1. MMW Mathematics In Nature .pdf

MATHEMATICS IN OUR WORLD ❑ The Fibonacci Sequence ❑ Patterns and Numbers in Nature and the World ❑ Mathematics for Humankind ❑ Ethnomathematics LEARNING OUTCOMES: ✓ Explain the nature of Mathematics ✓ Discuss how Mathematics exhibited in nature ✓ Apply the principles of Mathematics to resolv...

MATHEMATICS IN OUR WORLD ❑ The Fibonacci Sequence ❑ Patterns and Numbers in Nature and the World ❑ Mathematics for Humankind ❑ Ethnomathematics LEARNING OUTCOMES: ✓ Explain the nature of Mathematics ✓ Discuss how Mathematics exhibited in nature ✓ Apply the principles of Mathematics to resolve issues that pertain to human activities, natural occurrences and socio-cultural practices.  “[The universe] is written in the language of mathematics, and its characteristics are triangles, circles, and other geometric figures.”  Fibonacci Sequence was invented by the Italian Leonardo Pisano Bigollo (1180-1250), known by several names: “Leonardo of Pisa (Piscano) and Fibonacci (Son of Bonacci). A page of Fibonacci’s Liber Abaci from the Biblioteca Nazionale di Firenze, showing the sequence (in the box on the right). 1 At the start, there is just one pair. 1 After the first month, the initial pair mates, but have no young. 2 After the second month, the initial pair gives birth to a a pair of babies. 3 After the third month, the initial pair give birth to a second pair, and their first-born mate but have not yet given birth to any young. 5 After the fourth month, the initial pair give birth to another pair and their first-born pair also produces a pair of their own. After the fifth month, the initial pair give birth to another pair. Their first pair produces another pair, and the second-born pair produce a pair of their own. The process continues... 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… PUZZLE FORM A RECTANGLE USING ALL OF THE SQUARES 1 3 2 1 8 5 FIBONACCI SEQUENCE The first two numbers in the Fibonacci sequences are 1 and 1, and each subsequent number is the sum of the previous two. The sequence Fn is of Fibonacci numbers is defined by the recurrence relation Fn = Fn-1 + Fn-2 With seed values F1=1 and F2=1.  DNA consist of two spirals and molecules. Between two spirals’ length there are 34 angstroms, and its width is 21 angstroms (21 and 34 in Fibonacci series). The Great Wave off Kanagawa FRACTAL Applying the operation of shrinking and moving applied many times Involved symmetry of magnification (called dilation). It is a shape that you could zoom in on a part of it an infinite number of times and it would still look the same. FRACTAL PROPERTIES Self-similarity Appear the same under magnification Fractional Dimension Are not limited to integer dimensions, it can have fractional dimension. Formation by Iteration Form by a repeating process. KOCH SNOWFLAKES SIERPINSKI TRIANGLE CANTOR SET Gaston Maurice Julia Benoit Mandelbrot In the Mandelbrot set, points remaining finite through all iterations are in white; values diverging to infinity are darker. This partial view of the Mandelbrot set, possibly the world's most famous fractal, shows step four of a zoom sequence: The central endpoint of the "seahorse tail" is also a Misiurewicz point Mathematics of Patterns Isometries, Symmetries, and Patterns Transformation Process which shift points of the plane to possibly new locations on the plane. Translation (or a slide) moves a shape in given direction by sliding it up or down, sideways, or diagonally. Reflection (or a flip) has a point about which the rotation is made and an angle that’s says how far to rotate. Rotation (or a turn) can be thought of a getting a mirror image. It has a line of reflection where the distance between the image and the mirror line is the same as that between the original figure and the mirror line. Dilation a transformation which changes the size of an object. Rigid Transformations leave the dimensions of the object and its image unchanged. Non-Rigid Transformations can change the size and shape or both size and shape of the preimage. An Isometry of the plane is a mapping that preserves distance (and therefore shape): 𝑑 𝑓 𝑥 , 𝑓 𝑦 = 𝑑 𝑥, 𝑦 “iso” : Greek for “the same” “metria”: Greek for “measure” It is possible to combine isometries to produce other isometries. Reflect then Translate… Translate then Reflect… Glide Reflection A figure hassymmetry if there is a non-trivial transformation that maps the figure onto itself. Symmetries of a Square Adesign is a figure with at least one non-trivial symmetry. A pattern is a design that has a translation symmetry. A plane pattern has symmetry if an isometry of the plane that preserves it. ARosette Pattern consist of taking a motif or an element and rotating and / or reflecting that element. Mandalas are Buddhist devotional images often deemed a diagram or symbol of an ideal universe. A symbol of the universe in its ideal form, and its creation signifies the transformation of a universe of suffering into one of joy. Used as an aid to meditation, helping the meditator to envision how to achieve the perfect self. A cyclic has n fold rotational symmetry and no “reflectional symmetry”. A dihedral has n fold rotational symmetry and “reflectional symmetry”. A Frieze Pattern is an indefinitely long strip imprinted with a design given by a repeating pattern motif. HOP ▪ Translation- the pattern is unchanged if you slide it along STEP ▪ Translation. ▪ Glide reflection- the pattern is unchanged if you slide it along and reflect it in a horizontal line. SIDLE ▪ Translation. ▪ Vertical reflection- the pattern is unchanged if you reflect it in a vertical line. SPINNING HOP Translation. Rotation- the pattern is unchanged if your spin it by a half turn. SPINNING SIDLE ▪ Translation. ▪ Vertical reflection. ▪ Rotation. JUMP ▪ Translation. ▪ Horizontal Reflection. SPINNING JUMP ▪ Translation. ▪ Horizontal and Vertical Reflection. ▪ Rotation. FULL CREDITS TO THE OWNER OF THE IMAGES/ TEXT USED FOR ACADEMIC PURPOSES.