Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Final Paper: All About Fibonacci Numbers All About Fibonacci Numbers Miesha Wilson MAT-135 Professor Aldo Maldonado 10/9/16 Final Paper: Fibonacci Numbers 2 Introduction Born from humble beginnings, Leonardo of Pisa used his unique educational background and knowledge to inform and bring forth a better future and new perspective of mathematics to the world. Without the spread of this ancient Indian knowledge, the western world would have little understanding of the mathematical link between science, math, and nature. In addition to the lack of understanding, we as a civilization would have continued to use ancient European mathematical concepts, achieving little to no expansion on theoretical and numerical ideas and theories. Before Fibonacci numbers were introduced performing simple mathematics was a very timely chore. The introduction of a place value and ‘0’ proved to be extremely helpful. The invention of zero meant that they could make numbers infinitely as large or small as they wanted. The Fibonacci sequence even introduced the concept of using a single digit to represent a numerical value. Such as 1,2,3 and 4. Instead of the previous roman numerals like I which represented one and V which represented four. As we seek to grasp an understanding of mathematical patterns and their rules, we can better comprehend the natural patterns found in nature and the blueprint used to create this planet and every living thing in it. Final Paper: Fibonacci Numbers 3 History of Leonardo Bonacci Leonardo Bonacci was born in Italy in 1170 to Alessandra and Guglielmo Bonaccio. While his father worked as a customs officer, Leonardo traveled with him across North Africa. There, the Moors taught Leonardo extensively about mathematical studies, concepts, and ideas. He was learning much more about the vastness of mathematics and the power of numbers than he could have ever hoped to learn in Italy. At the age of 32, he wrote and published the book titled Liber Abaci, which is translated to mean “the book of calculations.” This book consisted of the mathematical knowledge that he acquired by the Moors while stationed in North Africa. After the publication of Liber Abaci, Leonardo was later given the nickname “Fibonacci” due to a mistranslation in the title thus giving birth to his new surname. No matter what he’s called, Fibonacci’s spread of the decimal system to west Europe has greatly influenced our current society and the way we understand the relationship between numbers. Mathematical concepts in Liber Abaci While reading the example that Leonardo gave on the mating rabbits, I realized that there seems to be a structured pattern in almost every aspect of nature as well as science and math. What seems to be a chaotic random occurrence in nature, has a definitively structured pattern. With the example of the mating rabbits, Leonardo dared to ask the question “How fast will rabbits multiply in an ideal circumstance of events?” The ideal scenario would be if no rabbits ever died and every rabbit mated when they reached puberty (1 month). He used a formula to help simplify this phenomenon. The formula was Xn+1=Xn+Xn-1. This formula simplifies the idea that if the original pair of rabbits continues to produce offspring each month and the offspring produce more offspring after they hit the stage of puberty then an Final Paper: Fibonacci Numbers 4 “x” amount of rabbits would be produced. By adding 2 to the first number to get the next number. For instance, with the numeric set “2,4,6,8… “by increasing the previous number by two the next possible number becomes apparent. Fibonacci in real-life occurrences On the website plus math.org, the author gives another example to help the reader better understand Fibonacci numbers in the natural world. “Honeybees provide an example. In a colony of honeybees, there is one special female called the queen.” (Knott, 2013) If only a queen bee can produce a female bee, but the male bee is produced by both the female (queen) bee and a male bee, how many parents does the male bee have? The sequence to help explain this occurrence is 1,2,3,5,8 with the one being the original queen mother and the eight being the three times great grandparents. The most fantastic aspect of Fibonacci numbers is that it occurs in every aspect of nature! “Flowers often have a Fibonacci number of petals; daisies can have 34, 55 or even as many as 89 petals!” (Knott,2013) From flower petals to leaves and seeds, Fibonacci numbers are running rampant. In the given illustration on the website, patterns in the number of seeds growing in the direction left and right in a single flower head follow a sequence. In one of the illustrated flowers, there were 34 seeds going left and 21 going right. The sequence of seeds packed inside one head is the perfect amount