MMW-L1-The-Nature-of-Mathematics.pdf

MATHEMATICS IN THE MODERN WORLD LESSON 1: THE NATURE OF MATHEMATICS Learning Outcomes: At the end of this lesson, the student must be able to: 1. Know the subject and the class. 2. Identify patterns in nature and regularities in the world. 3. Articulate the importance of mathematics in one’s l...

MATHEMATICS IN THE MODERN WORLD LESSON 1: THE NATURE OF MATHEMATICS Learning Outcomes: At the end of this lesson, the student must be able to: 1. Know the subject and the class. 2. Identify patterns in nature and regularities in the world. 3. Articulate the importance of mathematics in one’s life. 4. Argue about the nature of mathematics, what it is, how it is expressed, represented, and used. 5. Express appreciation for mathematics as a human endeavor. Definition:  Mathematics is the study of assumptions, its properties and applications. ✓ 3 steps: first define the basic terms followed by the properties with proof and make a table of the formulae and then solve at least five examples on each formulae or properties. ✓ In this way students learn the definition, formulae and understand the basic structure in application.  Mathematics is the science that deals with the logic of shape, quantity and arrangement. ✓ Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports. 15 INCREDIBLE EXAMPLES OF MATHEMATICS IN NATURE 15. Snowflakes The tiny but miraculous snowflake, as an example of symmetry in nature, exhibits six- fold radial symmetry, with elaborate, identical patterns on each arm. Snowflakes form because water molecules naturally arrange when they solidify. It’s complicated but, basically, when they crystalize, water molecules form weak hydrogen bonds with each other. 14. Sunflowers Bright, bold and beloved by bees, sunflowers boast radial symmetry and a type of numerical symmetry known as the Fibonacci sequence, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so forth. Scientists and flower enthusiasts who have taken the time to count the seed spirals in a sunflower have determined that the number of spirals adds up to a Fibonacci number. 13. Uteruses According to a gynecologist, doctors can tell whether a uterus looks normal and healthy based on its relative dimensions – dimensions that approximate the golden ratio. When women are at their most fertile, the ratio of uterus length to its width is 1.6. This is a very good approximation of the golden ratio. 12. Nautilus Shell A nautilus is a cephalopod mollusk with a spiral shell and numerous short tentacles around its mouth. Although more common in plants, some animals, like the nautilus, showcase Fibonacci numbers. A nautilus shell is grown in a Fibonacci spiral. The spiral occurs as the shell grows outwards and tries to maintain its proportional shape. 11. Romanesco Brocolli Romanesco broccoli has an unusual appearance, and many assume it’s another food that’s fallen victim to genetic modification. However, it’s actually one of many instances of fractal symmetry in nature. In geometric terms, fractals are complex patterns where each individual component has the same pattern as the whole object. In the case of Romanesco broccoli, the entire veggie is one big spiral composed of smaller, cone-like mini-spirals. 10. Pinecones Pinecones have seed pods that arrange in a spiral pattern. They consist of a pair of spirals, each one twisting upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a three–to–five cone meets at the back after three steps along the left spiral and five steps along the right. This spiraling Fibonacci pattern also occurs in pineapples and artichokes. 9. Honeycombs Honeycombs are an example of wallpaper symmetry. This is where a pattern is repeated until it covers a plane. Other examples include mosaics and tiled floors. Mathematicians believe bees build these hexagonal constructions because it is the shape most efficient for storing the largest possible amount of honey while using the least amount of wax. Shapes like circles would leave gaps between the cells because they don’t fit perfectly together. 8. Tree Branches The Fibonacci sequence is so widespread in nature that it can also be seen in the way tree branches form and split. The main trunk of a tree will grow until it produces a branch, which creates two growth points. One of the new stems will then branch into two, while the other lies dormant. This branching pattern repeats for each of the new stems. A good example is the sneezewort, a Eurasian plant of the daisy family whose dry leaves induce sneezing. 7. Milky Way Galaxy Symmetry and mathematical patterns seem to exist everywhere on Earth – the Milky Way Galaxy was discovered, and, by studying this, astronomers now believe the galaxy is a near-perfect mirror image of itself. Having mirror symmetry, the Milky Way has another amazing design. Like nautilus shells and sunflowers, each ‘arm’ of the galaxy symbolizes a logarithmic spiral that begins at the galaxy’s center and expands outwards. 