GECMAT Mathematics In The Modern World PDF

Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Introduction Mathematics is a useful way to think about nature and our world. The nature of mathematics underscores the exploration of patterns (in nature and the env...

Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Introduction Mathematics is a useful way to think about nature and our world. The nature of mathematics underscores the exploration of patterns (in nature and the environment). Mathematic exists everywhere and it is applied in the most useful phenomenon. Even looking by just at the ordinary part of the house, the room and the street, mathematics is there. Mathematics is an integral part of daily life; formal and informal. It is used in technology, business, medicine, natural and data sciences, machine learning, industry, engineering, and even in social sciences. It helps organize patterns and regularities in the world, predict the behavior of nature and phenomenon in the world., control nature and occurrences in our world for our end. Mathematics has numerous applications in the world making it indispensable. Mathematics is exhibited not only in the technologies that has dominantly influenced man’s daily pursuits. It is not only practiced by professionals like teachers, scientists, engineers, and economists. Mathematics is practically everywhere and for everyone. It is a thing that perpetually exists in nature and propels development at varying degrees of usefulness. The heart of mathematics is more than just numbers, numbers which many of us supposed to be meaningless and uninteresting. “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). Mathematics is everywhere because it finds many practical applications in our daily lives. Lesson 1: Pattern and Numbers in Nature and in the World Mathematics is a useful way to think about nature and our world. The nature of mathematics underscores the exploration of patterns (in nature and the environment). Mathematics exists everywhere and it is applied in the most useful phenomenon. Mathematics is an integral part of daily life; formal and informal. It is used in technology, business, medicine, natural data sciences, machine learning, and construction. Mathematics is not all about numbers. Rather, it is more about reasoning, of making logical inferences and generalizations, and seeing relationships in both visible and invisible patterns in nature and in the world. One cannot simply base a person’s potential in mathematics based on numeric skills in the same way that a good writer is not judged from his or her penmanship. Mathematics goes beyond arithmetic, and this lesson is devoted to depicting mathematics as a language by which the universe is elegantly designed, the value of which transcends the intellectual, the practical and even the aesthetic values. Patterns and counting are correlative. Counting happens when there is pattern. When there is counting, there is logic. Consequently, pattern in nature goes with logic or logical set-up. There are reasons behind a certain pattern. That’s why, oftentimes, some people develop an understanding of patterns, relationships, and functions and use them to represent and explain real-world phenomena. In this world, a regularity (Collins, 2018) is the fact that the same thing always happens in the same circumstances. While a pattern is a discernible regularity in the world or in a man-made design. As such, the elements of a pattern repeat in a predictable manner. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. According to Ian Stewart (1995), we live in a universe of patterns. Every night the stars move in circles across the sky. The seasons cycle at yearly intervals. No two snowflakes are ever exactly the same, but they all have six-fold symmetry. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature’s patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. Nature embraces mathematics completely. There are many different things around us that have a deep sense of awareness and appreciation of patterns. Nature provides numerous examples of beautiful shapes and patterns, from the nightly motion of the stars and the rainbow that we see in the sky. Some animals show pattern in their body like the tiger’s stripes and hyena’s spots. Snails make their shells, spiders design their webs, and bees build hexagonal combs. The structured formation of parts of human beings, animals and insects, and the beautiful pattern of plants and flowers are