Golden RatioDate: 01/03/98 at 22:32:18 From: Paul Park Subject: Golden Ratio Dr. Math, do you have any topics that I can use in my term paper about the golden ratio? There are so many things about it that I don't know what to write about. Thanks. Date: 01/04/98 at 02:43:02 From: Doctor Pete Subject: Re: Golden Ratio Hi, Here are some ideas: Geometry: If you have a regular pentagon ABCDE, with side length 1, what is the length of AC? (To find out, draw AC and BE to intersect at F, and compare the similar triangles ACF, BEF.) Trigonometry: Prove that Cos[Pi/5] = G/2, where G is the golden ratio (1+Sqrt[5])/2. Recursions: Let F[n] be the n(th) Fibonacci number, i.e., F[n] = F[n-1] + F[n-2] F[0] = 0, F[1] = 1. This is a recursive definition. Prove that F[n] = (A^n - B^n)/Sqrt[5], where A > B are the two roots of the polynomial x^2 - x - 1 = 0, and Sqrt[5] is the square root of 5. (Hint: Show that A = G. Then clearly G^2 = G+1, G^3 = G(G^2) = G(G+1) = G^2+G = (G+1)+G = 2G+1, and so successively compute G^4, G^5, and so forth in terms of G. You should see a pattern, and prove it by induction, which gives G^n. Similarly, do this for B.) Continued fractions: Find the value of the continued fraction 1 1 + ------------ 1 1 + -------- 1 1 + ---- 1 ... . (Hint: Call the above value p. Then p = 1 + 1/p. Why?) Continued roots: Find the value of Sqrt[1 + Sqrt[1 + Sqrt[1 + ...]]] . (Hint: Call the above value q. Then q = Sqrt[1 + q].) There are many more curiosities and unexpected places where the golden ratio pops up, but that should give you a nice overview. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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