6. Faces Humans possess bilateral symmetry. Faces, both human and otherwise, are rife with examples of the Golden Ratio. Mouths and noses are positioned at golden sections of the distance between the eyes and the bottom of the chin. Comparable proportions can be seen from the side, and even the eye and ear itself, which follows along a spiral. For example, the most beautiful smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. 5. Orb Web Spiders Orb web spiders create near-perfect circular webs that have near-equal distanced radial supports coming out of the middle and a spiral that is woven to catch prey. Orb webs are built for strength, with radial symmetry helping to evenly distribute the force of impact when a spider’s prey contacts the web. This would mean there’d be less rips in the thread. 4. Crop Circles Crop circles are a sight to behold because they’re so geometrically impressive. A study conducted by physicist Richard Taylor revealed that, somewhere in the world, a new crop circle is created every night, and that most designs demonstrate a wide variety of symmetry and mathematical patterns, including Fibonacci spirals and fractals. 3. Starfish Starfish or sea stars belong to a phylum of marine creatures called echinoderm. Other notable echinoderm include sea urchins, brittle stars, sea cucumbers and sand dollars. The larvae of echinoderms have bilateral symmetry, meaning the organism’s left and ride side form a mirror image. Sea stars or starfish are invertebrates that typically have five or more ‘arms. These radiate from an indistinct disk and form something known as pentaradial symmetry. 2. Peacocks The peacock takes the earlier principle of using symmetry to attract a mate to the nth degree. Male peacocks utilize their variety of adaptations to seduce sultry peahens. These include bright colors, a large size, a symmetrical body shape and repeated patterns in their feathers. 1. Sun-Moon Symmetry The sun has a diameter of 1.4 million kilometers, while its sister, the Moon, has a meager diameter of 3,474 kilometers. With these figures, it seems near impossible that the moon can block the sun’s light and give us around five solar eclipses every two years.  By sheer coincidence, the sun’s width is roughly four hundred times larger than that of the moon, while the sun is about four hundred times further away. The symmetry in this ratio causes the moon and sun to appear almost the same size when seen from Earth, and, therefore, it becomes possible for the moon to block the sun when the two align.  Earth’s distance from the sun can increase during its orbit. If an eclipse occurs during this time, we see what’s known as an annular or ‘ring’ eclipse. This is because the sun isn’t completely hidden.  Every one to two years, though, the sun and moon become perfectly aligned, and we can witness a rare event called a total solar eclipse. Every year, though, our moon drifts roughly four centimeters further from Earth. This means that, billions of years ago, every solar eclipse would have been a total eclipse.  Sun-moon symmetry is the special factor that makes life on Earth possible. FIBONACCI NUMBERS Discovered by Fibonacci, a great European mathematician of the Middle Ages. His full name in Italian is Leonardo Pisano, which means Leonardo of Pisa, because he was born in Pisa, Italy around 1175. Fibonnaci observed numbers in nature. His most popular contribution perhaps is the number that is seen in the petals of flowers. A calla lily flower has only 1 petal, trillium has 3, hibiscus has 5, cosmos flower has 8, corn marigold has 13, some asters has 21, and a daisy can have 34, 55 or 89 petals. Surprisingly, these petal counts represent the first eleven numbers of the Fibonacci Sequence. Not all petal numbers of flowers, however, follow this pattern discovered by Fibonacci. Some examples include the Brassicaceae family having four petals. Astoundingly, many of the flowers abide by that pattern observed by Fibonacci.  The principle behind the Fibonacci numbers is as follows:  Let 𝑥𝑛 be the 𝑛th integer in the Fibonacci sequence, the next (𝑛 + 1)th term 𝑥𝑛+1 is determined by adding 𝑛th and the (𝑛 − 1)th integers.  Consider the first few terms below: Let 𝑥1 = 1 be the first term, and 𝑥2 = 1 be the second term, the third term 𝑥3 is found by 𝑥3 = 𝑥1 + 𝑥2 = 1 + 1 = 2.  The fourth term 𝑥4 is 2 + 1 = 3, the sum of the third and second terms.  To find the new 𝑛th Fibonacci number, simply add the two numbers immediately preceding this 𝑛th number.  These numbers arranged in increasing order can be written as the sequence {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …}. THE GOLDEN RATIO The golden ratio is so fascinating that proportions of the human body such as the face follow the so called Divine Proportion.  The closer the proportion of the body parts to the Golden Ratio, the more aesthetically pleasing and beautiful the body is. Many painters, including the famous Leonardo da Vinci were so fascinated with the Golden Ratio that they used it in their works of art. The photo illustrates the following golden ratio proportions in the human face: Center of pupil : bottom of teeth : bottom of chin Outer and inner edge of eye : center of nose Outer edges of lips : upper ridges of lips Width of center tooth : width of second tooth Width of eye ; width of iris Questions ? Clarifications? THANK YOU!