examples of patterns that possess utility and beauty. The patterns that we see are also the keys to understanding the processes of biological growth. It is indeed true that the place we live is a world of patterns. Honeycombs and Snowflakes Spiral patterns of leaves and flowers 1 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Coat patterns of different species of animals In the general sense of the word, patterns are regular, repeated, or recurring forms or designs. We see patterns every day – from the layout of floor tiles, designs of skyscrapers, to the way we tie our shoelaces. Studying patterns help students in identifying relationships and finding logical connections to form generalizations and make predictions. Patterns indicate sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. Example 1: Let’s take a look at this pattern below. What do you think will be the next face in the sequence? Solution: It should be easy enough to note that the pattern is made up of two smiling faces – one without teeth and one with teeth. Beginning with a toothless face, the two faces then alternate. Logically, the face that should follow is Example 2: What is the next figure in the pattern below? ? A B Solution: Looking at the given figures, the lines seem to rotate at 90-degree intervals in a counterclockwise direction, always parallel to one side the square. Hence, either A or B could be the answer. Checking the other patterns, the length of the lines inside the square follow a decreasing trend. So again, either A or B could be the answer. Finally, looking at the number of the lines inside the box, each succeeding figure has the number of lines increase by 1. This means that the next figure should have five lines inside. This leads to option A as the correct choice. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem- solving ability and even effective communication skills. Mathematics is the cradle of all creations, without which the world cannot move an inch. Be it a cook or a farmer, a carpenter or a mechanic, a shopkeeper or a doctor, an engineer or a scientist, a musician or a magician, everyone needs mathematics in their day-to-day life. Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth. Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society. 2 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Exercise 1 a. From the following pictures below, tell or describe the patterns that you observed in one word. https://www.thesmartteacher.com/exchange https://theconversation.com/fractal-patterns-in-nature-and- https://www.slideshare.net/esmemc/fibonacci /resource/333/Abstract_Patterns_in_Nature art-are-aesthetically-pleasing-and-stress-reducing-73255 -presentation __________________________ __________________________ _________________________ b. Give three examples of objects or animals in your house (within your locality) that depicts a pattern similar to the three pictures above and discuss the pattern within each object or animal. First object (or animal): ____________________________ Second object (or animal): ____________________________ Third object (or animal): ____________________________ c. Make a short response or an essay for each of the following questions. After that, create a small group of four members and share your answer or thoughts about the question. Then synthesizing the answers of your group, choose one representative to present the answer to the class. i. What is mathematics? ii. Where is mathematics? iii. What role does mathematics play in your world? iv. Why is mathematics called a science of patterns? v. Are the patterns that appear in the natural world a coincidence? Why or why not? d. Watch the following video clips and answer the questions below. Video #1. Nature by Numbers by Cristobal Vila (https://www.youtube.com/watch?v=kkGeOWYOFoA) Video #2. Why Honeybees Love Hexagons by Zack Patterson and Andy Peterson. (https://www.youtube.com/watch?v=QEzlsjAqADA) Video #3. Mathematics is the queen of Sciences (https://www.youtube.com/watch?v=8mve0UoSxTo) i. Three things that I significantly learned from the video clips ii. Three things that are still unclear to me iii. Complete the statement: I used to think that…. e. Look for the following items below and observe. Then find out and describe the pattern on each item. i. Animals ii. Plants or Fruits iii. Things or man made iv. Natural Phenomenon f. State five (5) reasons why mathematics is important with clear description of application. State disadvantage if a person does not know and understand mathematics. Reasons Setbacks 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 3 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department g. Provide concise answers (maximum of 5 sentences) to the following questions. i. Why are numbers important in our life? ii. What new ideas about mathematics did you learn? iii. What is it about mathematics that might have changed your thoughts about it? iv. What is most useful about mathematics for humankind? h. