Nature of Mathematics PDF

MATHEMATICS IN THE MODERN WORLD CHAPTER 1 LESSON 1.1 NATURE OF MATHEMATICS Natures are really beautiful and yet perfect gift from God. They were said and believed to be created according to the ideal designs and creativeness of the Almighty. Each kind has its own purpose and distinct struc...

MATHEMATICS IN THE MODERN WORLD CHAPTER 1 LESSON 1.1 NATURE OF MATHEMATICS Natures are really beautiful and yet perfect gift from God. They were said and believed to be created according to the ideal designs and creativeness of the Almighty. Each kind has its own purpose and distinct structures that provides them an identity that make them stands out among the rest. Mathematics had been considered to be the science of pattern since it exemplifies orders of structures that cannot be observed by the naked eye but through logical and mathematical analysis. Some structures of nature have been known and visible after it was studied and described by science, but there are still some other patterns that mathematics have found and numerically studied to reveal its particularity. PATTERNS Patterns were defined as a rule that describes the whole process of the making of the series or objects. This is usually governed by a rule that demonstrates what must come along the way. The human brain has the power to anticipate the rule that runs into a pattern upon observing the whole structure and order of the events. When the rule of the pattern was determined, what comes next after the last or what comes before the first event can easily be concluded. There are some kinds of patterns that we might aware of. These are: 1. Logical Pattern Characterizing and sorting objects are one of the activities that everybody could do easily. One kind of logical pattern are the one usually seen in an aptitude test, where observing the changes happened per event is the key to identify what must be the next after the last scene. Example: https://www.google.com/search?q=logical+pattern&rlz=1C2TSNH_enPH695PH695&source=lnms&tbm=isch&sa=X&ved=0ahUKEwid_73G_OLdAhUIA4gKHS93CWgQ_AUIDigB&biw=1024&bih=496#imgrc=FeXlCo_vvYTDhM: 2. Numerical Pattern It is a sequence of pattern that follows a certain rule. This is the kind of pattern where most of the students are aware of since they are studying mathematics in school already. Most of the time, the students are used to find the right formula that best describes whole sequence by being particular with the changes that happened to the number over time. By this, they can easily determine the possible nth term they want or need to know as required by the questions. The set of even numbers is one of the most common numerical patterns that most of the students has known. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 3. Geometric Pattern This pattern depicts nonrepresentational shapes like lines, shapes or polygons that typically repeat in the process to form a design that may seem look like wallpaper. Examples: Jai Deco Geometric Design 139 4. Word pattern It focuses on the patterns found in syntax and uses of words that can also direct towards to the study of language in general and digital communication. Morphological rules in pluralizing words are one of the examples of word patterns. Example: Add s on the end of the word when pluralizing words ends with a vowel+ y Day – Days Key – Keys Guy – Guys Donkey – Donkeys LESSON 1.2 PATTERNS IN NATURE In nature, there are also patterns that might not be obvious but existing in its structure. It maybe because the point of view of our observation commonly focuses on the superficial attributes of the things that provides us only shallow information about what we can see into it. Deepening our realization can make us realize what was going through beyond what we see which lead us to a discovery of pattern that are really interesting. If we will try to be particular with the different forms of things in the world, both living and non-living, we can notice different forms of astounding patterns that can not only delight our imaginations but also challenges our understanding to how does a particular pattern happened with that kind of creations. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 PATTERNS IN NATURE We can observe some regularities, repetitive, distinct and recurring designs in the nature which you may seem normal but fascinating once you pay attention to. 1. Symmetry. This pattern is said to be present in the object once an imaginary straight line was drawn across the center then the other side appeared to be identical to the other side. 2. Spirals. Continuous circular pattern beginning at center. This pattern usually creates circular lines that goes around the center as it moves outward. 