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 2dimensional 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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 Spring '09
 SaadMneimneh
 Computer Science, Fibonacci number, Golden ratio, /F ib, golden number

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