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Section 1 .1: Mathematics in our World
What is Mathematics? Where is Mathematics? What is it for? What role does mathematics play in our world?
1.1.1 Patterns and Numbers
All around us, we see diverse types and forms of patterns from the different objects. Patterns are structures or designs that are arranged and that repeat, or we can also think of anything that is arbitrary. Our diverse universe and nature displays striking designs and patterns like the symmetry of snowflakes, the spiral of a snail’s shell, the hexagonal shape of
MODULE 1: THE NATURE OF MATHEMATICS
WORLD
cells of honeycombs. Some other natural patterns include spots, mosaics, waves, trees, spirals, cracks, and stripes. While the beauty of these patterns serves every individual well, a distinctive innate desire to comprehend these occurrences has gotten forth a variety of opportunities for man to better realize and exploit the universe where he exists.
Numbers are also everywhere in nature. Mathematicians noticed that numbers appear in many different patterns in nature: bird’s two wings, clover’s three leaflets, deer’s four hooves, buttercup’s five petals, insect’s six legs, rainbow’s seven colors, octopus’ eight arms , and many others. As men of science studied numbers, they also realized their significance in everyday life. Truly, mathematics has set the system and processes by which man can understand and predict the behavior and phenomena in nature and the world.
1.1.2. Fibonacci Sequence and the Golden Ratio
The Fibonacci Sequence The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on is known as the Fibonacci sequence. It is a recursive sequence such that each number is the sum of the two preceding ones. This means, 0+1 = 1, 1+1 = 2, 1+2=3 and so on, and this sequence can carry on indefinitely. If we denote 𝐹 1 = 1 ; 𝐹 2 = 1 ; 𝐹 3 = 2 ; 𝐹 4 = 3 and so on, the Fibonacci numbers, 𝐹𝑛 obey the following relationship: 𝐹 1 = 𝐹 2 = 1 𝐹𝑛 = 𝐹𝑛− 1 + 𝐹𝑛− 2 , 𝑛 ≥ 3 𝐹𝑛 is the sum of the two previous numbers
Fibonacci and the Nature The Golden Ratio and the World The mystical Golden Spiral and Golden Ratio are to be found throughout nature. For instance, the number that is seen in the petals of some flowers like a calla lilly has 1 petal, iris has 3 petals, gumamela has 5 petals, clematis has 8 petals, corn marigold has 13 petals. The reason why Fibonacci patterns are found on some flowers generally revolve around a functional explanation. An example is the packing of seeds in the seed heads of sunflowers where the seed heads grow in a specific outward manner and they usually possess 34, 55 or 89 spirals. objects in the natural world. Of course, not all things found in the nature follow the Fibonacci numbers but on the average, this interesting resemblance covers to a varied series of plants, fruits and other things in the natural world. The Golden ratio was used in Ancient Greek architecture to determine attractive dimensional connections between the width of a building and its height, the size of the portico and even the position of the columns supporting the structure. The Golden ratio is also evident in some of the famous structural design like The Great Sphinx in Egypt, , Notre Dame in Paris, Taj Mahal in India. It appears to use it in some aspects of its design to achieve beauty and balance.
Many artists have incorporated the Golden Rectangle into their works because of its aesthetic appeal. With deeper understanding of the Golden Section, one can craft aesthetics and visual harmony in any branch of the design arts. This concept was used extensively by Leonardo Da Vinci’s “The Last Supper”. Note how the Divine Proportion were evident in all the significant dimensions of the room, the table and ornamental shields. This also true with the paintings of Michelangelo, “The Creation of Adam”, and Raphael’s “The School of Athens”. Fibonacci and Golden mean are also found in musical scales, structure of compositions and musical patterns, and designs of musical instrument. For example, scales along a piano keyboard are composed of thirteen keys in the span of a full octave which consists of eight white keys and five black keys that are arranged in groups of two and three along the keyboard. Fifth Symphony in which the opening of the piece appears at the golden section point and at the appears at the recapitulation, which is Phi of the way through the piece. The golden ratio is also at the heart of many of the proportions in the human body. For instance, the dimension relationships between the eyes, ears ,mouth, and nose are said to have ratios equal to 𝜑. The shape of the perfect face and also the ratio of the height of the navel to the Another, Pentatonic scale has five notes, the Diatonic scale has eight notes, and the Chromatic scale has thirteen notes. In addition, the 1st, 3rd, and 5th notes in any scale create the basic foundation of chords. The climax of many songs or even an important measure where the song changes significantly such as the bridge often occurs near the songs Phi point. Beethoven used the golden section in his famous
