# The concepts of Fibonacci Sequence Project

Please complete 4 separate assignments (strand 2, Strand 3, Reference, and Final document). Strand 1 is already completed and attached here, please piggy back off of the work that has already been done. Again these need to be separate assignments with the final project being everything all put together.

__Strand #1__: ALREADY COMPLETED AND ATTACHED

__Strand #2__: Mathematics Outline contains at least three main points for the mathematics strand Outline does not contain at least three main points for the mathematics strand

__Strand #3__: Real-World Applications Outline contains at least three main points for the real-world applications strand Outline does not contain at least three main points for the real-world applications strand

__References for Outline:__ Provides at least three credible resources to support the selected topic Does not provide at least three credible resources to support the selected topic

__Final Project Document__

# The concepts of Fibonacci Sequence Project

Fibonacci Sequence: Milestone 2

Chris Fronapfel

Southern New Hampshire University

The concepts of Fibonacci Sequence are applicable in every aspect of science, nature, and art. Fibonacci Sequence is a concept that involves a series of numeric in which the next series f numbers is found by summing two numbers before it. The sequence begins by either 0 or 1, and the consequence follows 0, 1, 2…..10…….24…..50. The first digit is found by adding ‘0+1′ and 2 ‘1+1,’ and the sequence continues ((Ghose, 2018). The historical importance of this concept can be discussed into various perspectives ranging from its time of discovery and the people involved in its development.

Different academic articles and journals have claimed that Leonardo Bonacci discovered the concept. Born in Pisa, Italy, in the year of 1175, Leonardo Fibonacci was recognized by his contribution in his Fibonacci. He was a son of tax police who dealt with taxation issue within Pisa and Bugai, an Arab Port City located in the Northern Horn of Africa. Due to the influence of his father, who had an interest in understanding numbers, he emphasized on his son to be schooled in mathematic courses in Bugai. It was out of this that during his time that Leonardo showed interest in understanding the Hindu-Arabic numeric system that is the mother of the concept of his work, Fibonacci sequence. In the Hindu-Arabic order, their texts could predate Leonardo’s concept by centuries. As early as 1202, Leonard had published a book that was explaining and teaching about Hindu-Arabic system in different institutions. His book was titled ‘Liber Abaci,’ and his book was considered as the onset of the concepts of calculations in learning (Ghose, 2018). His work provided a significant solution to the ‘rabbit population puzzle’ in his work titled ‘Liber Laci.’ Leonardo considered ‘a hypothetical situation where there is a pair of rabbits that mate at the end of each month, and by the end of the second month, the female produces another pair. In this situation, the rabbits never die, they mate exactly after one month, and the females always produce a pair where one is male, and one is female’ (Fibonacci, 2015, September 10).

The concept of Fibonacci Sequence has had a significant mathematical interest despite the late discoveries of most of its implications and relationship. For quite several years, different mathematic scholars have shown interest and have so far discovered the sequence throughout nature. It was established that ‘in the spirals of pineapples, pinecones and the seeds of sunflower heads. In the 1750s, Robert Simson noted that the ratio of each term in the sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1:1.6180339887’ (Matson, 2010). This is what is referred to as ‘Golden Ratio. Within the concept of the Fibonacci Sequence, the Golden Ratio has already created a perfect spiral number that is placed a squarely measured in all dimensions. In 1877, a French mathematician Eduardo Lucas named the ‘rabbit problem’ the Fibonacci Sequence.

Thus, throughout its discovery, the concept of Fibonacci Sequence has proved to be one of the most significant ancient discovery that led to the development of mathematics through the help of the work of Leonardo Fibonacci. Its importance can be felt nature, science, and art. The concept was first introduced to the world through ‘Leonardo’s book “Liber Abaci” when Leonardo questioned a situation with a pair of mating rabbits and how many pairs of rabbits would be after one year’ (Fibonacci, 2015, September 10).

References

Fibonacci. (2015, September 10). Retrieved November 17, 2018, from https://www.famousscientists.org/fibonacci-leonardo-of-pisa/

Ghose, T. (2018, October 24). What Is the Fibonacci Sequence? Retrieved November 17, 2018, from https://www.livescience.com/37470-fibonacci-sequence.html

Matson, L. (2010). MEDIEVAL MATHEMATICS – FIBONACCI. Retrieved November 17, 2018, from https://www.storyofmathematics.com/medieval_fibonacci.html

