Updates to Chapter 1, " `Dat Pussle' ",
in Dissections: Plane & Fancy, by Greg N. Frederickson

Leonardo da Vinci's dissections

It was a revelation to learn that Leonardo da Vinci had studied geometric dissections (among many other interests) in the notebooks of his known as the Codex Atlanticus. These are available as:
  1. Leonardo da Vinci, Il Codice Atlantico: della Biblioteca Ambrosiana di Milano, Firenze: Giunti-Barbera, 1975.
  2. Leonardo da Vinci, Il Codice Atlantico, Firenze: Giunti-Barbera, 1973.
The first is a set of 12 volumes that contains a transcription of Leonardo's notes (in Italian). The second is a set that contains a facsimile reproduction of the notebooks.

Earlier source for Figure 1.2

In Ernest Freese's manuscript, Geometric Transformations (completed before his death in 1957), which I got access to in February 2003, there is a diagram of the same 9-piece dissection of a dodecagon to three squares as in Figure 1.2, which I had attributed to C. Stuart Elliott.

Book by Evans Valens

I have belatedly learned of a book by Evans Valens that has a lot of material on dissections in it:
  1. Evans G. Valens, The Number of Things, New York: Dutton, 1964.
The author gives a highly accessible discussion of geometric dissections, especially with respect to the Pythagorean theorem. He gives C. Dudley Langford's dissection of two octagons to one, without attribution. Evans has a nice dissection of dodecagons that I had not seen before, of side lengths 1, sqrt(2), and sqrt(3) in 12 pieces.

Additional references

Here are some additional books that I would have cited, if I had been aware of them when I wrote my book.
  1. Vladimir Nikolaevich Belov, ed., V Labirintakh Igr i Golovolomok (In the Labyrinths of Games and Puzzles), St. Petersburg: Lenizdat, 1992, ISBN 5-289-01150-1.
  2. Leonid Petrovich Mochalov, Golovolomki (Puzzles), Moscow: "Nauka", 1980. Revised and expanded: 1996, ISBN 5-09-005131-3.
  3. Roger B. Nelsen, Proofs Without Words, Washington: The Mathematical Association of America, 1993, ISBN 0-88385-700-6.
The first book (in Russian) contains a 10-page section (pp. 79-88) by Leonid Petrovich Mochalov on geometric dissections. It presents a number of dissection problems that I have not seen elsewhere. The second book (also in Russian) contains similar material, with the 1996 edition containing more dissections than the 1980 edition. The third uses dissection methods to demonstrate a number of numeric identities.
Vladimir Belov sent me sections from the following puzzle books (in Russian) that deal with geometric dissections.
  1. E. I. Ignat'ev, V Tsarstve Smekalky (In the Kingdom of Quick Wits), Moscow: `Nauka' publishing house, 1981 (reissue of the 1908 edition).
  2. N. N. Amenitskiy and I. P. Sakharov, Zabavnaya Arifmetika (Amusing Arithmetic), Moscow: `Nauka', 1992 (reissue of the 1910 edition).
  3. Ya. I. Perelman, Veselye Zadachi (Funny Problems), Moscow: Rusanov, 1997 (reissue of the 1914 edition).
  4. Ya. I. Perelman, Zhivaya Matematika (Lively Mathematics), Moscow: `Nauka', 1967 (reissue of the 1934 edition).
The dissection material in these books seems to derive from sources in the West.

Yet more references

Joe Malkevitch (at CUNY) suggested that I might want to alert readers to some books that would take them into related areas:
  1. Russell V. Benton, Euclidean Geometry and Convexity, New York: McGraw-Hill, 1966.
  2. Chih-han Sah, Hilbert's Third Problem: Scissors Congruence, San Francisco: Pitman Advanced Publishing, 1979, Research Notes in Mathematics, vol. 33.
  3. Stan Wagon, The Banach-Tarski Paradox, Cambridge, England: Cambridge University Press, 1985, Encyclopedia of Mathematics and its Applications, vol. 24.
The first of these books has a nice presentation of the Bolyai-Gerwien theorem (in section 3) and of some of Dehn's work (in section 15). The second book is an advanced monograph. The third book concerns paradoxes based in measure theory and is accessible to first- or second-year graduate students in mathematics. A warning: These books have little flavor of recreational mathematics.
Jean-Paul Delahaye (at Universite des Sciences et Technologies de Lille) suggested the following reference:
  1. Miklós Laczkovich, "Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem", Journal für die reine und angewandte Mathematik, volume 404 (1990), pages 77-117.
Related to the Banach-Tarski theorem, the results in this article do not apply when you want to use, for example, scissors and paper. The previous warning seems also to apply to this article.

Copyright 1997-2003, Greg N. Frederickson.
Permission is granted to any purchaser of Dissections: Plane & Fancy to print out a copy of this page for his or her own personal use.

Last updated March 22, 2003.