Math 01 Lecture Notes PDF

MATH 01 Mathematics in the Modern World Chapter 1. Nature of Mathematics I. Mathematics in Our World HON...

MATH 01 Mathematics in the Modern World Chapter 1. Nature of Mathematics I. Mathematics in Our World HONEYLOU F. FARINAS Department of Mathematics Math 01 – Mathematics in the Modern World College of Arts and Sciences  Learning Outcomes: At the end of the chapter, the students must have: 1. identified patterns in nature and regularities in the world 2. articulated the importance of mathematics in one’s life (V) 3. argued about the nature of mathematics, what it is, how it is expressed, represented, and used (K) 4. expressed appreciation for mathematics as a human endeavor (V) Math 01 – Mathematics in the Modern World  PATTERNS AND NUMBERS IN NATURE AND THE WORLD Definition A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. Math 01 – Mathematics in the Modern World  SYMMETRY Definition Symmetry is a geometrical or other regularity that is possessed by an object and is characterized by the operations that leave the object invariant. Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: REFLECTION Definition Reflection is a symmetry consisting of a mirror line or axis on the plane and maps points from one side of the line to the other side at the same distance from the axis. Reflection is also called line symmetry or mirror symmetry. Math 01 – Mathematics in the Modern World BILATERAL SYMMETRY Definition Bilateral symmetry is the simplest form of reflection involving only one mirror axis. A type of symmetrical arrangement of the parts of an organism along a central axis, so that the organism can be divided into two equal halves. Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: ROTATION Definition Rotation is a symmetry fixing one point (the center or rotocenter) and rotates everything by the same amount (angle) around that point. Note: The number of times an image appear in its original form is called the order of rotation. Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: ROTATION Order 3 Order 4 Order 5 Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: TRANSLATION Definition Translation is a symmetry operation which moves everything by a certain distance in a particular direction. Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: TRANSLATION Definition Translation is a symmetry operation which moves everything by a certain distance in a particular direction. Math 01 – Mathematics in the Modern World  SYMMETRY OPERATIONS: GLIDE REFLECTION Definition Glide reflection is a combination of a reflection with a translation along the direction of the mirror line. Math 01 – Mathematics in the Modern World  PATTERNS IN NATURE Honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honeybees in their nests to contain their larvae and stores of honey and pollen. The structural design encloses regions of maximum area but with minimum perimeter, suggesting that honeybees are economists and mathematicians. Recommended video: Why do bees build hexagonal honeycombs? by BBC Math 01 – Mathematics in the Modern World Link: https://www.bbc.co.uk/programmes/p03zn0bp  PATTERNS IN NATURE Phyllotaxis/Phyllotaxy Leaf arrangements known as phyllotaxis or phyllotaxy come in many forms. Leaves of plants may have the same arrangement on opposite sides where pairs of leaves are attached at a node. The arrangement of leaves may also alternate along the stem. Recommended reading: https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/phyllotaxis Math 01 – Mathematics in the Modern World  PATTERNS IN NATURE Tiger’s Stripes The stripes on the tiger’s skin are patterns that are actually part of the animal’s skin so it appears below the fur as well. Hyena’s Spots Tiger stripes have probably The base color on the fur of spotted hyenas developed because the pattern are irregular pattern of roundish spots. The helps hide them and protect spots may be reddish, deep brown or almost them in their natural blackish and like the tiger, the spots of the environment. hyena serves to hide them from prey. Recommended reading: How animals got their spots and stripes – according to maths Link: http://theconversation.com/how-animals-got-their-spots-and-stripes-according-to-maths-85053 Math 01 – Mathematics in the Modern World  ASSIGNMENT “Photography patterns in nature and the world.” Take a good picture of nature, an object, an architecture or scenery around you that represents the four symmetry operations discussed. Indicate in the picture taken the observed symmetry operation. One photo for each of the 4 operations. Math 01 – Mathematics in the