# The concepts of Fibonacci Sequence Project

MAT 135 Milestone 1 Final Project Topic Selection and Outline Guidelines and Rubric Overview: The final project for this course is the creation of a comprehensive final paper that includes the following main components: Introduction Strand 1: Historical Significance Strand 2: Mathematics Strand 3: Real-World Applications Conclusion Prompt: In Module Two, you will submit your chosen topic to the instructor for approval. You will research three areas of this topic and submit an outline for each “strand” for grading and feedback. Your submission should include potential reference sources. Take note that this milestone rubric requires three resources for your outline at this point in time so that you can receive valuable feedback from your instructor. However, additional research and resources are expected as you complete your paper. The topic may come from the list provided below or it can be self-designed, but it still must be approved prior to writing the outline. The final paper will not be accepted without topic approval. Final Project Topic List: Choose one Logic/Proof Irrational Numbers Euclidean Geometry Non-Euclidean Geometry Penrose Tiles Mobius Bands Klein Bottles Self-Selected Topic* Hilbert’s Hotel Power Set Theory Koch Curve Sierpinski Triangle Pythagorean Theorem Pythagoras Musical Scale Prime Numbers Fractals Linear Algebra Fibonacci Sequence Number Theory Mandelbrot Set Chaos Theory Infinity Pascal’s Triangle Knot Theory Graph Theory Continuum Hypothesis Euler’s Königsberg Bridge Problem *NOTE: Instructors will only approve topics for which they know sufficient research and information exists for the student to complete the final paper. In addition to the points you receive based on the grading rubric on the following page, you will also receive formative feedback from the instructor. Apply this feedback to the strands as you complete them. Rubric Guidelines for Submission: Your topic selection and outline must be submitted as a 1- to 2-page Microsoft Word document with double spacing, 12-point Times New Roman font, one-inch margins, and at least three sources cited in APA format. Your total points will be based on the rubric below. Your instructor will provide you full points on each row of the requirements listed below that you submit. You will receive instructor feedback in the comment column of the integrated rubric on areas that need work and feedback on areas where you are exceeding standards. Critical Elements Topic Selection Proficient (100%) Selects a topic from the provided list or a viable self-selected topic Strand #1: Historical Significance Outline contains at least three main points for the historical significance strand Strand #2: Mathematics Outline contains at least three main points for the mathematics strand Strand #3: Real-World Applications Outline contains at least three main points for the real-world applications strand References for Outline Provides at least three credible resources to support the selected topic Not Evident (0%) Does not select a topic from the provided list or a viable self-selected topic Outline does not contain at least three main points for the historical significance strand Outline does not contain at least three main points for the mathematics strand Outline does not contain at least three main points for the real-world applications strand Does not provide at least three credible resources to support the selected topic Total Value 20 20 20 20 20 100% Running Head: FIBONACCI SEQUENCE Fibonacci Sequence: Milestone 2 Chris Fronapfel Southern New Hampshire University 1 FIBONACCI SEQUENCE 2 The concepts of Fibonacci Sequence are applicable in every aspect of science, nature, and art. Fibonacci Sequence is a concept that involves a series of numeric in which the next series f numbers is found by summing two numbers before it. The sequence begins by either 0 or 1, and the consequence follows 0, 1, 2…..10…….24…..50. The first digit is found by adding ‘0+1′ and 2 ‘1+1,’ and the sequence continues ((Ghose, 2018). The historical importance of this concept can be discussed into various perspectives ranging from its time of discovery and the people involved in its development. Different academic articles and journals have claimed that Leonardo Bonacci discovered the concept. Born in Pisa, Italy, in the year of 1175, Leonardo Fibonacci was recognized by his contribution in his Fibonacci. He was a son of tax police who dealt with taxation issue within Pisa and Bugai, an Arab Port City located in the Northern Horn of Africa. Due to the influence of his father, who had an interest in understanding numbers, he emphasized on his son to be schooled in mathematic courses in Bugai. It was out of this that during his time that Leonardo showed interest in understanding the Hindu-Arabic numeric system that is the mother of the concept of his work, Fibonacci sequence. In the Hindu-Arabic order, their texts could predate Leonardo’s concept by centuries. As early as 1202, Leonard had published a book that was explaining and teaching about Hindu-Arabic system in different institutions. His book was titled ‘Liber Abaci,’ and his book was considered as the onset of the concepts of calculations in learning (Ghose, 2018). His work provided a significant solution to the ‘rabbit population puzzle’ in his work titled ‘Liber Laci.’ Leonardo considered ‘a hypothetical situation where there is a pair of rabbits that mate at the end of each month, and by the end of the second month, the female produces another pair. In this situation, the rabbits never die, they mate exactly after one FIBONACCI SEQUENCE 3 month, and the females always produce a pair where one is male, and one is female’ (Fibonacci, 2015, September 10). The concept of Fibonacci Sequence has had a significant mathematical interest despite the late discoveries of most of its implications and relationship. For quite several years, different mathematic scholars have shown interest and have so far discovered the sequence throughout nature. It was established that ‘in the spirals of pineapples, pinecones and the seeds of sunflower heads. In the 1750s, Robert Simson noted that the ratio of each term in the sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1:1.6180339887’ (Matson, 2010). This is what is referred to as ‘Golden Ratio. Within the concept of the Fibonacci Sequence, the Golden Ratio has already created a perfect spiral number that is placed a squarely measured in all dimensions. In 1877, a French mathematician Eduardo Lucas named the ‘rabbit problem’ the Fibonacci Sequence. Thus, throughout its discovery, the