Golden Patterns

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Golden Patterns
It is a common misconception to believe that mathematics is only found in books, written on paper, expressed as variables, functions, shapes and alphanumerical characters or as a real world application of a mathematical problem found in real life now worded and written into text. As a student progresses through grade school a student can’t help but feel no connection between the world’s nature and mathematical work, this concept however could not be further from the truth. As the student journeys through college and higher division mathematics, the student finds that mathematical patterns and sequences can be found naturally and without the need of human influence and that furthermore, patterns influence nature heavily. Patterns have always been of interest in mathematics, after all it can be said that mathematics is the science of patterns. In nature there are many simple yet elegant patterns present. Let’s consider the patterns composed by the scales on a pineapple, or the ones found on an acorn, is there a possible way to model these patterns mathematically? We can most certainly answer this question with a “yes” given that patterns have always been of interest as stated above. Furthermore, the amount of research and time that has gone into finding mathematical representations for these patterns is phenomenal and I will be using just a few of these works to give some insight on some of the most important patterns in mathematics, that is, the Fibonacci sequence and the golden ratio. A Mathematical Medley: Gleanings from the Globe and Beyond by John A Adams, a very simply but efficiently worded book on patterns such as the golden ratio, will help us understand the fundamentals of what the golden ratio is along with its golden geometrical shapes which include the golden triangle and the golden rectangle. The Fibonacci Numbers Exposed by Dan Kalman will aid us in further definitions and understanding some interesting facts about the Fibonacci numbers. In Mathematics in Nature, Modeling Patterns in the Natural World, again another great book by John A. Adams, we will learn about some very magnificent patterns that happen before our very eyes. Let us begin then, with patterns.

Patterns in nature are visible in form and occur naturally in our world; these patterns can be viewed and modeled mathematically. In order to obtain a proper understanding for what will be discussed, let us begin by defining what a pattern is and what patterns mean to mathematicians. A common definition for the term pattern is “a decorative design, a set of lines, shapes, or colors that are repeated regularly.” (MacMillan Dictionary). To the general public, a pattern may be something that has a repetitive behavior or nature with some order. Decorative patterns can be found in objects such as flowers, pinecones, pineapple scales and other natural objects like light travel and wave motion, the two most fundamental and widespread phenomena that occur in nature. “Both may occur almost anywhere given the right circumstances, and both may be described in mathematical terms at varying levels of complexity.” (Adam et al, 2). It is the scattering of these light particles, and of course some aerosols that give us our beautiful sunsets and sunrises. Another observable example involving light particles are rainbows; a rainbow is appealing to the eye, yet what might be even more eccentric and beautiful might be the precise mathematics necessary for such a phenomenon to even occur. “The rainbow is formed by sunlight scattered in preferential directions by near-spherical raindrops: scattering in this context means refraction and reflection.” A sum or series can be created to describe the way light bounces inside a water droplet or at the angles that the light comes out as it exits the droplet. Simply by experiencing the manifestation of a rainbow we have witnessed elaborate mathematics that involve space, light motion, geometrical shapes,...
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