Chapter 1-Nature of Mathematics and Fibonacci Sequence PDF

Mathematics in the Modern World Weeks 1 – 2 The Nature of Mathematics The Nature of Mathematics Stewart’s Nature’s Number Form Measures Patterns Mathematics is a Tool Study of Patterns art...

Mathematics in the Modern World Weeks 1 – 2 The Nature of Mathematics The Nature of Mathematics Stewart’s Nature’s Number Form Measures Patterns Mathematics is a Tool Study of Patterns art language Set of Process of problem thinking solving tools Mathematics In Our World Different Kinds of Pattern Patterns of Visuals Visual patterns are often unpredictable, never quite repeatable, and often contain fractals. These patterns are can be seen from the seeds and pinecones to the branches and leaves. They are also visible in self-similar replication of trees, ferns, and plants throughout nature. Patterns of Flow The flow of liquids provides an inexhaustible supply of nature’s patterns. Patterns of flow are usually found in the water, stone, and even in the growth of trees. There is also a flow pattern present in meandering rivers with the repetition of undulating lines. Different Kinds of Pattern Patterns of Movement In the human walk, the feet strike the ground in a regular rhythm: the left-right-left-right-left rhythm. When a horse, a four- legged creature walks, there is more of a complex but equally rhythmic pattern. This prevalence of pattern in locomotion extends to the scuttling of insects, the flights of birds, the pulsations of jellyfish, and also the wave-like movements of fish, worms, and snakes. Different Kinds of Pattern Patterns of Rhythm Rhythm is conceivably the most basic pattern in nature. Our hearts and lungs follow a regular repeated pattern of sounds or movement whose timing is adapted to our body’s needs. Many of nature’s rhythms are similar to a heartbeat, while others are like breathing. The beating of the heart, as well as breathing, have a default pattern. Different Kinds of Pattern Patterns of Texture A texture is a quality of a certain object that we sense through touch. It exists as a literal surface that we can feel, see, and imagine. Textures are of many kinds. It can be bristly, and rough, but it can also be smooth, cold, and hard. Geometric Patterns A geometric pattern is a kind of pattern which consists of a series of shapes that are typically repeated. These are regularities in the natural world that are repeated in a predictable manner. Geometrical patterns are usually visible on cacti and succulents. Patterns Found in Nature Waves and Dunes Patterns Found in Nature Spots and Stripes Patterns Found in Nature Spiral Symmetries Reflection symmetry sometimes called line symmetry or mirror symmetry, captures symmetries when the left half of a pattern is the same as the right half. Symmetries Rotations also known as rotational symmetry, captures symmetries when it still looks the same after some rotation (of less than one full turn). The degree of rotational symmetry of an object is recognized by the number of distinct orientations in which it looks the same for each rotation. Symmetries Translations This is another type of symmetry. Translational symmetry exists in patterns that we see in nature and in man-made objects. Translations acquire symmetries when units are repeated and turn out having identical figures, like the bees’ honeycomb with hexagonal tiles. Sequence Sequence refers to an ordered list of numbers called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. There are different types of sequence and the most common are the arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. Sequences Arithmetic Sequence Arithmetic sequence. It is a sequence of numbers that follows a definite pattern. To determine if the series of numbers follow an arithmetic sequence, check the difference between two consecutive terms. If common difference is observed, then definitely arithmetic sequence governed the pattern. Sequences Geometric Sequence It is a sequence in which every term (except the first term) is multiplied by a constant number to get its next term. i.e., To get the next term in the geometric sequence, we have to multiply with a fixed term (known as the common ratio), and to find the preceding term in the sequence, we just have to divide the term by the same common ratio. Sequences Harmonic Sequence In the sequence, the reciprocal of the terms behaved in a manner like arithmetic sequence. Sequences Fibonacci Sequence The Fibonacci numbers are a series of numbers that often occur in nature. This number sequence was developed in the middle ages, and it was named after Leonardo Pisano Bigollo, a famous Italian mathematician who also happened to discover Fibonacci. Fibonacci is the short term for the latin filius bonacci, which means “the son of Bonacci”.​