Module 1: Patterns in Nature

Table of Contents

Introduction 1
Paulinian Essential Elements 2
Paulinian A rmation 2

Topic 1: Fibonacci Sequence and the Golden Ratio 3

Learning Outcomes 3
Learning Activities 3
Processing 3
Fibonacci Numbers 3
The Golden Ratio 5
Formation 7
Activity 1: Let’s Investigate! 7
Activity 2: Mathematics in Rabbits! 8
Activity 3: Movie Watching! 10
Synthesis 11
Assessments 11
Assignment 12

Re ection Guide 12

Expanded Opportunity 12

References 13
Module 1: Patterns in Nature

Introduction

Fig. 1. A Fibonacci spiral that resulted from plotting the Fibonacci numbers on a graph

The heart of mathematics is more than just numbers, numbers which many supposed to be
meaningless and uninteresting. Have you gone for beach trips or did mountain climbing
perhaps and noticed in awe the beautiful world around you? The degree of changing hues
of color has to be exact measurement to appear pleasing and harmonious to the human

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eye. And it is mathematics that reveals the simplicities of nature and permits us to
generalize from simple examples to the complexities of the real world. It took many people
from many different areas of human activity to turn a mathematical insight into a useful
product (Stewart, 1995).

We look at mathematics as a useful way to think about the nature and the world in general;
thus, patterns and numbers that are useful in this world will be dealt with- to think about
Fibonacci sequence and other arrays of numbers in order to predict and control the
behavior of nature and phenomena in this world. In the long run, we think about the
numerous applications of mathematics as aids in decision making.

Paulinian Essential Elements

Engaging, Trustworthy, Team Builders, and Mentors who

● clarify at the outset the substance and intent of all agreements and commitments
made, making every effort to fulfill them and supporting others to do the same;
● publicly explain the purpose and potential benefits of all group endeavors, enlisting
explicit agreement and support for them from participants before proceeding;
● initiate and develop jointly beneficial and sustainable projects with colleagues in
which plans and responsibilities are equitably shared, reliably carried out, and
honestly evaluated by all; and
● willingly and openly share relevant insights, observations, expertise, guidance and
support with less experienced colleagues.

Paulinian A rmation
“As a Christ-centered Paulinian, I am an engaging, trustworthy, team builder, and
mentor, fostering community through active collaboration.”

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Topic 1: Fibonacci Sequence and the Golden Ratio

Learning Outcomes
During the learning engagement, you should be able to
● critically examine the Fibonacci sequence in nature and in art,
● investigate common patterns in nature as explained through some mathematical
concept or model, and
● explain the nature of number as a language through which relationships in nature
can be explored.

Learning Activities

Processing

Fibonacci Numbers
Leonardo Pisano Fibonacci (1170–1250) from Pisa, Italy was most famous for his
description of the number sequence, which, in the 19th century, was given the name
“Fibonacci numbers” after its inventor. He was sometimes called Bigollo, which means
“traveller,” and it is very probable that he got this nickname because he was traveling a lot.
But Bigollo also means “good-for-nothing”—so who knows…

Leonardo Pisano is better known as Fibonacci which is

shortened word for the Latin term “filius Bonacci” which stands
for “son of Bonaccio,” and his other name Pisano comes from
the place where he lived most of his life, Pisa, which is famous
for its leaning tower. Building of the tower began when
Fibonacci was three years old.

As a young man, Fibonacci traveled with his father who was a

big merchant. Fibonacci’s father was a representative of the

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merchants of Pisa, who imported and exported materials from and to northern Africa,
mainly a place called Bugia, now in Algeria. During these travels, Fibonacci learned
mathematics as it was known there.

He wrote a very famous book Liber Abaci in 1202 to describe the mathematics he learned.
He mentioned in this book that in Algeria he learnt Indian’s number symbols and also what
he called Indian accounting. It was in Liber Abaci that Fibonacci first described his now
famous numbers, through the problem on rabbits.

Fibonnaci observed numbers in nature. His most popular contribution perhaps is the
number that is seen in the petals of flowers. Why Fibonacci numbers? This number
sequence is found everywhere in nature and has other real-life applications too. The
sequence may not be a law of nature, but it is a fascinating prevalent tendency! In fact,
these numbers come up many times in nature. Scientists who study plants have found that
the Fibonacci numbers show up often in plants that have multiple parts (such as leaves,
petals, or seeds) arranged around a single stem. A study found that two successive
Fibonacci numbers appeared in more than 90% of such plants. The frequent appearance of
Fibonacci numbers in nature has been a puzzle for a long time. Recently, scientists and
mathematicians have been able to reproduce the patterns in laboratory studies and have
offered new ideas about why the numbers arise.