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Among the natural patterns, choose three patterns, discuss its structure and characteristics, give an example, and its role in our nature. i. Describe in details the importance of mathematics in the following areas. Cite specific and real-life example for each. i. Government ii. Medicine iii. Music and Arts iv. Society v. In your field of specialization (Education, Engineering, Business, Industrial Technology, Information System, Hospitality Management, Psychology, Accountancy, Business Management, Entrepreneurship, Office Administration, Fisher Lesson 2: Fibonacci Sequence, Golden Ratio and Golden Rectangle The pictures below depict the different species of flowers. Count the number of petals for each flower. Flower Asiatic Candy Name Calla lily Flag Iris Dahlia Pink daisy Sunflower dayflower flower Number of petals What sequence of numbers formed from the number of petals? ___, ___, ___, ___, ___, ___, ___ Solution: 1, 2, 3, 5, 8, 13, 21 The sequence of numbers formed from the number of petals of the different species of flower is a Fibonacci sequence. 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 led 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. Fibonacci sequence derived from a problem in the Liber Abaci, which was about how fast rabbits could breed in ideal circumstances. a) 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? b) Beginning with a male and female rabbit, how many pairs of rabbits could be born in a year? The problem assumes the following conditions: c) Begin with one male rabbit and female rabbit that have just been born. d) Rabbits reach sexual maturity after one month. e) The gestation period of a rabbit is one month. f) After reaching sexual maturity, female rabbits give birth every month. g) A female rabbit gives birth to one male rabbit and one female rabbit h) Rabbits do not die. 4 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department This is illustrated in the diagram. After one month, the first pair is not yet at sexual maturity and can't 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. A recursive definition for a sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. If we sue the mathematical notation 𝐹𝑛 to represent 𝑛𝑡ℎ Fibonacci number, then the numbers in the Fibonacci sequence are given by the following recursive definition 𝐹1 = 1, 𝐹2 = 1, and 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 for 𝑛 ≥ 3. Example 1: Use the definition of Fibonacci numbers to find the seventh and eight Fibonacci numbers. Solution: The first six Fibonacci numbers are 1, 1, 2, 3, 5, and 8. The seventh Fibonacci number is the sum of the two previous Fibonacci numbers. Thus, 𝐹7 = 𝐹6 + 𝐹5 =8+5 = 13 The eight Fibonacci number is 𝐹8 = 𝐹7 + 𝐹6 = 13 + 8 = 21 Binet's Formula Binet's formula is an explicit formula used to find the nth term of the Fibonacci sequence. It is so named because it was derived by mathematician Jacques Philippe Marie Binet. Formula If 𝐹𝑛 is the nth Fibonacci number, then 1 1 + √5 𝑛 1 − √5 𝑛 𝐹𝑛 = [( ) −( ) ] √5 2 2 Example 2: Using the Binet’s formula, determine the eight Fibonacci number. By substitution, 𝑛 = 8, we have 1 1 + √5 8 1 − √5 8 𝐹8 = [( ) −( ) ] √5 2 2 1 (1+√5)8 (1−√5)8 = [ 8 − ] √5 2 28 Simplifying the expression, we have 1 (1+√5)8 −(1−√5)8 = [ ] √5 28 8 1 (18 +8(1)7 (√5)+28(1)6 (√5)2 +56(1)5 (√5)3 +70(1)4 (√5)4 +56(1)3 (√5)5 +28(1)2 (√5)6 +8(1)1 (√5)7 +(√5) )−(18 +8(1)7 (−√5)+ = [ √5 28 8 28(1)6 (−√5)2 +56(1)5 (−√5)3 +70(1)4(−√5)4 +56(1)3 (−√5)5 +28(1)2 (−√5)6 +8(1)1 (−√5)7 +(−√5) ] 28 1 (1+8√5+140+280√5+1750+1400√5+3500+1000√5+625)−(1−8√5+140−280√5+1750−1400√5+3500−1000√5+625) = [ ] √5 28 1 1+8√5+140+280√5+1750+1400√5+3500+1000√5+625−1+8√5−140+280√5−1750+1400√5−3500+1000√5−625 = ( ) √5 28 1 1+8√5+140+280√5+1750+1400√5+3500+1000√5+625−1+8√5−140+280√5−1750+1400√5−3500+1000√5−625 = ( ) √5 28 1 16√5+560√5+2800√5+2000√5 = ( ) √5 28 1 5376√5 = ( 8 ) √5 2 1 5376√5 = ( 8 ) √5 2 = 21 Hence, the eight Fibonacci number is 21. 