There is another 3-piece dissection that converts a triangle to its mirror image: Orient the triangle so that its base is a longest side. Identify the point of the base directly beneath the apex. Make cuts from the midpoints of the two other sides to this point on the base. The earliest source that I know of for this dissection is the article by Taylor (1905) in the bibliography. This dissection has the merit of being simple and hingeable. The dissection that I give in Fig. 3.9 has the merit of being generalizable to a 6-piece dissection of a tetrahedron to its mirror image, which I give in Fig. 20.1-20.3.
I have found two sources for Fig. 3.11. The earliest source that I have found with an actual diagram is the article by Böttcher (1921) in the bibliography. His Fig. 8 shows the two right triangles hinged, with their rotational motion indicated by curved arrows. The triangle on the right is drawn in the wrong orientation, but it is clear what the orientation should be by examination of his Fig. 7. An earlier source, without a diagram, is by Kelland (1864), where he writes:The modification which I gave of the demonstration of [the 47th proposition of Euclid's first book] in the notes to my edition of Playfair's Geometry (edition 1846, p.~273), has had the honor of being exhibited in two mechanical forms. The first by rotations without sliding, whereby the two squares on the sides, when placed together, are converted into the square on the hypothenuse; the second, by two transpositions (slidings) without rotation, whereby the same change is effected. The former is obvious enough, and could have escaped nobody. The latter is described by Professor DeMorgan in the ``Quarterly Journal of Mathematics,'' vol. i. p. 236.
The earliest source that I have found for Fig. 3.12 is the book by Donald L. Bruyr, Geometrical Models and Demonstrations, J. Weston Walch, Portland, Maine, 1963. His Fig. 163 gives the hinging that I show in my Fig. 3.12.
I have written a book on hinged dissections, Hinged Dissections: Swinging & Twisting, Cambridge University Press, 2002. The goal for each dissection is to have as few pieces as possible, subject to the pieces being hinged. I have created and re-engineered several techniques for producing hinged dissections.
I have just received a copy of a paper in Japanese, "Dudeney Dissection of Polygons", by Jin Akiyama and Gisaku Nakamura, at Tokai University in Tokyo. Judging from their figures (since I don't read Japanese!), they describe how to produce hinged dissections of:
Some of their techniques use the same general approach as some of those in my manuscript.
- a quadrilateral to another quadrilateral whose tessellation is related
- a quadrilateral to a parallelogram whose tessellation is related
- a triangle to a parallelogram (the same as Henry Martin Taylor's)
- a parhexagon to a parallelogram whose tessellation is related
- a parhexagon to a quadrilateral whose tessellation is related
- a parhexagon to a triangle whose tessellation is related
- a trapezoidal pentagon to a trapezoid whose tessellation is related
Copyright 1997-99, Greg N. Frederickson.
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Last updated January 27, 1999.