Donald B. Wagner (University of Copenhagen) has concluded that the third-century Chinese commentator Liu Hui gave a dissection proof of the Pythagorean theorem. His conclusion appears in "A proof of the Pythagorean Theorem by Liu Hui (third century AD)", Historia mathematica 12 (1985), pp. 71-73. Since the diagram to which Liu Hui refers is no longer extant, the conclusion is based on the wording on page 241 in Qian Baocong (ed.), 1963, Suanjing shi shu [Ten mathematical classics], Beijing: Zhonghua shuju.
The University of Michigan has made available an electronic copy of Paul Mahlo's 1908 dissertation.
Bill Casselman (at the University of British Columbia) asked me about the source for Fig. 4.5, which I had neglected to include in the text. The earliest article that I found that contains this figure is "Der Pythagoräische Lehrsatz", by Felix Bernstein (in Göttingen), Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 55 (1924), pp. 204-207.
In Ernest Freese's diagrams (from the 1950's), which I got access to late in 2002, there is a diagram for a 6-piece dissection of two unequal squares to two different unequal squares. Thus he discovered the dissection in Figure 4.8 at least 40 years before I published my book.
In Figure 4.15, I use two uncentered versions of Perigal's dissection. As it turns out, Arthur W. Siddons noted in "1020. Perigal's Dissection for the Theorem of Pythagoras", Mathematical Gazette, vol. 16, no. 217 (February 1932), p. 44, that uncentered versions of Perigal's dissection exist. His Figure 1 illustrates a case when 5 pieces still suffice, while his Figures 2 and 3 show how additional pieces must be used if the version is too uncentered.
- Thanks to David Singmaster for sending me a copy of Siddons' note.
The existence of Theon's class discussed in the updates to chapter 8 suggests a relationship that I missed for three squares to one. The relationship is x (w - x - y) = z (z - w + x) for x < y < z. The resulting 7-piece dissections seem closely related to those for relationship 4.5.
In their 1998 note, "Three Proofs Without Words for the Sine of the Sum," Volker Priebe and Edgar A. Ramos, at the Max-Planck-Institut für Informatik, Saarbrücken, Germany, have found several nice dissection proofs of the addition formula for sines:
sin(alpha+beta) = sin(alpha) cos(beta) + sin(beta) cos(alpha)
Their dissection proofs are generalizations of dissection proofs for the Pythagorean theorem. In place of an x-square and a y-square they have a (sin(alpha) x cos(beta))-rectangle and a (sin(beta) x cos(alpha))-rectangle. In place of the z-square they have an (alpha+beta)-rhombus of sidelength 1. Below on the left is a variation of their second figure, which shows the superposition of tessellations. On the right is the corresponding dissection, which is a generalization of Thabit's dissection. These are analogous to Figures 4.1 and 4.2.
There is also a dissection that is related to Perigal's dissection in the same way that the one by Priebe and Ramos is related to Thabit's dissection. While it is less self-evidently a proof of the addition formula for sines, it is a more symmetric dissection and is even hingeable. The superposition of tessellations and the corresponding dissection below are analogous to Figures 4.5 and 4.6.
As luck would have it, it is the third proof of Priebe and Ramos that is published in "Proof Without Words: The Sine of a Sum," Mathematics Magazine, volume 73, number 5, December 2000. This one is not based on tessellations, and the (sin(alpha) x cos(beta))-rectangle, (sin(beta) x cos(alpha))-rectangle, and (alpha+beta)-rhombus are not cut, but are left when two right triangles of angle alpha and two right triangles of angle beta are rearranged within a ((sin(alpha+sin(beta))) x (cos(alpha)+cos(beta)))-rectangle.
Erwin Dintzl was not the originator of the tessellation of squares and parallelograms that appears in solid edges in Figures 4.25 and 4.27. Instead, the tessellation is in Figure 10 of "Die regelmässigen ebenen Punktsysteme von unbegrenzter Ausdehnung," by Leonhard Sohncke, of Carlsruhe, in Journal für die reine und angewandte Mathematik (Crelle's Journal), volume 77, first part, 1873.
Günter Rote has pointed out that a "Promotion" is the act by which a doctoral degree is bestowed, and also the associated academic ceremony. So Erwin Dintzl would have been awarded a Dr. phil.
(This is corrected in the paperback edition.)
Copyright 1997-2003, Greg N. Frederickson.
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Last updated October 26, 2004.