5 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Determine the quotient of the following two consecutive Fibonacci numbers. Two consecutive 2 3 5 8 13 21 34 55 89 Fibonacci numbers 1 2 3 5 8 13 21 34 55 Quotient or ratio (in three decimal places) If we continue the sequence, we still have 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765… If we continue down the ratio of Fibonacci numbers, what number does it approach or converge upon? Solution: 1.618… As the numbers in the sequence gets larger and larger, the ratio will eventually become the same number, and that number is the Golden Ratio. 𝐹𝑛+1 lim =𝜑 𝑛→∞ 𝐹𝑛 The limit of the ratio of two consecutive Fibonacci numbers is phi. Golden ratio phi “𝝋” The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Since the ratio is basically a fraction, we will find the ratios of these numbers by dividing the larger number by the smaller number that fall consecutively in the series. The golden ratio is the division of a given unit of length into two parts such that the ratio of the whole to the longer part is equals the ratio of the longer part to the shorter part. It is also known as the golden proportion, golden mean, golden section, golden number, and divine proportion. It is a number often encountered when taking the ratios of distances in simple geometric figures, such as the pentagon, pentagram, decagon and dodecahedron. It is a ratio or proportion defined by an irrational number phi (approximately equal to 1.618033988749895…). It is expressed algebraically as, a+b a = =φ a b where: a is the longer side and b is the shorter side. It has its unique positive solution with a value 1 + √5 φ= ≈ 1.6180339887 … 2 One more interesting thing about phi is its reciprocal. If you take the ratio of any number in the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will approach the approximation 0.618. This is the 1 reciprocal of Phi: = 0.618 …. It is highly unusual for the decimal integers of a number and its reciprocal to be exactly 1.618 the same. 1+√5 Why φ equal to ? 2 a+b a To derive the exact value of φ, we will use the equation = , a b a+b a = a b a b a + = a a b b a 1+ = a b a Since = φ, we have b 1 1+ =φ φ Multiply both sides by 𝜑, we have 6 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department 1 φ(1 + ) = (φ)φ φ 𝜑 + 1 = 𝜑2 𝜑2 − 𝜑 − 1 = 0 Using quadratic formula, solve for 𝜑, we have −b±√b2 −4ac 𝜑= , where a = 1, b = -1, and c = -1 2a By substitution, −b ± √b 2 − 4ac −(−1) ± √(−1)2 − 4(1)(−1) 1 ± √1 + 4 1 ± √5 𝜑= = = = 2a 2(1) 2 2 1−√5 Since is negative, hence we have 2 1 + √5 𝜑= 2 Golden rectangle Look at the following rectangles: which of them seems to be the most naturally attractive rectangle? If you were to measure each rectangle's length and width, and compare the ratio of length to width for each rectangle you would see the following: Rectangle one: Ratio 1:1 Rectangle two: Ratio 2:1 Rectangle Three: Ratio 1.618:1 Golden rectangle is a rectangle whose side lengths are in the golden ratio. 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 one that has a certain length to width ratio and is most pleasing to the eye. 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. The Divine Proportion The Divine Proportion is often represented by the golden spiral. This is the tool used by artists and sculptors to achieve remarkably accurate proportion and aesthetic composition. In the golden spiral, the ratio of the length of the side of each square to the length of the side of the next smallest square is in the golden ratio; the rectangles formed by the combination of squares are golden rectangles; and at every quarter-turn, the spiral gets wider by a factor of Phi. 7 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department Generalization The Fibonacci sequence is the sequence 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , … which has its first two terms 𝐹1 and 𝐹2 both equal to 1 and satisfies thereafter the recursion formula 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2. When continued indefinitely, the sequence encountered in the rabbit problem 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, … is called the Fibonacci sequence and its terms the Fibonacci numbers. Binet's formula is an explicit formula used to find the nth term of the Fibonacci sequence. If 𝐹𝑛 is the nth Fibonacci number, then 1 1 + √5 𝑛 1 − √5 𝑛 𝐹𝑛 = [( ) −( ) ] √5 2 2 The ratios of sequential Fibonacci numbers approach the golden ratio. In fact, the higher the Fibonacci numbers, the closer their relationship is to 1.618. With one number 𝑎 and another smaller number 𝑏, the ratio of the two numbers is 𝑎 found by dividing them. Their ratio is. Another ratio is found by 𝑏 adding the two numbers together 𝑎 + 𝑏 and dividing this by the larger 𝑎+𝑏 number 𝑎. The new ratio is. If these two ratios are equal to the 𝑏 same number, then that number is called the golden ratio. The Greek letter 𝜑 (phi) is usually used to denote the golden ratio. The golden ratio is sometimes called the "divine proportion," because of its frequency in the natural world. The golden ratio 𝜑 (𝑝ℎ𝑖) is equal to 1+√5 or approximately equal to 1.6180339887. 