Modern World  FIBONACCI SEQUENCE Definition The Fibonacci sequence is a list of ordered natural numbers: 1, 1, 2, 3, 5, 8, 13, 21,... The first two numbers are both equal to 1 and the succeeding numbers in the sequence are the sum of the two consecutive numbers immediately preceding them. Math 01 – Mathematics in the Modern World  FIBONACCI SEQUENCE Math 01 – Mathematics in the Modern World  FIBONACCI’S PROBLEM The original problem that Fibonacci investigated (that was posed in his book Liber Abaci (The book of counting) in the year 1202) was about how fast rabbits could breed in ideal circumstances. Math 01 – Mathematics in the Modern World  If a pair of rabbit breeds once a month, and each pair they produce can also breed new pair at one month old, how many pairs of rabbits will be bred in a year, starting with the one pair? Math 01 – Mathematics in the Modern World  FIBONACCI SEQUENCE The sequence 𝐹𝑛 of Fibonacci numbers is defined by the recurrence relation 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 with initial values 𝐹1 = 1 and 𝐹2 = 1. 1, 1, 2, 3, 5, 8, 13, 21,... Math 01 – Mathematics in the Modern World  FIBONACCI IN NATURE Sunflowers boast radial symmetry and an interesting type of numerical symmetry known as the Fibonacci Sequence. 21 counterclockwise 34 clockwise Math 01 – Mathematics in the Modern World SUNFLOWER  FIBONACCI IN NATURE FLOWERS NUMBER OF Two-petalled Three-petalled PETALS One-petalled White calla lily Euphorbia Trillium Lilies 3 Buttercups 5 Delphiniums 8 Five-petalled Columbine Eight-petalled Bloodroot Thirteen-petalled Black-eyed Susan Marigolds 13 Asters 21 Daisies 34, 55, 89 21-petalled 34-petalled Shasta daisy Field daisy Math 01 – Mathematics in the Modern World FLOWER PETALS  FIBONACCI IN NATURE Counting spirals on sunflowers can be difficult, so if you want to test this principle yourself, try counting the spirals on bigger things like pinecones Math 01 – Mathematics in the Modern World PINECONES  FIBONACCI IN NATURE The shell closely follows the pattern of a spiral drawn in Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers. Math 01 – Mathematics in the Modern World SNAIL’S SHELL  FIBONACCI IN NATURE The plant Sneezewort also called Harangan also possesses Fibonacci property in terms of their leaves and branches. Math 01 – Mathematics in the Modern World BRANCHES AND LEAVES  FRACTALS Definition Fractals are geometric figures like triangles, squares, circles and rectangles that repeat their structure on ever-finer scales. They are never-ending patterns, infinitely complex patterns that are self- similar across different scales. This means that one can take a small extract of the shape and it looks the same as the entire shape. Math 01 – Mathematics in the Modern World  FRACTALS IN NATURE Neurons Fern Leaf Trees’ branches Nervous System Math 01 – Mathematics in the Modern World  GEOMETRICS FRACTALS These geometric fractals have the property of self-similarity. Geometric figures are similar if they have the same shape. Self-similar objects appear the same under magnification, hence, in some fashion, they are composed of similar copies of themselves. Sierpinski Triangle – one of the simplest fractal shapes named after Polish mathematician Waclaw Sierpinski Math 01 – Mathematics in the Modern World  GEOMETRICS FRACTALS These geometric fractals have the property of self- similarity. Geometric figures are similar if they have the same shape. Self-similar objects appear the same under magnification, hence, in some fashion, they are composed of similar copies of themselves. Sierpinski Triangle Koch Snowflake Math 01 – Mathematics in the Modern World  GEOMETRICS FRACTALS Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Sierpinski Triangle) Constructing the Sierpinski triangle using the “method of successive removals” Level 1: Draw an equilateral triangle. It serves as the base shape. Base Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Sierpinski Triangle) Constructing the Sierpinski triangle using the “method of successive removals” Level 2: Mark the midpoint of each side of the triangle and draw three lines that connect the marks. This results in four new smaller congruent triangles. Shade the middle triangle. This is now the motif shape. Motif Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Sierpinski Triangle) Constructing the Sierpinski triangle using the “method of successive removals” Level 3: Notice that there are three new white triangles, one in each corner in Level 2. Apply the motif shape to each of these three triangles. Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Sierpinski Triangle) Constructing the Sierpinski triangle using the “method of successive removals” Level 4: Repeat this process for at least one more level. Note: When a process is repeated over and over, each repetition is called an “iteration”. Level 2 is the iteration rule for the Sierpinski triangle Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Koch Snowflake) Constructing the Koch Snowflake using the “copies of copies method” The Koch Snowflake has the following base and motif: Motif Base Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Koch Snowflake) Constructing the Koch Snowflake using the “copies of copies method” Level 1: Draw an equilateral triangle. It serves as the base shape. Level 2: Replace all the sides of the triangle with the motif. Make sure that the motif shape is placed such that the small triangle “bump” is facing outward. This should make the overall area increase. The shape now looks like a star. Math 01 – Mathematics in the Modern World  MAKING FRACTALS (Koch Snowflake) Constructing the Koch Snowflake using the “copies of copies method” Level 3: Each of the twelve small segments in level 2 will be replaced with the motif shape. The fractal will now begin to look more like a snowflake. Level 4: Continue this process one more time by replacing each segment with the motif. Math 01 – Mathematics in the Modern World  ASSIGNMENT “Cross Fractal Construction” Create the cross fractal using the following base and motif: Base Motif Level 1. Draw a square. Level 2. Replace all four sides of the square with the motif. Make sure that the “cross” in the motif is facing outward. The motif has thirteen small segments to it. Each of those smaller segments is 1/3 of the size of the square that had been replaced. Level 3. Replace each of the small segments from level 2 with the motif. Each of those segments is 1/3 the size of the segments used in the previous level. Math 01 – Mathematics in the Modern World  SPIRALS Definition Spirals are common patterns in nature that we see more often in living things. They are curves which emanate from a point, moving farther away as it revolves around the point. Spirals are “open” curves compared to circles and ellipses which are closed curves. Math 01 – Mathematics in the Modern World  SPIRALS Math 01 – Mathematics in the Modern World SPIRALS IN NATURE The Milky Way galaxy is an example of a spiral galaxy. It has logarithmic spirals in its arms beginning at the center of the galaxy and expanding outwards. Interestingly, spiral galaxies follow the familiar Fibonacci pattern. Recently, Thomas Dame and Patrick Thaddeus of the Harvard-Smithsonian Center for Astrophysics made a discovery which suggests that the Milky Way galaxy has a mirror-like symmetry. If true, the Milky Way galaxy will be a near-perfect mirror image of itself with two major arms: the Perseus and the Scutum-Centaurus. Math 01 – Mathematics in the Modern World SPIRALS IN NATURE The shell of a nautilus exhibits Fibonacci numbers. It is grown in almost a Fibonacci spiral. That is, it closely follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers Mosquito coils (mosquito repellants) are shaped like Fermat’s spiral Math 01 – Mathematics in the Modern World  GOLDEN RATIO Definition The Golden Ratio, also known as the Divine Proportion, Golden Section or Golden Mean, denoted by 𝜑 (read as “phi ”), is approximately equal to 1.618033988749895… Math 01 – Mathematics in the Modern World  GOLDEN RATIO There is a special relationship between the Golden Ratio and the Fibonacci numbers. If any two successive Fibonacci Numbers are taken, their ratio approximates the Golden Ratio. One can observe that the higher the numbers in the sequence, the closer they match the golden mean. CONSECUTIVE FIBONACCI NUMBERS a/b b a 2 3 1.5 3 5 1.666… 5 8 1.6 8 13 1.625 13 21 1.615384615… … … … 144 233 1.618055556… 233 377 1.618025751 … … … Math 01 – Mathematics in the Modern World  GOLDEN RATIO In a line segment, the golden ratio can be found by dividing the segment into two parts so that the longer part (a) divided by the smaller part (b) is also equal to the whole length (a+b) divided by the longer part. 𝑎+𝑏 𝑎 = = φ = 1.618 … 𝑎 𝑏 Math 01 – Mathematics in the Modern World  GOLDEN RATIO Consider a line segment with length 100 centimeters. How long should a and b be for them to satisfy the Golden Ratio? 𝑎+𝑏 Recall that = 1.618 and 𝑎 + 𝑏 = 100 𝑐𝑚 is the length of the line segment. 