3. Waves and Ripples. Waves and ripples are usually patterns created once the wind pass over any large body of water or sand. It is more likely the traces created by the wind once it passed since wave is a kind of disturbances that carries energy as it moves. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 4. Meander. These are sinuous bends in rivers or other channels, which form as a fluid, most often water flows around bends. 5. Spots and Stripes. Usually seen on animals, e.g. leopards, ladybirds : spotted and zebras, angelfish: stripes. These patterns are said to have functions to the animals that increases the chances that their offspring will survive to reproduce, a function to hide from its predator – camouflage and a function to signal whenever they are about to attack by the predator. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 6. Cracks. These are linear opening that usually forms to relieve stress. It happens once an elastic materials have reach its breaking point then create a linear pattern that falls in a certain directions, depending on the elasticity of the materials. 7. Tessellations. This patterns are made up of one or more shape that meets together having no space in between them. 8. Branching. This is a natural pattern that usually happen in plants where its vein, leaves, branches or roots follows a certain systematic pattern as they grow. Alternate branching is one of the common example of how leaves and branches grow in the stem of a plant. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Fractal. It is a pattern that commonly associated in branching. Fractals is a an infinitely complex patterns that have the scheme of using self-similarity in an iterative system. Benoit Mandelbrot was the one who introduced the fractals and considered to be the father of fractals. He coined the term the fractal in 1975 from a Latin word Fractus which means “fragmented” as he referred that there are fractional components within every fractals. A shape can only be classified as fractals once it display inherent and repeating similarities. Some of the common examples of fractals in nature as follows: Fractal Tree Fractal in Animal Body By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Snowflakes Romanesco Brocolli Koch Snowflakes is a 2D fractal that have infinite perimeter but with a finite area. Sierpinski Triangle (also known sierpinski gaset or sierpinski sieve) is a fractal that consist of a self- similar equilateral triangle. It was named after Polish mathematician, Wacław Franciszek Sierpiński (1882 – 1969). By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Lesson 1.2 : FIBONACCI SEQUENCE AND THE GOLDEN RATIO FIBONACCI SEQUENCE Fibonacci sequence is one of the most popular numerical sequences that best described the numerical pattern that presents in the nature. It was believed that this might be the blue prints of every nature that the God Almighty had been used in creating the world. This number sequence was introduced in 1202 by Leonardo Pisano who was popularly known with his pseudo name as Fibonacci which means “the son of Bonacci”. He had successfully observed this fascinating pattern of the nature through mathematics. He began his study with this sequence by posing a rabbit population problem which was written in his book Liber Abaci. In this problem, he was about to determine “How many pairs of rabbits will be produced in a year beginning with a single pair, if in every month each pair bears a new pair which becomes productive form the second month on?” In this problem, every pair of rabbit produces another pair every month. However, a newly born pair of rabbits should have to wait for a gestating period of one month to become old and ready to mate. It means that a newly born rabbit can only reproduce a new pair after two months. Moreover, rabbits in this problem were anticipated not to die The production of pairs of rabbits each month was illustrated below using a diagram. https://www.google.com.ph/search?q=rabbit+problem&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjkjbbKwuXdAhVMc3AKHXqKBJwQ_AUIDigB&biw=1365&bih=596&dpr =0.75#imgrc=MJgU4cBHueZc2M Notice that the count of pairs of rabbits each month provides a sequence 1, 1, 2, 3, 5. If we will try to observe each term, it leads to a pattern where each next term, if the sequence will be continued, can be obtained by adding the preceding two terms (e.g. 1 + 1 =2, 1+ 2 = 3). By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 By its definition, this sequence started with 1 , 1 and the subsequent terms would be the sum of the two previous terms. That is Fn = Fn-1 + Fn-2 having a seed values F1 = 1 and F2 = 1. The Binet’s formula, 𝐹! = & & !"