Dissections: Plane & Fancy

This book promises to serve as the bible of recreational dissections for at least a generation, just as Harry Lindgren's Recreational Problems in Geometric Dissections and How to Solve Them was the classic source from 1964 to the present. Moreover, the two books are intimately connected, for Frederickson edited the 1972 revision of Lindgren's for Dover. In his preface to that revision he wrote, "Several years ago I picked up the first edition of this book. It immediately fascinated me ... I was just enticed by the diagrams. Off and on I would pull the book down off my bookshelf and spend an hour marveling at the beauty of the dissections." This is the reaction I expect people to have to his own book.

What is a "recreational dissection?" Bolyai in 1832 and Gerwein in 1833 proved that a given set of polygons can be cut up into a finite number of pieces which can then be reassembled to form another prescribed set of polygons, as long as the two sets have the same total area. (Frederickson also credits Lowry in 1814 and Wallace in 1831.) The proofs are constructive but produce dissections that tend to have many pieces. That raises a challenge: find economical dissections, ones that use just a few pieces and accomplish the same purpose. The game is devising such dissections, and ideally, one with as few pieces as possible. Though there are a few algorithms for constructing economical dissections, there is no algorithm for deciding whether one has discovered a dissection with the minimum number of pieces.

One may also insist on using only translates of the pieces, or on avoiding flips. In their elegance and ingenuity the solutions often resemble beautiful mathematical proofs. What makes them "recreational" is that the results seem to have no application to other areas of mathematics and are accessible to amateurs.

To indicate the flavor of the problems I cite one: Since 5² + 12² = 13², a square of side 13 can be dissected into pieces which can form a square of side 5 and a square of side 12. The obvious solution is simply to cut it into 169 unit squares, but on page 64 is a dissection into only four pieces, one of which is the square of side 5.

Some other typical problems: transform a square to a triangle, one regular polygon to another or several others, a regular polygon to a star, a cross to a square.

I hope every teacher of high school mathematics will use this book as a source of enrichment in their algebra and geometry classes. It even applies some trigonometry and is a fine introduction to regular tilings of the plane, which are put to use to make economical dissections.

That the book is a monumental labor of love is proved not only by the enthusiastic style, the abundance of clear diagrams, the lengthy list of references, the thorough index, but also by the biographies of 48 contributors to the field (among whom are architects, a chemist, an astronomer, a banker, a businessman, in addition to engineers and mathematicians).

The biographies raise three questions: Why are no women to be found among the contributors? Why do recreational dissectors tend to live to a ripe old age? Could geometry be therapeutic?

I should mention a couple of quibbles. On page 62 Brahmagupta is credited with showing that every basic solution of x² + y² = z² has the form x = m² - n², y = 2mn, and z = m² +n². However, Dickson's History of the Theory of Numbers, Volume 2, page 167, credits Koerbero for this in the year 1738. Also, at the bottom line of page 272 the word "tournay" appears; I assume that this should be "tourney."

Frederickson maintains a web page for the book, which already presents improved dissections, a sample from the book, a few corrections, etc.

It is clear by browsing through the book and the web page that the field of recreational dissections is alive and well. Frederickson's book will lead it well into the next century.

Sherman Stein is the author of Mathematics: The Man-made Universe (now available from Dover), Strength in Numbers (a "popular" book, from Wiley), Algebra and Tiling (with S. Szabo, an MAA book), Geometry, a Guided Inquiry (with C. Crabill and G. D. Chakerian, published by Morton), and Calculus and Analytic Geometry (with T. Barcellos, published by McGraw Hill).