𝑎 100 ⟹ = 1.618 𝑎 1.618 𝑎 = 100 𝑎 = 61.8 𝑐𝑚 Solving for 𝑏: 𝑏 = 100 − 61.8 = 38.2 𝑐𝑚. Thus, 𝑎 = 61.8 𝑐𝑚 and 𝑏 = 38.2 𝑐𝑚. Math 01 – Mathematics in the Modern World  GOLDEN RECTANGLE Definition The Golden Rectangle is a rectangle whose side lengths are in the golden ratio 1+ 5 1: 𝑜𝑟 1: 𝜑 2 Math 01 – Mathematics in the Modern World  GOLDEN RECTANGLE A golden rectangle (in blue) with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship 𝑎+𝑏 𝑎 = = φ = 1.618 … 𝑎 𝑏 Math 01 – Mathematics in the Modern World  APPLICATION OF THE GOLDEN RATIO AND THE GOLDEN RECTANGLE: ARTS Three distinct sets of Golden Rectangles: one set for the head area torso legs Vitruvian Man Math 01 – Mathematics in the Modern World  APPLICATION OF THE THE GOLDEN RECTANGLE: ARTS Mona Lisa by da Vinci It is believed that Leonardo, as a mathematician tried to incorporate mathematics into art. This painting seems to be made purposefully line up with golden rectangle. Math 01 – Mathematics in the Modern World  APPLICATION OF THE GOLDEN RATIO AND THE GOLDEN RECTANGLE: ARCHITECTURE Parthenon The Great Pyramid of Giza in 4700 B.C. Math 01 – Mathematics in the Modern World  GOLDEN ANGLE Definition The golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc. Math 01 – Mathematics in the Modern World  GOLDEN ANGLE circle Let a + b be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that 𝑎+𝑏 𝑎 = = 1.61803 … 𝑎 𝑏 subtended The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.50776° Math 01 – Mathematics in the Modern World  GOLDEN ANGLE IN NATURE Math 01 – Mathematics in the Modern World  IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Helps Organize Patterns and Regularities in the World Numerical Patterns Kepler, fascinated with his search for mathematical patterns in nature (though accidental) was able to discover that the orbital period of a planet (time it takes a planet to go once around the Sun) and the distance of the planet from the sun are related! How did Kepler do it? Let d = distance of any planet from the Sun o = orbital period 𝑑3 Then = constant 𝑜2 This constant, a very elegant number, was the same for all the six planets that precisely existed during his time; Mercury, Venus, Earth, Mars, Jupiter and Saturn! Amazing isn’t it? Math 01 – Mathematics in the Modern World  IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Helps Organize Patterns and Regularities in the World Geometric Patterns Did you know that the rainbow as we normally view as arcs is a collection of circles, one for each color? These circles can be seen from the air! Circles can also be seen in the ripples on a pond, in the human eye and butterflies’ wings. Many viruses assume the shape of an icosahedron – a regular solid that is formed out of twenty equilateral triangles. Math 01 – Mathematics in the Modern World  IMPORTANCE OF MATH IN ONE’S LIFE https://www.quantamagazine.org/the-illuminating-geometry-of-viruses-20170719/ Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Helps Predict the Behavior of Nature and the World Lunar and solar eclipses and the return of comets as predicted by astronomers are achieved by understanding the heavenly bodies’ motion. For example, the positions of the Sun and Moon relative to the Earth can predict tides for many years ahead. In contrast, weather is a phenomenon that is much harder to predict. However, meteorologists can make effective short- term predictions of weather patterns, at most 5 days in advance. Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Helps Control Nature and Occurrences in the World for our Own Ends Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE In the medical field, headlines on epidemics are often seen and heard over the media and publications. Mathematicians, with their knowledge on mathematical modelling are able to control the spread of diseases to the population. Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE https://imsp.cas.uplb.edu.ph/uncategorized/dr-jomar-f-rabajante/ Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Has Numerous Applications in the World Making it Indispensable Flight Computer, electrical scheduling and telephone lines that designing optimizes routines Math 01 – Mathematics in the Modern World IMPORTANCE OF MATH IN ONE’S LIFE Mathematics Has Numerous Applications in the World Making it Indispensable Windmills located along the beach to maximize the storing of energy Math 01 – Mathematics in the Modern World  https://web.facebook.com/photo/?fbid=5179326305412220&set=a.87800563887 7663 Math 01 – Mathematics in the Modern World  References: 1. Module in Mathematics in the Modern World by the Faculty of the Department of Mathematics 2. Mathematics in the Modern World by Rodriguez, et. al. 3. Essential Mathematics for the Modern World by Rizaldi Nocon and Ederlina Nocon Math 01 – Mathematics in the Modern World