2 If the length of a rectangle divided by its width is equal to the golden ratio, then the rectangle is called a "golden rectangle.” If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and 𝑎 pink together) is a golden rectangle because = 𝜑. The blue part (B) 𝑏 is a square. The pink part by itself (A) is another golden rectangle 𝑏 because = 𝜑. (𝑎−𝑏) Exercise 2: a. List man-made objects that were designed based on the Fibonacci sequence. b. Fibonacci exists on many things in the natural world like the resemblance of the Fibonacci spirals on the seashells. Explain how the Fibonacci sequence exists on the following natural objects. i. Leaves ii. Storms iii. Tress iv. Human beings c. List down 3 objects that resemble a golden ratio in the natural world and explain the existence of the golden ratio in each object. Present each object using a picture or a drawing. d. Measure the following body parts (in cm). You may ask someone for you to assist in measuring your body parts. Round answer up to two decimal places. Then, write down the proportions of each body parts. Body parts Measurement Ratio a = Top-of-head to chin a = ______ 𝒂 𝟏) = _________ b = Top-of-head to pupil b = ______ 𝒈 𝒃 c = Pupil to nose tip c = ______ 𝟐) = _________ 𝒅 d = Pupil to lip d = ______ 𝒊 e = Width of nose e = ______ 𝟑) = _________ 𝒋 f = Outside distance between eyes f = ______ 𝒊 𝟒) = _________ g = Width of head g = ______ 𝒄 h = Hairline to pupil h = ______ 𝒆 𝟓) = _________ 𝒍 i = Nose tip to chin i = ______ 𝒇 j = Lips to chin j = ______ 𝟔) = _________ 𝒉 k = Length of lips k = ______ 𝒌 𝟕) = _________ l = Nose tip to lips l = ______ 𝒆 Which ratio of your body part is nearest to the golden ratio? Which ratio of your body part is the farthest to the golden ratio? 8 Revision 02 Chapter 1 Mathematics in Our World GECMAT (Mathematics in the Modern World) CHMSU – CAS Mathematics Department e. Are you beautiful? Dr. Kendra Schmidt, an assistant professor of biostatistics, uses Golden ratio to study facial sex appeal. She conjectures that beauty or sex appeal is related to the proportions of facial features which happen to follow the golden ratio. Based on this conjecture, let us check how close your facial proportions are to the golden ratio. Measure the length and width of your face. Then divide the length by the width. What is the result? Is the result roughly the golden ratio? f. Do you have the “Greek god body?” The golden ratio (shoulder to waist) is the most important ratio for achieving the Greek god body. Now, measure your shoulder circumstances (𝑠) and then your waist size (𝑤). Then divide 𝑠 by 𝑤. Is the result roughly the golden ratio? If not, then what must be your ideal waist size to get the golden ratio? Provide concise answers (maximum of 5 sentences) to the following questions. g. Did the lesson present change your perspective in Mathematics? Why? h. What is the most fascinating information that you have learned about Fibonacci sequence, golden ratio and golden rectangle? i. There have been debates on how these patterns that appears in the natural world are regularities that can help us understand the world or just mere coincidence. Which side are you on, and why? j. Give two examples where Fibonacci sequence or the Golden ratio is used in relation to your major field of specialization. k. For each of the following, describe one thing aside from the examples listed in the lesson. a) that show the Fibonacci sequence b) that illustrates the Golden Ratio c) that follows a Golden Rectangle l. Develop a one-page synthesis paper focusing on the following aspects of mathematics. a) Mathematics helps organize patterns and regularities in the world. b) Mathematics helps predict the behavior of nature and phenomena in the world. c) Mathematics is a tool to quantify, organize, and control nature and occurrences in the world for our own ends. 9 Revision 02