By L S Dent Glasser; International Union of Crystallography
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Paul Mahlo (1908) realized that he could produce this dissection by superposing tessellations, as did Percy MacMahon (1922). 2 also applies to similar rectangles. This gives a 5-piece dissection whenever the long side of the smaller rectangle is shorter ^ / '*. / 1 . 4: Two attached squares than the short side of the larger rectangle, and a 4-piece one when those sides are equal. Superposing tessellations gives this dissection too. These dissections are all translation al. 3. This was given in Plato's Menon and also in his Timaeus.
He also located point F on side CD so that line segment FD is the desired length of the edge on the left. Finally, he let point G be the midpoint of line segment AE and H be the midpoint ofCF. He then made a cut from G to H, a second cut from E that is parallel to AD, and a third cut from F that is parallel to BC. He rotated the two triangles created by the second and third cuts 180° degrees, to match with a side of another piece. 5. 13. The dissection works for any set of dimensions within a certain range.
Kelland published papers in both physics and mathematics, most notably on the motion of waves in canals, various problems in optics, non-Euclidean geometry, and algebra. He was president of the Royal Society of Edinburgh at the time of his death. 14 shows how to use Pslides in conjunction with each other. We view the given figure as two rectangles joined on an edge, so that we can apply a P-slide to each of these rectangles. The nontriangular pieces remain adjacent in each of the transformed rectangles, so that they remain joined along the dotted lines in the resulting 5-piece dissection.