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Additional Learning Materials for Module 1

The Nature of Mathematics
(Additional Learning Materials)
Fibonacci Numbers
The Fibonacci sequence exhibits a certain numerical pattern
which has turned out to be one of the most interesting ever written
down. Its method of development has led to far-reaching applications
such as to model or describe an amazing variety of phenomena, in
mathematics and science, and even more fascinating is its surprising
appearance in Nature and in Art, in classical theories of beauty and
proportion. The mathematical ideas of the Fibonacci sequence leads
to the discovery of the golden ratio, spirals and self- similar curves,
and have long been appreciated for their charm and beauty, but no
one can really explain why they are echoed so clearly in the world of
art and nature.
Leonardo Pisano Bigollo was born late in the twelfth century in Pisa, Italy: Pisano in Italian
indicated that he was from Pisa. His father was a merchant called Guglielmo Bonaccio and it's
because of his father's name that Leonardo Pisano became known as Fibonacci. The name came
from "filius Bonacci" meaning "son of Bonaccio" – as his surname, and “Fibonacci” was born.
Math was incredibly important to those in the trading industry, and his passion for numbers was
cultivated in his youth.
Fibonacci numbers were first introduced in his Liber abaci in 1202. Knowledge of numbers
is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied
while growing up in North Africa. Prior to the publication of Liber abaci, the Latin-speaking world
had yet to be introduced to the decimal number system. He wrote many books about geometry,
commercial arithmetic and irrational numbers. Except for his role in spreading the use of the
Hindu-Arabic numerals, Leonardo’s contribution to mathematics has been largely overlooked. His
name is known to modern mathematicians mainly because of the Fibonacci sequence derived
from a problem in the Liber abaci, which was about how fast rabbits could breed in ideal
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs
of rabbits can be produced from that pair in a year if it is supposed that every month each pair
begets a new pair which from the second month on becomes productive?
Beginning with a male and female rabbit, how many pairs of rabbits could be born in a year?
The problem assumes the following conditions:
a. Begin with one male rabbit and female rabbit that have just been born.
b. Rabbits reach sexual maturity after one month.
The gestation period of a rabbit is one month.
d. After reaching sexual maturity, female rabbits give birth every month.
e. A female rabbit gives birth to one male rabbit and one female rabbit.
Rabbits do not die.
This is illustrated in the following diagram.
After one month, the first pair is not yet at sexual maturity and can not mate. At two months,
the rabbits have mated but not yet given birth, resulting in only one pair of rabbits. After three
months, the first pair will give birth to another pair, resulting in two pairs. At the fourth month mark,
the original pair gives birth again, and the second pair mates but does not yet give birth, leaving
the total at three pairs. This continues until a year has passed, in which there will be 233 pairs of
The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (Leonardo himself omitted
the first term), is the first recursive sequence (in which the relation between two or more
successive terms can be expressed by a formula) known in Europe. The Fibonacci sequence is
a series of numbers where a number is found by adding up the two numbers before it. Written as
a rule, the expression is xn = xn-1 + xn-2.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…
The Golden Rectangle
The Golden Rectangle is famous concept relating
aesthetics and mathematics that is found in many natural
and man-made things on Earth. A golden rectangle is The
ancient Greeks considered the Golden Rectangle to be the
most aesthetically pleasing of all rectangular shapes.
A classic example is the front of the Parthenon that
is comfortably framed with a Golden Rectangle.
Golden section continues to be used
today in modern architecture just like
the design of Notre Dame in Paris
The Golden Ratio in Arts
For centuries, designers of art and
architecture have recognized the significance of the
Golden Ratio in their work. The ratio of the width of
Mona Lisa’s forehead to the length from the top of
her head to the chin, displays a perfect golden
rectangle. The Golden Section was used
extensively by Leonardo Da Vinci in his work, “The
Last Supper”.
Fibonacci Spirals
Fibonacci spirals and Golden Mean ratios appear
everywhere in the universe. A golden spiral is a logarithmic spiral
with a growth factor of ‘Phi’, which is the golden ratio – that
means it gets wider by a factor of Phi for every quarter turn it
makes. Some examples are the natural flow form of water when
it is going down the drain and the natural flow form of air in
tornadoes and hurricanes. A beautiful example of a Fibonacci
spiral in nature is the Nautilus shell.
Examples of Fibonacci Numbers in Nature
1. Flower petal
4. Fruits: Pineapple, banana
7. Spiral galaxies
2. Seed head
5. Succulent plants
8. Hurricane
3. Pine cone
6. Tree branches
9. Human ear
The Divine Proportion
The photo below illustrates the following golden ratio proportions
in the human face:
1. Center of pupil : Bottom of teeth : Bottom of chin
2. Outer & inner edge of eye: Center of nose
3. Outer edges of lips : Upper ridges of lips
4. Width of center tooth : Width of second tooth
5. Width of eye : Width of iris
The Proportions in the Body
1. The white line is the body’s height.
2. The blue line, a golden section of the white line, defines the
distance from the head to the finger tips.
3. The yellow line, a golden section of the blue line, defines the
distance from the head to the navel and the elbows.
4. The green line, a golden section of the yellow line, defines the
distance from the head to the pectorals and inside top of the
arms, the width of the shoulders, the length of the forearm and the
shin bone.
5. The magenta line, a golden section of the green line, defines the
distance from the head to the base of the skull and the width of
the abdomen. The sectioned portions of the magenta line
determine the position of the nose and the hairline.
6. Although not shown, the golden section of the magenta line (also
the short section of the green line) defines the width of the head and half the width of the
chest and the hips.
The Ratio of your Forearm to Hand is Phi
Your Hand Shows Phi and the Fibonacci Numbers