√$ !'√$ " # $" # % % can also be used to find the nth of a Fibonacci number. √& Example: Find the 20thand 35th number of the Fibonacci. %( %( )$ )$ !"√$ !'√$ !"√$ !'√$ " # $" # " # $" # % % % % 𝐹'( = 𝐹)& = √& √& 𝐹'( = 𝐹)& = Later on, the presence has not become limited to the rabbit population problem only and this had also been discovered in some mathematical forms as well as in nature. One example is the Pascal’s Triangles of Blaise Pascal which was used and popular for the expansion of the binomial (a+b)n. See the figure below: Notice that by adding the numbers in diagonals of the Pascals’ Triangle, its sum has obviously follows the Fibonacci Sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … The Fibonacci Sequences has become present also in the nature. This can be observed in the petal counts. Try to observe the number petals of Calla lily, euphorbia milli, trillium, hibiscus, clematic and Cineraria (See photos below). They have one, two, three, five, eight and thirteen petals respectively which are Fibonacci numbers. Although there are other kinds of flowers that have number of petals that are not in fibonacci numbers but at least in these kinds of species of flowers, Fibonacci exists. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Calla Lily Euphorbia Milli Calla Lily Hibiscus Photo from http://www.viveropilmaiquen.cl Photo from https://worldofsucculents.com Photo from https://durak.org Photo from https://deserthorizonnursery.com Clematic Cineraria Photo from http://diggingdog.com Photo from http://www.makemebloom.com Furthermore, once we count the number of spirals in sunflowers seeds in counterclockwise and clock wise manner separately, we will found the 8th to 12th Fibonacci numbers depending on the species of the sunflower. With the sunflower shown in the photos, it has 21 spirals clockwise, 34 spirals in counterclockwise direction and another more 55 spirals if we will count the outer spirals. These three numbers are three consecutive numbers from the sequence. Photo retrieved from https://www.daringgourmet.com Fibonacci numbers can also show up in the pineapple and in the bottom of a pinecone. Through counting the spirals that of the same way in the scales of a pineapple, we may found the number 8 – 13 - 21 which are three consecutive Fibonacci numbers. For the pinecone, with counting the spirals on the scales on its bottom, we will found another two adjacent Fibonacci number which are 8 and 13. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Astoundingly, Fibonacci Sequences has significantly shown in many aspects in nature and as time passes by, the connection between the nature and mathematics goes run down deep. Another interesting thing about Fibonacci is its connection with the Golden Ratio. Taking the ratio of two consecutive terms in the Fibonacci Numbers, its value is approaching the value of the Golden Ratio (𝜙) which is approximately equal to 1.61803. * =1 8 * = 1.6; 5 ' =2 13 * = 1.625; 8 ) = 1.5 ' 21 = 1.615 5 13 = 1.67 3 By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 What is the Golden Ratio? *+√& It is a ratio denoted as 𝜙 and has a value equal to 1.61803… or which is irrational number. It is also ' known as golden proportion, golden section or golden mean. It appears many times ancient arts, architecture and even in some areas in geometry. Its value comes from the ratio of the sides of a golden rectangle. It is the rectangle that many artists and architects believe makes their master piece in pleasing, perfect and pleasing appearance. With a golden rectangle, if a square will be cut off from it, the leftover rectangle would have the same proportion with the original rectangle and their value is equal to 𝜙. (See illustration below) a b ,+- , , = - =𝜙 a The derivation of the value of the golden ratio can be generated using ratio obtained from the sectioning of the golden rectangle. That is, ,+- , = =𝜙 , - By taking a fact, each ratio is equal to 𝜙, that is ,+- , =𝜙 =𝜙 , - Considering ,+- , =𝜙 Then, - 𝜙 =1+, , Since 𝜙 = - , hence * 𝜙 =1+. Consequently, by multiplying both sides by 𝜙, it gives a quadratic equation in terms of 𝜙. