GEC 4 (MIDTERM)

• Patterns - patterns in nature is not only for us to appreciate, but it is a source of knowledge that gives us vital clues in understanding the world around.
• Patterns In Nature Fractals Symmetry Spirals Stripes Cracks Tessellation Bubbles and Foam Waves/Dunes
• Fractals - Derived from the latin word "Fractus" means (Fragmented or broken) Coined by Benoit B. Mandelbrot (American-French-Polish Mathematician).
• Fractals - are any various extremely irregular curves or shapes for which any suitable chosen part is similar in shape to a large smaller part when magnified or reduced to the same size. It possess the property of self similarity where component parts resembles the whole. Fractals can be seen in some plants, trees, leaves and others.
• Symmetry - a sense of harmonious and beautiful proportion of balance or an object is in variant to any of various transformations such as reflections, rotation or scaling.
• 2 Types Of Symmetry 1. Bilateral Symmetry - formed when organism is divided into two and the left side approximately mirrors the right side along the midline. 2. Radial Symmetry or Rotational Symmetry - a type of symmetry around a fixed point known as the center and it can be classified as either cyclic or dihedral.
• Spirals - a spiral (logarithmic spiral or growth spiral) is a plane curve that winds around a central point while moving further from it. It is a self-similar curve which often appears in nature.
• Spiral - it was first described by Descartes and investigated by Jacob Bernoulli, who called it "Spira Mirabilis" which means the marvelous spiral.
• Stripes - stripes consists of a line or long narrow section differing in color and texture from parts adjoining. They have functions within trees the chances that the offspring of the pattern animal will survive to reproduce. The function of this animal pattern is for camouflage.
• Cracks - are linear openings that form in materials relieve stress. The pattern of cracks indicates whether the material is elastic or not. When an elastic materials stretches or shrinks uniformly, it eventually reaches its breaking strength and it fails suddenly in all directions, creating cracks with 120 degrees joints, so three cracks meet at a node. Conversely, when an inelastic material fails, straight cracks form to relieve the stress.
• Tessellations - are patterns formed by repeating tiles all over a flat surface. While common in art and design, exactly repeating tiling are less easy to find in living things. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples.
• Bubbles and Foam - A bublles is a tiny, round ball of air or gas inside a liquid. When two bubbles join together, they form a more complex shape where the other surfaces are spherical. These surfaces are joined together by a third spherical surface as the smaller bubble bulges slightly into the larger one. A mass of bubbles is called a foam.
• Waves/Dunes - waves are distubances that carry energy as they move. Ripples are created when waves in water in water or wind pass over sand. On the other hand, dunes are formed when wind blows over a large bodies of sand. We usually see ripples when a stone is thrown on a still lake, while we see huge waves in the seashore during storm surge.
• Fibonacci Sequence - In the series of numbers: 0,1,1,2,3,5,8,13,21,34,55... The next number is found by adding up the two numbers before it.
• Fibonacci numbers were discovered by Leonardo Pisano Bigollo or Fibonacci which translates to "filius bonacci" meaning "son of bonaccio".
• Fibonacci, in his book "Liber Abaci", investigated how fast rabbits could multiply in an ideal surrounding. 
• Golden Ratio - also known as the golden number, golden proportion, or the divine proportion, is a ratio between two numbers that equals approximately to 1.618. Usually written as the greek letter "Phi" symbol.
• Nocon & Nocon (2016) as cited in Piramide (2018) defined Mathematics as a set of problem-solving tools, a language, a process of thinking, a study of patterns and an art. Mathematics involves the study of numbers and arithmetic operations.
• Mathematics - is a science that involves logical reasoning, drawing conclusions from assumed premises and strategic reasoning based on accepted rules, laws and possibilities.
• Application of Mathematics In The World

  1. Forensic Science
  2. Medicine
  3. Fluid Dynamics
  4. Information Technology
  5. Cryptography
  6. Archaeology
  7. Social Science
  8. Political Science
  9. Music and Arts
• Forensic Science (Crime Investigation and Evidence Examination) - in this field, mathematics, specifically the differential and integral calculus is applied to clarify the blurred image to clear image.
• Medicine - in this field, much of a function of protein is determined by its shape and how the pieces move. Many drugs are designed to change the shape or motions of a protein by modeling using geometry and related areas.
• Fluid Dynamics (Describes the flow of fluids such as liquid and gases) - in this field, engineers use numerical analysis in phenomena involving heat, electricity and magnetism, relativistic mechanics, quantum mechanics and other theoretical constructs.
• Information Technology - in this field, modern computers are invented through the help of mathematics. An important area of application of mathematics is in the development of formal mathematical theories related to the development of computer science. Computer science development includes logic, relations, functions, basic set theory, counting techniques, graph theory, combinatorics, discrete probability, recurrence relations, and number theory, and computer-oriented numerical analysis.
• Cryptography (Hiding or Coding Information) - it is a combination of both mathematics and computer science and it is affiliated closely with information theory, computer security and engineering. It is used in applications present in technology advanced societies; examples include the security of ATM cards, computer passwords and electronic commerce.
• Archaeology (Material Remains) - In this field, archaeologist uses a variety of mathematical and statistical techniques to present the data from archaeological surveys and try to find patterns to shed light on past human behavior and in carbon dating artifacts.
• Social Science (Economics, Sociology, Psychology, Linguistics, etc.) - In this field, calculus, probability, game theory, and network theory were used in economics, sociology, psychology, and linguistics.
• Political Science - In this area, political analyst study past election results to see changes in voting patterns and the past influence of various factors on voting behavior, on switching of both among political parties and mathematical models for conflict resolution using game theory and statistics
• Music and Arts - Here, the rhythm that we find in all music notes is the result of innumerable permutations and combinations. Music theories understand musical structure and communicate new ways of hearing music by applying set theory, abstract algebra and number theory