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.The pleasing design in Figure 2.59 rightmay be readily produced, in which the ratio of the sides of the larger to thesmaller triangles is equal to Æ.This intriguing result was first observed by George Odom, a resident of theHudson River Psychiatric Center, in the early 1980s [254, p.10].Upon com-municating it to the late H.S.M.Coxeter, it was submitted to the AmericanMathematical Monthly as Problem E3007, Vol.90 (1983), p.482 with thesolution appearing in Vol.93 (1986), p.572.(a) (b) (c)Figure 2.60: (a) Bination of Equilateral Triangle.(b) Male Equilateral Spiral.(c) Female Equilateral Spiral.[83]Property 75 (Equilateral Spirals).E.P.Doolan has introduced the notionof equilateral spirals [83].Successive subdivision of an equilateral triangle by systematically connect-ing edge midpoints, as portrayed in Figure 2.60(a), is called (clockwise) bi-nation.(Gazalé [139, p.111] calls the resulting configuration a  whorledequilateral triangle.) Retention of the edge counterclockwise to the new edgeproduced at each stage produces a (clockwise) male equilateral spiral (Figure2.60(b)).Replacement of each edge of such a male equilateral spiral by thearc of the circumcircle subtended by that edge produces the corresponding(clockwise) female equilateral spiral (Figure 2.60(c)).Doolan has shown that the female equilateral spiral is in fact C1.Moreover,both the male and female equilateral spirals are geometric in the sense that,for a fixed radius emanating from the spiral center, the intersections with thespiral are at a constant angle.Note that this is distinct from equiangularity inthat this angle is different for different radii.In addition, he has investigatedthe  sacred geometry of these equilateral spirals and shown how to constructthem with only ruler and compass. 70 Mathematical Properties(b)(c)(a)Figure 2.61: Padovan Spirals: (a) Padovan Whorl [138].(b) Inner Spiral [293].(c) Outer Spiral [139].Property 76 (Padovan Spirals).Reminiscent of the Fibonacci sequence,the Padovan sequence is defined as 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21,., whereeach number is the sum of the second and third numbers preceding it [138].The of succesive terms of this sequence approaches the plastic number,ratio3 31 1 23 1 1 23p = + + - H" 1.324718, which is the real solution of2 6 3 2 6 3p3 = p + 1 [139].The Padovan triangular whorl [139] is formed from thePadovan sequence as shown in Figure 2.61(a).Note that each triangle sharesa side with two others thereby giving a visual proof that the Padovan sequencealso satisfies the recurrence relation pn = pn-1 + pn-5.If one-third of a circleis inscribed in each triangle, the arcs form the elegant spiral of Figure 2.61(b)which is a good approximation to a logarithmic spiral [293].Beginning with agnomon composed of a  plastic pentagon (ABCDE in Figure 2.61(c)) withsides in the ratio 1 : p : p2 : p3 : p4, if we add equilateral triangles that grow insize by a factor of p, then a truly logarithmic spiral is so obtained [139].Property 77 (Perfect Triangulation).In 1948, W.T.Tutte proved that itis impossible to dissect an equilateral triangle into smaller equilateral trianglesall of different sizes (orientation ignored) [309].However, if we distinguish be-tween upwardly and downwardly oriented triangles then such a  perfect tilingis indeed possible [310].Figure 2.62, where the numbers indicate the size (side length) of the com-ponents in units of a primitive equilateral triangle, shows a dissection into 15pieces, which is believed to be the lowest order possible.E.Buchman [35] hasextended Tutte s method of proof to conclude that no planar convex regioncan be tiled by unequal equilateral triangles.Moreover, he has shown that Mathematical Properties 71Figure 2.62: Perfect Triangulation [310]any nonequilateral triangle can be tiled by smaller unequal triangles similar toitself.Figure 2.63: Convex Tilings [294]Property 78 (Convex Tilings).In 1996, R.T [ Pobierz caÅ‚ość w formacie PDF ]

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