Book reviews for Recreational Problems in Geometric Dissections & How to Solve Them


Book note in the Journal of Recreational Mathematics

The following note came on page 150 of the Journal of Recreational Mathematics, vol. 6, no. 2 (Spring 1973), following a series of book reviews by Charles W. Trigg:
"Those who know of Harry Lindgren's classic Geometric Dissections, published in 1964, … will be glad to learn that Dover Publications … has reprinted the book as Recreational Problems in Geometric Dissections and How to Solve Them. … The reprint, however, is a revised and enlarged (by about 20 pages) version by Greg Frederickson who has broken many of Lindgren's dissection records. J. Recreational Math., readers will recognize Mr. Frederickson as a regular contributor of dissection material."

"If you haven't delved into geometric dissections you simply can't afford to pass up the opportunity to learn about them from this excellent revision of a classic in the field."

Short notice in the Mathematical Gazette

The following notice came on page 364 of the Mathematical Gazette, vol. 57, no. 402 (December 1973):
"`An unusual and stimulating book' said our reviewer of the first edition in Gazette XLIX, 455 (No. 370, December 1965). It is now re-issued at about half the original price, revised by `the one who rendered parts of the work out of date'. This has been done by adding an appendix Eight years after and amending two other appendices."

Review in the Mathematical Spectrum

A review by Edward Harrington Lockwood (E. H. Lockwood), formerly a scholar of St. Johns College, Cambridge, and the Senior Mathematics Master at Felsted School in Felsted, England, appeared on pages 74 and 75 of the Mathematical Spectrum, volume 6, issue 2 (Spring 1974). Here are some excerpts:
"Many of us are apt to think of dissections as isolated puzzles, ingenious and perhaps amusing: but here is a systematic account of methods for finding them which gives new interest to the subject."
"At this point (if not before) the reader begins to feel the urge to do it himself; it becomes a true recreation, that is to say, something to do, with freedom to choose the material and to do it one's own way. The requirements are geometrical instruments, plenty of tracing paper and lots of time---just the thing for anyone making a long stay in a hospital or a prison!"
"The real addict wants to do a dissection with the minimum number of pieces and, as it is rarely possible to prove that the minimum has been reached, it is always intriguing to try to reduce the number."
"There are several useful appendices: problems, with solutions, dimensions of regular polygons, an ingeniously arranged index of dissections, a list of sources and a number of clever dissections invented by the reviser."
"The book is well produced and good value at the price."

Review in the Mathematics Teacher

A review by James D. Bristol, a teacher at Shaker Heights High School in Shaker Heights, Ohio, appeared on page 738 of the Mathematics Teacher, vol. 66, no. 8 (December 1973).
James Bristol, who passed away in 2001, published several books with the D.C. Heath and Co., including:
Here are excerpts from Bristol's review:
"This 1972 booklet is a revised, enlarged, and corrected version of the 1964 publication, Geometric Dissections. It promises only recreation. The topic is single-minded, having 184 pages of concentration on this one topic---dissections."
"The reader must be challenged by such as this: `Can I cut a geometric figure (say a rectangle) into pieces and reassemble them to form another geometric figure (say a square)? Then, if a given dissection required n cuts, can I now re-do the dissection with n - 1 cuts?' This booklet is filled with such examples and exercises: one parallelogram to another, two similar triangles to a third, a heptagon to a square."
"Many are not simple. Most have mathematical ramifications. Perhaps recreation is all that is sought. As a fringe benefit, however, comes the sharpening of one's analytic processes as better solutions are sought---and much mathematics is viewed and reviewed."
"This booklet could be used for diversionary, yet related, study in a geometry or trigonometry class or as a source of challenge to members of a mathematics club."

Review in Monatshefte für Mathematik

A review by Peter Schmitt, in Vienna, Austria, appeared on page 94 of Monatshefte für Mathematik, vol. 78, issue 1 (February 1974). As of Summer 2010, Peter Schmitt was still an "ausserordentlicher Professor" at the University of Vienna! His specialities have included discrete geometry, computational geometry, and tiling theory. His review:

"Eine beliebte Art von Denksportaufgaben sind Zerlegungsaufgaben, bei denen eine geometrische Figur (oder ein Schachbrett .... ) geeignet zerschnitten und die erhaltenen Teile zu einer neuen Figur zusammengesetzt werden sollen. Nun hat schon HILBERT (nach einer anderen Quelle waren es BOLYAI und GERWIEN) gezeigt, dass jedes ebene Polygon auf diese Weise in jedes beliebige flaechengleiche Polygon transformiert werden kann, aber diese Aufgabenstellung war dadurch nicht entwertet, denn es blieb dabei offen, eine oekonomische, d. h. wenig Teile benutzende, oder gar die optimale Zerlegung zu finden. Der Autor hat in diesem Buch alles erreichbare Material zu diesem Thema zusammengetragen (die vorliegende zweite Auflage wurde durch einen Anhang auf den neuesten Stand gebracht). Hauptthema sind die regelmaessigen Polygone, aber auch Sterne und Buchstaben, krummlinige Figuren und Zerlegungen im Raum werden behandelt. Der Autor beschreibt Methoden, die das Finden von Zerlegungen erleichtern. Passionierte Raetselfreunde moegen das vielleicht ablehnen, es regt aber andererseits an, selbst nach neuen Zerlegungen zu suchen. -- Kein mathematisches Buch, sondern ein Buch fuer den Laien und fuer den Mathematiker (man beachte die Beziehungen zur Theorie der Lagerungen und Packungen!)."
Translation: "A popular type of brain teasers are cutting tasks where a geometrical figure (or a checkerboard ....) gets suitably cut and the resulting pieces assembled into a new figure. Now already HILBERT (or according to another source BOLYAI and GERWIEN) showed that each plane polygon can be transformed in this way to any equal-area polygon, but this task is not then cheapened, because it remains open to find an economical, i.e. few pieces used, or even the optimal decomposition. The author has collected in this book all material accessible on this topic (The second edition was brought up to date by an appendix). The main topics treated are not only the regular polygons, but also stars and letters, curvilinear figures and dissections in three dimensions. The author describes methods that facilitate finding decompositions. Passionate puzzle enthusiasts might refuse to like it, but on the other hand it stimulates the search for new dissections. - Not a mathematical book, but a book for both the layman and the mathematician (note the connections with the theory of layout and packing!)."