so that the flower head isn’t too dense nor very sparse. From the number of seeds present inside of a sunflower to the number of leaves along a flower stem, Fibonacci patterns will always be detectable. Final Paper: Fibonacci Numbers 5 Fibonacci Patterns Explained Leonardo Fibonacci’s most respected contribution to the mathematic world was the spread of Fibonacci numbers. Fibonacci numbers are a sequence or pattern of numbers that follow a single rule. In the example of the reproduction of bees, the male is reproduced with only one parent bee (male) while two parents are needed to produce a female bee. If the number of bees from each generation is summed up it will reflect the infamous Fibonacci sequence 0,1,1,2,3,5,8,13,21,34,35 and so on. Another great example of a Fibonacci sequence is the pattern of spirals in a flower head or pinecones. To count the spirals in both a flower head and pine cone, first start by counting the spirals going in one direction, next count the spirals going in the opposite direction. The number of spirals occurring in any given direction follows a single rule. Rules of the Game No matter the sequence, all Fibonacci numbers have a given rule. To reveal the rule of a Fibonacci sequence, you must first discover the hidden pattern. In the example of the Fibonacci pattern 0,1,1,2,3,5,8,13 there is a “rule” to uncover. For this pattern the rule is xn = xn-1 + xn-2. (Knott,2013) Final Paper: Fibonacci Numbers 6 In this mathematical equation, xn represents the unknown number or the number that will follow the previous numbers. The xn-1 represents the previous number, and xn-2 represents the term before that. To simplify this equation, I’ll use the number 2 in the previously given equation. To get the number “2”, we added, “1” (xn-1; previous number) +”1” (xn-2; number before that). (Knott,2013) Finite and infinite sequence When dealing with finite sequences, it is important to keep in mind that both finite and infinite sequences both have a beginning but only one sequence has a “fixed” ending. This numbering pattern has come to be known as a finite sequence. Because this series of numbers have an ending point, it doesn't continue forever. Infinite sequences are a little trickier. Like a finite series, it has a beginning, but an infinite sequence has no barriers or boundaries; it keeps continuing forever. In a number sequence of all even numbers in a numeric system, there is no stopping point. Therefore, resulting in a countless series of numbers, continuing for eternity. A good way to explain this phenomenon of countless even numerical values is 2,4,6,8,10,12…∞ Examples of infinite sequences are N = (0, 1, 2, 3, ...) and S = (1, 1/2, 1/4, 1/8, ..., 1/2 n , ...). Infinity is represented by the three dots located after the last value in the series. (Rouse,2005) Infinite groups of numbers can go in either direction, both positive and negative. A clear representation of this would be. (…-5,-4-3-2-1 0 1,2.3,4,5…) Golden Ratio According to Wolfram math world, the golden ration is the divine proportion, golden mean, or golden section. It is a number often encountered when taking the ratios of distances in simple Final Paper: Fibonacci Numbers 7 geometric figures such as the pentagon, pentagram, decagon, and dodecahedron”. (Golden Ratio, 2016) The golden ratio represents aesthetically pleasing measurements, or what we know as perfection! German researcher Adolf Zeising found that the relation between a man’s upper and lower body has proportions that can be represented by the fraction 13/8 which equals to 1.625. This numerical value compared to that of a women’s body (1.6) is closer to the golden ratio. “If the numbers of the segments on a human body (arms, legs, fingers and toes) are written as a series of numbers, it would be Fibonacci numbers. Real World Applications Leonardo Fibonacci’s most respected contribution to the mathematic world was the spread of Fibonacci numbers. Fibonacci numbers are a sequence or pattern of numbers that follow a single rule. No matter the Fibonacci sequence every pattern has one. Both finite and infinite patterns have a beginning but only one contains an ending. A good example of a finite sequence is a series of numbers with limitations. These limitations could consist of even numbers ranging from 2 through 20. {2,4,6,8,10,12,14,16,18,20} Infinite numbers are limitless and boundless. An example of this occurrence would be a sequence of all even numbers. {2,4,6,8,10…∞} As explained in the first strand with the example of honey bees, in a colony of honeybees, there is one special female called the queen.” (Knott, 2013) If only a queen bee can produce a female bee, but the male bee is produced by both the female (queen) bee and a male bee, how many parents does the male bee have? The sequence to help explain this occurrence is 1,2,3,5,8 with