Henry Dudeney exhibited the dissection of an equilateral triangle to a square before the Royal Society at Burlington House on May 17, 1905. I have recently learned more about Henry Dudeney's exhibit: The 1906 Yearbook of the Royal Society contains on pages 140-144 a "Catalogue of objects and experiments exhibited at the conversazione held in the Society's apartments at Burlington House on May 17, 1905." Click here for more details.
Over the last decade, Gavin Theobald has identified several additional strip techniques, which he has now described on his webpage Methods. These new techniques allow the common area in a crossposition to be one half the area of each of the dissected figures. Right now there aren't too many examples of dissections created by these techniques, but I'm sure that there will be more in the future.
Most intriguing is Gavin's suggestion that the common area could be 1/3 or 1/4 of the area of each of the dissected figures. Of course, 1/n for any whole number n would be possible. How about 2/3?
On page 137, I wrote that the 4-piece dissection of an equilateral triangle to a square published by Dudeney (Dispatch, 1902) is the earliest example of a dissection that could have been discovered by the T-strip technique. This is wrong: Philip Kelland (1864) gave a number of dissections of a gnomon (consisting of 3 attached squares) to a square. His dissection XXII in that paper can be derived as a TT2 dissection.
On page 138, I noted that Henry Taylor (1905) had recognized that the dissection of an equilateral triangle to a square was a special case of a nonequilateral triangle to a parallelogram. But at the end of his paper, Taylor gave an equivalent procedure (not illustrated) for dissecting a trapezoid to a nonequilateral triangle. (For both dissections, the dimensions of the figures must not be too different.) This dissection is used in Fig. 5.9, where I should have referred to a trapezoid rather than a quadrilateral. Thanks to Anton Hanegraaf for identifying the attribution.
On page 138, I had asked the question whether everyone had missed the fact that Dudeney's legendary hinged model of a triangle to a square could be hinged in three other ways. In Fig. 157 of his book Geometrical Models and Demonstrations, published by J. Weston Walch, Portland, Maine, 1963, Donald L. Bruyr gave a 4-piece hinged dissection of a square into an isosceles triangle whose height equals the diagonal of the square. For this variation of the well-known dissection, each edge of the square is cut at its midpoint. Bruyr remarked that the pieces can be hinged at any three of the four midpoints of the square's sides.
Günter Rote has observed that the description of the triangle to square dissection in Hugo Steinhaus's book, Mathematical Snapshots, third American edition, Oxford University Press, New York, 1969, is inaccurate. Figure 2 on page 4 shows the bottom side of the triangle divided into three lengths in the ratio of 1 : 2 : 1. This is clearly wrong, but it is true that the middle segment is one half of the length of the side. I would also question the citation of Figure 1 and the attribution of Figure 3 as given in the "Notes" on page 297.
In fact, Donald W. Crowe and I. J. Schoenberg, identified the problem with Figure 2 in their paper, "On the Equidecomposability of a Regular Triangle and a Square of Equal Areas," Mitteilungen aus dem Mathematische Seminar Giessen, Heft 164, Coxeter-Festschrift, Teil II, Giessen, 1984.
Wolfgang Stöcher, an Austrian mathematician and software developer, has found a lovely dissection of a {3n} to an {n}, in 4n+1 pieces. The general dissection has a lovely 3-fold rotational symmetry, but does turn pieces over. It's not as good as known techniques for n = 3 or n = 4, but no one had even thought to tackle the general problem. And so elegantly!
After thinking about Wolfgang's dissection, I was able to make a simple change to it that reduced the number of pieces to 3n+1. Before he saw my message describing this, Wolfgang also discovered this improvement. Then he thought some more, and several days later found a way to reduce the number of pieces to ceiling(5n/2)+1. Wolfgang has posted a short discussion of his method.
An anchor point on the extreme righthand side of Figure 12.23 is slightly out of position.
(This is corrected in the paperback edition.)
Gavin Theobald sent me his dissections just a few months before a final version of my manuscript was due at the publisher. In addition, I was running up against the page limit that my publisher had set. So I never had full opportunity to include as many of his dissections as I would have liked. Now you can see a wonderful presentation of so many, many more of Gavin's dissections.
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Dissections: Plane & Fancy
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Last updated March 27, 2019.