Telegraphic review in American Mathematical Monthly

Paul J. Campbell, of St. Olaf's College, wrote a telegraphic review that appeared on page 304 of the American Mathematical Monthly, vol. 81, no. 3 (March 1974):
"Basically the 1964 edition with appendix bringing the subject up to date. … `The subject is nowhere near exhaustion', as the results since 1964 attest. Unfortunately, still no index."

Not really a review (but just as good) in Games & Puzzles

Games & Puzzles was a monthly magazine published in the United Kingdom by Edu-Games from 1972 to 1981. What was equivalent to a nice review appeared on page 34 of Games & Puzzles, Issue 53 (October 1976). It was in a feature "CHALLENGE," on "Puzzle Pages" written by David Wells.

David G. Wells has written and published a series of popular math books over the years. These include:
1982: Can You Solve These? Mathematics Problems to Test Your Thinking Powers
1991: What's the Point? Motivation and the Mathematics Crisis
1991: Penguin Dictionary of Curious and Interesting Geometry
1992: Penguin Book of Curious and Interesting Puzzles
1995: You Are a Mathematician: A Wise and Witty Introduction to the Joy of Numbers
1997: Penguin Book of Curious and Interesting Mathematics
1997: Penguin Dictionary of Curious and Interesting Numbers
2005: Prime Numbers: The Most Mysterious Figures in Math
2012: Games and Mathematics: Subtle Connections
2015: Motivating Mathematics: Engaging Teachers and Engaged Students
2018: Hidden Connections and Double Meanings: A Mathematical Exploration

Here is an excerpt (minus the diagrams, which you can fill in for yourself):
"On my left, the exciting and mysterious 12-gon, or dodecagon. From fifteen yards it looks like a circle, but isn't. On my right, a boring old square. (Yawn! Yawn!) Not at all similar, are they? If you were asked to cut the dodecagon into several pieces which would fit together without waste to form the square you would no doubt consider this a tricky problem, not dissimilar to a hole-in-one at golf or a double rook sacrifice at chess (which is actually much rarer than a hole-in-one)."

"No doubt you realize also that you would be quite wrong, or there would be no point to my story. Or perhaps your eyes have naughtily skipped ahead to the next figure. (Pause for applause.) Four straight line cuts and only six pieces are necessary to perform this minor miracle. (Pause for more applause.) It is one of the easier, yes, easier, easier, dissections in a brilliant book which should be on the shelf of every puzzle enthusiast, plus the art critic of the Times newspaper, the Director of the National Gallery and anyone else who is, like, man, into a heavy visual scene. It is called Recreational Problems in Geometric Dissections & How to Solve Them, is written by Harry Lindgren, an Australian patents lawyer, is published by Dover in cheap paperback, and is the bible of dissectors of the non-surgical variety."
. . .
"Our next pair of figures shows an even more dazzling transformation. In eight pieces a 12-pointed star becomes a simple hexagon. If you don't see how the cut-lines are obtained, lay a ruler along the lines and you will notice how they relate to the corners of the star. I was drawn again to browse through this delightful book . . ."
(The above-mentioned "even more dazzling transformation" was of my own creation, appearing in Figure H3 of the Appendix H that I wrote. I discovered most of the dissections in Appendix H, and would proudly classify 25 of them as dynamite, even today!)

Review in Zentralblatt fuer Mathematik

Oene Bottema (1901-1992), a retired full professor of pure and applied mathematics at the Technische Universiteit Delft, in the Netherlands, who worked in geometry and kinematics, wrote a nice review that appeared in Zentralblatt fuer Mathematik. Excerpting from the review, which is indexed as 261.50006 and appears on page 289:
"This Dover edition is a revised and enlarged republication of "Geometric Dissections". ... The main part of the text is not changed, but the reviser corrected some appendices and added a new one. Both authors write witty prose which increases the charm of this nice booklet. ..."
"The subject is on the border line of serious mathematics and the puzzle region, which is emphasized by the new title. It deals with the dissection of a given plane polygon into n pieces, which after rearrangement fill up another given polygon of the same area. The possibility of such a procedure is well-known since Hilbert's days, but a new feature of the problem is to minimize the number n. Many dissections have been found by trial and error, but this book by discussing some general principles tries to arrive at a more systematic approach. It seems, however, often impossible to prove that a certain dissection gives the least value of n. ... The number n varies for each individual case and general rules do not seem to exist."
"For these who know the first edition we mention the reviser's Appendix (`Eight years after'), with many interesting results."

Listing of known reviews for the original edition of Geometric Dissections

Clearly reviews of the original edition do not supply any direct information about the merits of the revision. They are listed only to compare the level of interest in the original edition vis-à-vis interest in the revised edition.

Last updated October 29, 2018.