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 𝜙' − 𝜙 − 1 = 0 Using quadratic formula, the value of 𝜙 can be obtained. In which *+√& *$√& 𝜙= ' = 1.618033... and 𝜙= ' = −0.618033... But since this value was taken from the measures of the sides of the golden ratio, the positive value would be the one to be considered. Another interesting fact about Golden Ratio is its value can also be express using a nested or continuous fractions. * Considering 𝜙 = 1 +., the value of 𝜙 can continuously expanded into fraction by constantly replacing the value of 𝜙 by its value itself as 𝜙 which provides nested fraction value for the golden ratio. 1 𝜙 =1+ 1 1+ 1 1+ 1 1+ 1 1+ 1 1+1+⋯ Astoundingly, this is also connected with a Fibonacci numbers where the ratio between two Fibonacci numbers can also be written into nested fraction which is similar with the 𝜙 value. 1=1 1 8 1+ = = 1.6; 1 5 * ' 1+ 1 1+*= =2 1+ * 1 1+1 * ) 1+ ! = ' = 1.5 *+ 1 13 ! 1+ = = 1.625; 1 8 1 5 1+ 1 1+ = = 1.67 1+ 1 3 1 1+ 1 1+ 1+1 1 1+1 By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 1 21 1+ = = 1.615 1 13 1+ 1 1+ 1 1+ 1 1+ 1 1+1 Another thing, the 𝜙 value can also be express as infinite radicals. Considering the quadratic equation 𝜙 ' − 𝜙 − 1 = 0, it gives a 𝜙 = 31 + 𝜙 By continuously replacing the value of 𝜙, it generates infinite radical 6 ⃓ ⃓ ⃓ ⃓ ⃓ ⃓ 7 𝜙=⃓ ⃓1 + 1 + 81 + 91 + :1 + 31 + √1 + ⋯ ⃓ ⎷ Golden ratio has become very popular in the field of arts and in mathematics. In assessing the value that expresses perfection for every master piece, nature and human body, this special number has always been used. The study for this number had started by Luca Pacioli (1445-1517) where he wrote it in his book “De Divina proportione” (The Divine Proportion) and illustrated by his friend Leonardo D’Vinci. The book studied dozens of geometric solids, architectures and the human body where the application of the Golden Ratio has been realized. On the first page of the book, Pacioli wrote “A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy perspective, painting, sculpture, architecture and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse question touching on a very secret science.” In his words, Pacioli is trying to reveal the essence of the Divine Proportion in creating harmonic forms and beauty every art. Da Vinci later called it “Sectia Aurea” or Golden Ratio. Since then, many artists have embraced this ratio in creating their master to create balance and proportions that makes it more pleasing and attractive. Many Rennaissance paintings and sculptures then have the application of the Golden Ratio. In most of the artworks of Da Vinci, the use of the Golden Ratio is very evident even before his collaboration with Pacioli. Two of the best examples are “The Last Supper” and “Mona Lisa”. Aside to the beauty of these paintings, it had been obvious to both of the master piece that Da Vinci had defined and maintained all the fundamental proportions in the said paintings using the Golden Ratio. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Photo retrieved from http://wcdf-france.com/golden-section-paintings/golden Photo retrieved from https://www.goldennumber.net/leonardo -section-paintings-a-guide-to-the-golden-ratio-aka-golden-section-or-golden-mean -da-vinci-golden-ratio-art/#jp-carousel-6612 Notice that the proportions of the tables, walls, backgrounds and even the position of the disciples in the Last Supper are in well-defined through the Golden Ratio. However, Mona Lisa has many elements to be used as preference points for the determination of the Golden Ratio. Vitruvian Man Photo from https://www.goldennumber.net The Vitruvian Man was another famous work of Leonardo da Vinci on 1940. Its official title is “Le Proporzioni del corpo umano secondo Vitruvio” which means “The proportions of the human body according to Vitruvius”. It was a man inscribed in a circle and shows some dimensions within the human body which connotes proportionality that is according to the Golden Ratio. It pointed out that the height of the man must be in the golden proportion with the ratio of the measures of the head from the navel and navel from the bottom of the feet. By: R.S. Villafuerte MATHEMATICS IN THE MODERN WORLD CHAPTER 1 Aside from the paintings of Leonardo da Vinci, there are also other Renaissance architectures were believed to be built according to Golden Ratio. Examples are Parthenon of Greece and the Great Pyramid of Giza. It was believed that Phidias that he used Golden Rectangles as a pattern in making the Parthenon of Greece during 440BC. Parthenon Photo retrieved from https://medium.com/@social_archi/does-the- golden-ratio-exist-in-architecture-c15a1b3edfba The ratio of the hypotenuse of the Triangle of the Egypt (the right triangle that comes from taking the cross section of the Great Pyramid) to its base is equal to the value of the Golden Ratio. Great Pyramid of Giza By: R.S. Villafuerte