers just one topic in the field. So the present book is really the first attempt to survey the whole field and nothing but the field of mathematical cranks. The book is organized into topics arranged alphabetically, though one could easily do some grouping into larger chapters. All the obvious topics are present: The best base for the number systems; Duplication of the cube; Fermat's last theorem; Euclid's fifth postulate; The four-color theorem; Phi (the golden ratio); Quadrature of the circle; Trisection of the angle. In addition, a truly surprising number of other mathematical topics have attracted the attention of cranks: Cantor's diagonal process; Circumference of an ellipse; Solving equations (e.g. of the fifth degree); Fermat's little theorem; Godel's theorem; The Goldbach conjecture; Magic squares; Casting out nines; Regular nonagons; Perfect numbers; Existence of infinitely many twin primes; Set theory; Parameter estimation in statistics; van der Pol's equation. Several of these topics, particularly the best base for the number system and the golden ratio, are quite reasonable topics on the fringes of mathematics and only become crankery in extreme cases. Several topics which are often considered as mathematical crankery are here omitted or just covered in passing, because they are really uses of mathematics rather than mathematics itself. Pyramidology occurs only in a short topic on Application of pyramid height to the alphabet. Numerology occurs briefly in a number of topics. Dubious appearances of the golden ratio, astrology, etc. are not mentioned at all. A number of topics are so individual that only one person is involved. Examples: The role of 57 in the American Revolution (coincidentally---or not coincidentally---this book has 57 topics. Does this reveal a profound and hitherto unsuspected connection with the American Revolution?); Legislating pi [D. Singmaster, Math. Intelligencer 7 (1985), no. 2, 69--72]; Conspiracy involving linear programming; Matrix prayer; Philosophy of the sphere. The most recurrent theme running through all topics is the nature of cranks. Several topics are primarily devoted to this theme---their titles are self-explanatory: Cranks' bitterness; Lack of consultation of cranks with experts; Case study of a crank; The making of a crank; The folly of encouraging cranks; Greed; Incomprehensibility of cranks' works; Insanity; Megalomania; Lack of money to be made in mathematics; Cranks' prolificity; Wasted time. The crank ranges across the spectrum of humanity---some are professors (even of mathematics), one was president of a reputable university (the reviewer recalls one not mentioned by the author who was a general in South America), but most seem to be relatively uneducated and many seem to have no mathematical background. In some cases, crankiness is a minor quirk in an individual who otherwise leads a normal life, but in the more conspicuous cases the quirk has become an obsession, consuming all their time, energy and finances and sometimes leading to genuine madness---several of those quoted were writing from prisons or hospitals. In a few cases, the author or a colleague was in touch with the crank for some time and the course of the disease can be followed. It is unfortunate that mathematics can make some men mad, but one rationalises that the mathematics is just a symptom rather than a cause. Although some cranks become extremely abusive in letters, none seem to have resorted to physical violence! The author's attitude, like that of De Morgan before him, is invariably human: a mixture of amazement and amusement with sympathy, sadness and despair. Although the number of cranks is proportionally small, every known department or mathematician will come in contact with some. The initial contact can be quite innocuous---indistinguishable from an ordinary request that one feels duty-bound to answer. Who wants to ignore another Ramanujan? Even a full-blown crank can appear to be a neglected genius to the nonexpert and he can get substantial coverage in newspapers, or a legislature, or even The Congressional Record. Such coverage is relatively harmless in the long run, but it does contribute to the public's distorted view of mathematics. This book thoroughly records this interesting aspect of mathematical folklore. More importantly, it is a valuable guide for the perplexed mathematician or journalist or legislator who comes in contact with a crank. Anyone likely to deal with a crank should read this in order to know what to expect. Reviewed by David Singmaster Previous Review _________________________________________________________________ 89a:51045 51M15 01A05 51-01 Dudley, Underwood A budget of trisections. (English) Springer-Verlag, New York-Berlin, 1987. xvi+169 pp. $29.80. ISBN 0-387-96568-8 _________________________________________________________________ This book does not present another proof of the impossibility of trisecting an angle with ruler and compass, but an exhaustive (and exhausting) list of attempts by amateurs to trisect an arbitrary angle. Amusingly written, it not only analyses numerous attempts at solving the problem, but also gives accurate profiles of typical trisectors. Its main use for mathematicians will be to ward off would-be trisectors. Reviewed by Cyril W. L. Garner Cited in: 94m:51028 93m:00003 (c) Copyright American Mathematical Society 1998