Five years after my first book, Dissections: Plane & Fancy, came out Jeff Oaks (Mathematics, University of Indianapolis) pointed out that it is more appropriate to say, for example, "Arabic-Islamic civilization" rather than "Arabian-Islamic civilization." Unfortunately, production on Hinged Dissections was already done, so that this type of error is in this book as well. On page 1, we should have "Arabic-Islamic" rather than "Arabian."
On page 3, I wrote:I have since discovered such an abundance of hinged dissections that it is tempting to ask: Given any pair of figures that are of equal area and bounded by straight line segments, is it always possible to find a swing-hingeable dissection of them?The origin of this question was in the spring of 1997, after I had delivered the manuscript of my first book, Dissections: Plane & Fancy to the publishers. I thought some about this question, but not getting much of anywhere, decided to try to find more hinged dissections. Within two years, I had a 240-page manuscript (tentatively entitled Dissections Too! Swingin'). Another year boosted the page count to 320 pages, and I formally identified the open problem in my paper, "Geometric Dissections That Swing and Twist", in Discrete and Computational Geometry, Japanese Conference, JCDCG 2000, Springer LNCS 2098, pp. 137-148.
With no one having resolved the question by 2006, I restated it on page 31 of my third book, Piano-Hinged Dissections: Time to Fold!. That must have done the trick, because within a year, the answer was in hand: Yes! The answer is in "Hinged Dissections Exist," by Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott D. Kominers, in Proceedings of 24th Annual Symposium on Computational Geometry, College Park, MD, (2008), pp. 110-119.
The argument is ingenious, although somewhat involved, and when suitably generalized seems to work for those pairs of 3-dimensional figures between which finite dissections exist. (Recall that such pairs of figures must satisfy the "Dehn invariant", which almost all pairs do not. Still, this is a lovely achievement!)
Even for 2-dimensional figures, the resulting dissections have a very large number of pieces. I haven't seen a diagram that shows a complete version of a hinged dissection generated by this method, and I doubt that anyone will really want to produce a working physical model of such a dissection. There are just so many pieces, many of which are extremely thin, upon which hinges must be mounted---a true nightmare for any craftsman. Nonetheless, it's a pretty mathematical result, and now we know that we can always find some hinged dissection.
On page 5, I noted that Johannes Böttcher (1921) had given a diagram illustrating a 3-piece hinged dissection of two attached squares to one. His hinging was different from that in Figure 1.6 and also had a mistake in the orientation of one of the triangles. Actually, he had published a different variation (and entirely equivalent to my figure) much earlier in:"Einfaches Modell zum Beweis des Pythagoreischen Satzes," by Dr. J. E. Böttcher, Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, vol. 17 (1886), pages 99-100.The editor of the journal noted in a footnote:"Herr Dr. Böttcher war so freundlich, uns das diesbezügliche Modell zur Verfügung zu stellen, und wir haben dasselbe ausserordentlich instruktiv gefunden. Es kann auch leicht von Schülern selbst angefertigt werden."Johannes Eduard Böttcher was born 1847 in Dresden, Germany. During 1865-70 he was a student at Leipzig and Königsberg. He earned a Ph.D. for work on particle dynamics at the University of Leipzig in 1875. Böttcher became a teacher in 1871, a master teacher in 1872, a professor in 1892, and rector in 1897, at the Realgymnasium in Leipzig. He published Beweise für die Heronsformel aus zwei jahrtausenden (1909) and Alles Jahreskalender auf einem Blatt (1913). Böttcher died in 1919.
Translation: Dr. Böttcher was so friendly as to put the relevant model at our disposal, and we found the same extraordinarily instructive. It can also be made easily by pupils.)
Donald Bruyr was an early proponent of hinged dissections, even illustrating a model that is related to piano-hinged dissections (see page 7 of Piano-hinged Dissections: Time to Fold!). Thus it seems appropriate to give some details of his life:
Donald Lee Bruyr was born Kansas in 1930. He earned a B.S. in mathematics from Pittsburg State College in 1951 and an Ed.D. in mathematics from Oklahoma State University in 1964. He taught high school math 1951-1960 and was a professor at Kansas State Teachers College (now Emporia State Univ.) from 1960 until his death in 1979. The annual high school math competition that he inaugurated at Emporia State is named the "Donald L. Bruyr Mathematics Day" in his memory.Bruyr had a great enthusiasm for life: His fourth child was born on his fifth wedding anniversary! He painted, taught himself to play banjo, guitar, and harmonica, built furniture, brewed beer (bursting a few bottles along the way), and collected beer cans. He also loved to travel with his family. Stopped at the Mexican border from bringing back filled beer cans on a vacation, he kept the family waiting a long time as he (laboriously) drained the cans from the bottom so as not to decrease their value by popping the tops.
Copyright 2002-2006, Greg N. Frederickson.
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Last updated June 16, 2008.