When you chart Fibonacci numbers on a graph, you get the Fibonacci spiral.

Fig. 2 The Fibonacci spiral

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Module 1: Patterns in Nature

The swirl of seashells, ridges of pinecones, curve of animals’ nails, centers of sunflowers,
even human teeth curve in the Fibonacci spiral.

The Fibonacci sequence occurs many times in nature. Take a look at sunflowers. In
particular, pay attention to the arrangement of the seeds in its head. Do you notice that
they form spirals? In certain species, there are 21 spirals in the clockwise direction and 34
spirals in the counterclockwise direction.

The Golden Ratio

The golden ratio is so fascinating that the proportions of the human body as the face
follows the so-called divine proportion. The closer the proportion of the body parts to the
golden ratio, the more aesthetically pleasing and beautiful the body is. Many painters,
including the famous Leonardo da Vinci, were so fascinated with the golden ratio that they
used it in their works of art. The amazing ratio is denoted by the Greek symbol phi (ϕ).

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The ratio of two consecutive Fibonacci numbers as n becomes large, approaches the golden
ratio; that is,

Fn
= 1.6180339887... .
→∞
lim F
n ­
n 1

This can be verified by measuring some parts of the human body: the length of the arm,
height, or the distance of the fingertips to the elbow. The ratio between the forearm and the
hand also yields a value close to the golden ratio!

The golden ratio denoted here by the Greek symbol ϕ is sometimes called the golden
mean or golden section.

1+√5
φ= 2 = 1.6180339887...

In mathematics and arts, two quantities are in a golden ratio if their ratio is the same as the
ratio of their sum to the larger of the two quantities. In symbols, a and b, where a > b > 0,
are in a golden ratio if a
b = a+b
a . As seen in the preceding section, Fibonacci numbers appear
in many places. The golden ratio does too. It shows up in art, architecture, music, and
nature.

After the first several numbers in the Fibonacci sequence, the ratio of any number to the
next higher number is approximately equal to 0.618, and the next lower number is 1.618.
These two figures (0.618 and 1.618) are known as the golden ratio or the golden mean. Its
proportions are pleasing to the human eyes and ears. Here are just a few examples of
shapes that are based on the golden ratio: playing cards, sunflowers, snail shells, the
galaxies of outer space, hurricanes, and even DNA molecules.

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William Hoffer, in the Smithsonian Magazine, wrote in 1975: “The continual occurrence of
Fibonacci numbers and the golden spiral in nature explain precisely why the proportion of
0.618034 to 1 is so pleasing in art. Man can see the image of life in art that is based on the
golden mean.” Nature relies on this innate proportion to maintain balance, but the financial
markets also seem to conform to this "golden ratio." Here, we take a look at some technical
analysis tools that have been developed to take advantage of it.

Two Fibonacci technical percentage retracement levels that are most important in market
analysis are 38.2% and 62.8%. Most market technicians will track a “retracement” of a price
uptrend from its beginning to its most recent peak. Other important retracement
percentages include 75%, 50% and 33%. For example, if a price trend starts at zero, peaks at
100, and then declines to 50, it would be a 50% retracement. The same levels can be applied
to a market that is in a downtrend and then experiences an upside “correction.”

Formation

Activity 1: Let’s Investigate! 15 minutes

This investigation activity will lead you towards understanding patterns in nature. You may
do the activity in groups of either four or five members. Choose one from the three
activities below.

Pattern 1. Measure your waistline and neckline individually in centimeters. Tally the
measurements and determine the ratio of your waistline to your neckline.

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Pattern 2. Measure the length of your palm and the length of your middle finger
individually in centimeters. Tally the measurements and determine the ratio
of the length of your palm to the length of your middle finger.

Pattern 3. You may proceed to the school clinic. With due courtesies and politeness,
request the clinic staff to allow your group members to use appropriate tool
to determine individually your weight and height. Tally the measurements and
determine the ratio of your height to your weight.

Guide Questions
Based on the results of your group investigation:
1. What pattern have you generated?