Final Paper: Fibonacci Numbers 8 the one being the original queen mother and the eight being the three times great grandparents. Real world examples of Fibonacci numbers can be seen in multiple aspects of the natural world. Flowers come in various colors, shades, and even number patterns! The petals on a flower can have a natural occurrence of the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21. “Buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.” (Knott,2016) According to the article titled Fibonacci in Nature by Nikhart Praveen “Phyllotaxis is the study of the ordered position of leaves on a stem.” The leaves on a plant grow in the most efficient way to maximize sun exposure. The number of leaves present on the stem of a plant are an excellent example of Fibonacci numbers. In the previous strand exercise, the occurrence of the Fibonacci sequence in flowers was explored. The sequence of seeds packed inside one head is the perfect amount so that the flower head isn’t too dense nor very sparse. From the number of seeds present inside of a sunflower to the number of leaves along a flower stem, Fibonacci patterns will always be detectable. Fibonacci numbers can even be found in the depths of the deep blue sea. Final Paper: Fibonacci Numbers 9 Sea stars and sand dollars have five points while octopus and squid have eight arms. “The distance between the tips of a starfish’s arms compared to the distance from tip to tip across the entire body is very close to the golden ratio, and the eye, fins and tail of dolphins all fall at points along the dolphin’s body that correspond to the ratio.” (Haelle.2014) “The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.” (Dvorsky,2013) With the prsecence of Fibonacci patterns found in the cosmos, it makes you wonder about the possible existence of a creator. Some mathematicians even use the natural prescence of Fibonacci sequences, to develop the argument that God does indeed exist. Scientist and mathematical geniuses have all wondered if the repetitive occurrence of Fibonacci numbers throughout the world is credible proof of a creator. No one knows why this phenomenal existence occurs on earth, or whether it exists on other planets or in another unidentified life form. As scientist make new discoveries, more unanswered questions will bubble to the service. Maybe one day we will have a scientific answer to explain the presence of the Fibonacci sequence or maybe some questions will never be answered by science alone. As I dove into Fibonacci numbers, sequences, and the golden ratios, I have learned that math isn’t just about solving equations, plotting points on a graph or finding the value for ‘x’, Mathematics has Final Paper: Fibonacci Numbers 10 a place in numerous aspects of life that I personally find interesting, like science, art and music. The Fibonacci sequence has yet to unlock all of the earth’s mysteries or answer all of humanity’s most challenging questions about the universe, space and time. I believe that one day, we will be able to uncover these mysteries. In the future I trust that as people become more aware of the connections and complexities of this universe, we as humans will gain a higher level of respect for the planet and solar system we reside in. I look forward to diving deeper into the pool of knowledge that is the heart of mathematics. Final Paper: Fibonacci Numbers 11 citations Dr R Knott: fibandphi (AT) ronknott.com. (n.d.). Fibonacci Numbers and Nature. Retrieved October 01, 2016, from http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html Dividing, B. (n.d.). Fibonacci in Nature. Retrieved October 01, 2016, from http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm The life and numbers of Fibonacci. (n.d.). Retrieved September 25, 2016, from https://plus.maths.org/content/life-and-numbers-fibonacci Golden Ratio. (n.d.). Retrieved September 25, 2016, from https://www.mathsisfun.com/numbers/golden-ratio.html S. (2011). Finite Sequence and Infinite Sequence - Types of Sequences & Series. Retrieved September 25, 2016, from https://www.youtube.com/watch?v=LjF8dGAOKnI A. (2015). What is Golden Ratio - easy explanation. Retrieved September 25, 2016, from https://www.youtube.com/watch?v=wdk37T8TltM [IMA Videos]. (2011, January 19). Finite sequence and infinite sequence-types of sequences and series. [video file] Retrieved from https://www.youtube.com/watch?v=LjF8dGAOKnI [Assignment Expert. (2015, March 26). What is Golden Ratio? [video file]. Retrieved from https://www.youtube.com/watch?v=wdk37T8TltM The life and numbers of Fibonacci. (n.d.). Retrieved September 11, 2016, from https://plus.maths.org/content/life-and-numbers-fibonacci Final Paper: Fibonacci Numbers Space and fibonacci numbers - Google Search. (n.d.). Retrieved October 10, 2016, from https://www.google.com/search?q=space and fibonacci numbers 12