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# note14 - CSCI 120 Introduction to Computation Computation...

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CSCI 120 Introduction to Computation Computation everyday (draft) Saad Mneimneh Visiting Professor Hunter College of CUNY 1 Introduction I had the intention of showing how we experience in our everyday life different aspects of math and computer science, without even knowing. My plan was to cover the following: Map coloring and the relation to the coloring of planar graphs (graph theory), in addition to an overview of the 4 color theorem which states that only 4 colors are needed to color any 2-dimensional map in a way that adjacent countries have different colors (google the 4 coloring theorem for very nice stories on the history of this theorem). Today, the only proof of this theorem is based on a computer program! Binary search trees and how they are used in everyday applications to search for things. The golden number and the Fibonacci sequence and the relation to all aspects of life including architecture, art, music, literature, biology, etc... Grammars and languages in the field of linguistics and the relation to programming languages. I previously talked about map coloring, graphs, planar graphs, the 4 color theorem, and binary search trees. I will concentrate here on the golden number and the Fibonacci sequence, and on grammars and languages. 2 The golden number and the Fibonacci sequence The golden number or the golden ratio is obtained in the following way: Consider a segment and a point on the segment. The point divides the segment in two parts, as illustrated in Figure 1 below:

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a b c Figure 1: Segment divided by point Referring to Figure 1 above, the placement of the point defines two ratios, namely a/b and b/c . When these two ratios are equal, we call the ratio the golden ratio, or simply the golden number. So how should we place the point in order to obtain a/b = b/c ? It turns out that there is only one solution in which a/b = b/c = 1 . 618033989 ... . This number is given the symbol φ and is pronounced phi . Like π , φ is simply another kind of proportion. While π is the ratio of the circle perimeter to its diameter, φ is the ratio obtained by making a/b = b/c for any line segment as shown in Figure 1. You are not responsible for the detail on how to obtain φ , but here’s a derivation for those of you who are interested: We need a/b = b/c . But c is nothing but a - b . Therefore, we need a b = b a - b But b a - b = 1 a/b - 1 Therefore, we need: a b = 1 a/b - 1 Now letting a/b = φ , we have: φ = 1 φ - 1 which gives the equation φ 2 - φ - 1 = 0. The only positive number that satisfies this equation is (1 + 5) / 2 = 1 . 618033989 ... . It is not really known when φ was first discovered. It was probably rediscov- ered many times throughout history. For instance, ancient egyptian civilizations used φ in the construction of pyramids as illustrated below.
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note14 - CSCI 120 Introduction to Computation Computation...

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