2. How did you do the pattern?

Activity 2: Mathematics in Rabbits! 30 minutes

The purpose of this worksheet is to investigate what mathematics has got to do with the
birth rates of rabbits. (Excerpt from Joseph Yeo)

Mathematical Investigation: Rabbit Problem

A man bought a pair of rabbits (one male and one female) in Jan 2004. The rabbits did not
produce any rabbits in Feb 2004 but they produced a new pair of rabbits (one male and one
female) every month from Mar 2004 onwards. Each new pair of rabbits followed the same
behaviour. For example, the pair of rabbits born in Mar 2004 did not produce any rabbits in
Apr 2004 but they produced a new pair of rabbits every month from May 2004 onwards.

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1. How many new pairs of rabbits were produced at the end of Mar 2004? ________
2. How many new pairs of rabbits were produced at the end of Apr 2004? ________
3. Draw a model or a diagram in the table below to show the number of new pairs of
rabbits produced at the end of each month from Jan to Sep 2004. Some parts have
been done for you. Complete the table also.

1st pair of rabbits Pairs produced by 2nd pair

Pairs produced by 1st pair Pairs produced by 3rd pair

For other pairs produced, you can use the same symbol.

Month Model Total New Total Pairs

Pairs at End at End of
of Month Month

Jan 0 1

Feb 0 1

Mar 1 2

Apr 1

May 2

Jun

Jul

Aug

Sep

4. The last column in the table above shows the total pairs of rabbits at the end of each
month. This number pattern is called the Fibonacci sequence. Explain how you get
the next term.
5. How many pairs of rabbits were there at the end of January 2005?

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Activity 3: Movie Watching! 20 minutes

The following videos show how patterns or relationships in nature are explained
mathematically:
1. Donald Duck – Mathmagic Land
2. Fingerprint of God
(Note: The above videos were last accessed on 18 March 2020.)

Guide Questions
Based on what are shown in the videos:

1. How do you consider mathematics as a discipline?

2. How can mathematics be a tool to explain patterns in nature?

3. Why do you think the blogger claimed that the concept of golden proportion is
analogous to the "fingerprint of God?"

4. Comment on what Galileo advocated: "Mathematics is the language by which God

has written the universe."

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Synthesis
Mathematics is universal. Mathematical concepts and influences can be seen in both nature
and human creations, such as art. Seeing patterns in nature has led many to believe that
such patterns are evidence of divine creation. The relationship between mathematical
concepts and aesthetics has influenced art; some even view mathematics itself as an art.
The two have become intertwined, leading Luca Pacioli to exclaim, “Without mathematics,
there is no art.”

Assessments
Direction: Choose the letter of the correct answer.
1. Which of the following explains that nature itself is a perfect ratio?
a. Through mathematics, patterns in nature can be explored.
b. Relationships in nature can be quantified and explored.
c. Patterns in nature have unified proportion.
d. Relationships or patterns in nature can be reduced into numbers.

2. Which of the following does not approximate a golden ratio or the number phi?
a. 5/3
b. 21/13
c. 8/13
d. 144/89

3. “Mathematics is the language by which God has written the universe.” This truth was
clearly articulated by ____________.
a. Pythagoras
b. Fibonacci
c. Euclid
d. Galileo

4. Which of the following shows a golden ratio?

a. Monalisa Painting by Da Vinci
b. Nautilus shell
c. Sunflower Petals
d. All of the above

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5. What is the 10th Fibonacci number on the sequence 5, 8, 13, ...?

a. 89
b. 233
c. 377
d. 610

Assignment
Research about mathematical languages and symbols, and identify on how to translate
them in English sentences.

Re ection Guide

1. What have I LEARNED this day that has helped me do all aspects of this better?

2. What have I DONE this week that has made me better at doing all aspects of this?

3. How can I IMPROVE at doing all aspects of this?

Expanded Opportunity

Research about the concept and properties of normal distribution. Explain how this
mathematical model can explain patterns in nature.

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References

Baltazar, Ethel, Carmelita Ragasa, and Justina Evangelista. Mathematics in the Modern
World. Quezon City: C and E Publishing Inc., 2018.

Kuepper, Justin. “Fibonacci and the Golden Ratio.” Retrieved from

https://www.investopedia.com/articles/technical/04/033104.asp (accessed 10 July
2020)

Nocon, Rizaldi and Ederlina Nocon. Essential Mathematics for the Modern World. Quezon
City: C and E Publishing, Inc., 2018.

Wyckoff, Jim. “Why Successful Traders Use Fibonacci and the Golden Ratio.” Retrieved
from https://www.traderslog.com/traders-fibonacci (accessed 10 July 2020)

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