By Emile Borel
Borel E. advent geometrique a quelques theories physiques (1914)(fr)(ISBN 1429702575)
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Additional info for Introduction géométrique à quelques théories physiques
This property outweighs all the elegancies! It is desirable to distinguish the two strips in a drawing of their superposition. Therefore broken lines are used for the MORE PP DISSECTIONS 21 sloping strip in every drawing of a superposition in this book. This harmonizes with the practice, adopted from Fig. 1 . 1 on wards, of indicating added cuts by broken lines. And the dis sected polygons, appearing on the left and right of each super position, are oriented so as to make it as easy as possibl e to identify the pieces therein with those in the superposition.
2. This then would be the first step in dissecting it. Dissections involving triangles, it seems, are minimal only if one uses T T2 . But this does not always apply to the dissection of two quadrilaterals. Each can be cut into three pieces to make a P-strip element, with a good chance that a PP superposition will give a minimal dissection. Figure 8. 6, due to W. H. Macaulay, shows one. The rearrangements are included, to show how they (as well as the common area) have some degree of symmetry as much as can be expected when the quadrilaterals are irregular.
5 . 3. The edges o f each s trip, a s can be seen, d o not pass through adj acent points in the other; instead (referring to Fig. 3) there is a midpoint K in between, where lines in both strips intersect at the center of the common area. The latter in effect contains a dissection of two crosses into two triangles. But as K) being a common midpoint, is a center of symmetry for each strip, it is also a center of symmetry for the common area. Hence, the pieces in half the common area give a dissection of one cross into one triangle.
Introduction géométrique à quelques théories physiques by Emile Borel