Sherman Stein asked if "tournay" should not in fact be "tourney". This may be the use of an archaic spelling in Fiske's book, and unfortunately I do not have photocopies of the relevant pages to check. The Oxford English dictionary lists "tournay" as an obsolete form of "tourney", and gives examples of the use of "tournay" in 1820 and 1878. I also ran the chapter from the manuscript through a spelling checker, which I must have forgotten to do back in 1996. The checker picked up misspellings "erronious" and "irresistable", which (of course) should be "erroneous" and "irresistible". Of the two, a copy editor appears to have caught the latter. It seems symbolic, and almost irresistible, that a chapter on dissection fakes should contain erroneous spelling.
(This is corrected in the paperback edition.)
A fascinating article has appeared that discusses Abu'l-Wafa's manuscript "On the Geometric Constructions Necessary for the Artisan". It is "Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World", by Alpay Özdural, has appeared in Historia Mathematica, vol. 27 (2000), pp. 171-201. Follow this link to view the abstract. In the manuscript, Abu'l-Wafa discussed two incorrect methods that Islamic artisans had been using to dissect three squares to one.
One of these methods used just six pieces: One square was left uncut, and one square was cut on its diagonal. The third square was cut into three pieces, of which one is an isosceles right triangle of area one quarter of the square, and the other two pieces are mirror images of each other. With a little bit of thought, you will see that these pieces assemble to give something close to, but not exactly, a square. For an artisan, this would be good enough, because he could hide the approximation with the grout lines in an actual tile floor.
Douglas Rogers has found that Hooper's "Geometric Money" paradox was apparently taken from the 1769-1770 collection Nouvelles récréations physiques et mathétiques by the French author Edmé Gilles Guyot. Actually, Guyot's work contained a mistake (a 3x6 rectangle rather than a 2x6 rectangle), that Guyot corrected before Hooper's first edition in 1774. Hooper's first edition still contained that original mistake, which was corrected in later editions. So it looks like eighteenth-century dissection enthusiasts got a double dose from William Hooper!
About a decade after Oskar Schlömilch published the chessboard paradox, V. Schlegel, in Waren, Germany, wrote an article that generalized the paradox, analyzed it in terms of recurrence relations, and elucidated the link to the Fibonacci numbers and (indirectly) the golden ratio:Schlegel, V. (1879). Verallgemeinerung eines geometrischen Paradoxons. Zeitschrift für Mathematik und Physik 24, 123-128.
Oskar Schlömilch was born in 1823 in Weimar, Germany. After study in Jena, Berlin, and Vienna, he earned a Ph.D. in Jena in 1842, where he became a privatdozent and then an ausserordentlicher professor. In 1849 he was appointed professor of higher mathematics and analytic mechanics at the polytechnic in Dresden. He was the founder (in 1856) and longtime publisher of Zeitschrift für Mathematik und Physik, and published a textbook with Osmar Fort on analytic geometry. Oskar Schlömilch died in 1901.
Torsten Sillke has prepared an interesting article with a substantial bibliography on bamboozlement: Jigsaw Paradox.
Copyright 1997-2003, Greg N. Frederickson.
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Last updated November 24, 2008.