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 circumstances. 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. c. 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. f. Rabbits do not die. 1 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 rabbits. 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… 2 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 and the United Nations Headquarters. 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. 3 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 4 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 5