Chapter 1
Mathematics in Nature
1.1
An Interesting Sequence
Let us begin with an activity. Consider two kinds of “trains," as shown below.
How many trains of length 3 can we form? There are three possible trains, which are:
Let us use the notation [1, 1, 1], [2, 1], and [1, 2], respectively, to represent each train drawn above.
On the next page, you will be asked to enumerate and count how many trains can be formed of a certain
length. Then you will be asked to find a pattern.
1
2
CHAPTER 1. MATHEMATICS IN NATURE
NAME:
Complete the following table, using the instructions from the previous page. One row has been done for
you.
Length
Trains that can be formed
Number of trains
[1, 1, 1], [1, 2], [2, 1]
3
1
2
3
4
5
6
7
Now, try to find a pattern formed by the numbers in the third column. Describe the pattern you observed.
Without listing all the trains of length 8, how many trains of length 8 can be formed based on the pattern
you observed?
Answer:
1.1. AN INTERESTING SEQUENCE
3
The numbers in the previous activity are part of the Fibonacci sequence.
Fibonacci Sequence
The Fibonacci sequence is a sequence that starts with a1 = 1 and a2 = 1. Then, each successive
term is obtained by adding the previous two terms (i.e., an = an−1 + an−2 for n ≥ 3).
A Fibonacci number is any number that occurs in this sequence.
Write! List the first ten Fibonacci numbers in the space provided below.
It is said that Fibonacci numbers occur very often in nature. For example, the number of spirals going
clockwise and counter-clockwise are often consecutive Fibonacci numbers.
The number of leaves or petals is also often a Fibonacci number. A clover often has 3 leaves. The raflesia
(the largest flower, found in the mountains of Mindanao) has 5 petals. So do the Philippine national
flowers, the waling-waling and sampaguita. In a pineapple, there are 13 spirals in one direction and 8 (or
21) spirals in another direction.
Raflesia flower
Waling-waling
And the arrangements of leaves or petals is governed by the plant’s needs (i.e., how to receive sunlight
without being covered by other leaves or petals).
4
1.2
CHAPTER 1. MATHEMATICS IN NATURE
The Golden Ratio
The Fibonacci sequence starts with
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
Compute the ratios of successive Fibonacci numbers and write your answers below (round off to four
decimal places).
1
=
1
2
=
1
3
=
2
5
=
3
8
=
5
13
=
8
21
=
13
34
=
21
55
=
34
√
1+ 5
Now calculate
. Notice that the ratios of successive terms of the Fibonacci sequence get closer and
2
closer to this number. This number is called the golden ratio.
The Golden Ratio
√
1+ 5
The golden ratio (usually denoted by φ) the number
≈ 1.618.
2
It is said that the most beautiful rectangle is the one whose length is φ times its width (we call such a
rectangle the golden rectangle). The Parthenon of ancient Greece is a classic example of how the golden
ratio appears in architecture.
Measure! Find any TV, computer, or cellphone screen. Divide the measure of the length by the width.
You’ll see that the answer is close to the golden ratio.
1.2. THE GOLDEN RATIO
5
NAME:
Below, you will find two small squares and a larger square, forming a rectangle. Call this rectangle 1.
• Above rectangle 1, construct a square with side length 3. This forms a new largest rectangle; call it
rectangle 2.
• To the left of this rectangle, construct a square with side length 5. This forms a new largest rectangle;
call it rectangle 3.
• Continue the process, constructing squares in a counterclockwise direction until you form rectangle
7.
You have just constructed the structure of the golden spiral, which is found in many shells in nature.
Now record the dimensions of each rectangle in the table below.
Rectangle
Dimensions
1
3×2
Length ÷ Width
3
= 1.5
2
2
3
8×5
4
5
6
7
State an observation about the numbers in the third column.
8
= 1.6
5