concept of Fibonacci Sequence has proved to be one of the most significant ancient discovery that led to the development of mathematics through the help of the work of Leonardo Fibonacci. Its importance can be felt nature, science, and art. The concept was first introduced to the world through ‘Leonardo’s book “Liber Abaci” when Leonardo questioned a situation with a pair of mating rabbits and how many pairs of rabbits would be after one year’ (Fibonacci, 2015, September 10). FIBONACCI SEQUENCE 4 References Fibonacci. (2015, September 10). Retrieved November 17, 2018, from https://www.famousscientists.org/fibonacci-leonardo-of-pisa/ Ghose, T. (2018, October 24). What Is the Fibonacci Sequence? Retrieved November 17, 2018, from https://www.livescience.com/37470-fibonacci-sequence.html Matson, L. (2010). MEDIEVAL MATHEMATICS – FIBONACCI. Retrieved November 17, 2018, from https://www.storyofmathematics.com/medieval_fibonacci.html MAT 135: Final Project Guidelines and Grading Guide Overview The final project for this course is the creation of a comprehensive final paper that includes the following main components: Introduction Strand 1: Historical Significance Strand 2: Mathematics Strand 3: Real-World Applications Conclusion The project is divided into five milestones, which will be submitted at various points throughout the course to scaffold learning and ensure quality final submissions. These milestones will be submitted in Modules Two, Three, Four, Five, and Seven. Outcomes To successfully complete this project, you will be expected to apply what you have learned in this course and should include several of the following course outcomes: 1. Communicate and problem solve in mathematics, without the constraints of formal mathematical notation 2. Demonstrate knowledge in fundamental areas of higher mathematics, including number theory, infinity, geometry, topology, fractals, and other topics 3. Experiment with viewing the world from a mathematical perspective Main Elements Students will select and research one of the following topics (students may also propose their own topic) and submit this for instructor approval. MAT 135 Final Project Topic List Logic Mathematical Puzzles Irrational Numbers Prime Numbers Number Theory Fibonacci Sequence Numbers in Nature Euclidean Geometry Non-Euclidean Geometry Tessellations Mathematical Patterns Symmetry Infinity Fractals Knot Theory Graph Theory Linear Algebra Chaos Theory Self-Selected Topic NOTE: Instructors will approve topics for which they know sufficient research and information exists for the student to complete the final paper. Students will research three areas of this topic and submit each “strand” for grading and feedback. The strands are as follows: 1. Strand 1: The Historical Significance: Students will research the historical development of the topic from inception through modern-day usages. Students may select the most significant developments and contributors to the topic. 2. Strand 2: Mathematics: Students will research and explain the mathematics of the topic chosen. This may include the most significant discovery, theory, or usage. Students will fully explain the mathematics of the topic. 3. Strand 3: Real-World Applications: Students will research and make connections between the topic and the usages in the real world. Students may make connections to other fields where appropriate. Final Paper: Students will write an introduction to this paper, briefly outlining the topic and explaining the three strands. Students will include all three strands as subsections to the final paper, taking into account specific instructor feedback and suggestions for improvement. Given the nature of specific topics, students may need to consolidate repetitive sections to make a cohesive paper. Students will write a conclusion paragraph, which will be a reflective analysis of what the student gained from researching this topic. The final paper will include a cover sheet and reference page, using proper APA formatting. Total paper length: no less than 10 pages (exclusive of cover page and references). The final paper should be submitted as one document with the following components: Cover Sheet Introduction Strand 1: Historical Significance Strand 2: Mathematics Strand 3: Real-World Applications Conclusion References Format Milestone One: Topic Selection & Outline In Task 2-2, you will submit your chosen topic to the instructor for approval and an outline of the three strands of the paper, including potential references. The topic may come from the list provided above or it can be self-designed. This milestone will be graded separately using the Final Project Topic Selection and Outline Rubric, and feedback will be provided for revisions to the final paper. Milestone Two: Strand 1—Historical Significance In Task 3-2, you will submit the paper for Strand 1: Historical Significance. This strand should be between 2–3 pages and fully explain the history of the topic. Proper APA citations and references are expected. This milestone will be graded separately using the Strand Paper Rubric, and feedback will be provided for revisions to the final paper. Milestone Three: Strand 2—Mathematics In Task 4-2, you will submit the paper for Strand 2: Mathematics. This strand should be between 2–3 pages and fully explain the mathematics of the topic chosen. This may include the most significant theory, theorem, or finding. Proper APA citations and references are expected. This milestone will be graded separately using the Strand Paper Rubric, and feedback will be provided for revisions to the final paper. Milestone Four: Strand 3—Real-World Applications In Task 5-2, you will submit the paper for Strand 3: Real-World Applications. This strand should be between 2–3 pages and fully explain the real-world applications of the topic chosen. This may include common usages of the topic, applications, and/or connections to other fields. Proper APA citations and references are expected. This milestone will be graded separately using the Strand Paper Rubric, and feedback will be provided for revisions to the final paper. Milestone Five: Final Paper In Task 7-2, you will submit the final paper. Include an introduction outlining the topic and what the reader can expect within each of the three strands. This paper will then consist of the three strands of research that has been conducted over the course of the term. Finally, write a conclusion, which will be a reflective analysis of what you learned from the research you conducted. The final paper should be cohesive and polished and take into consideration feedback provided by the instructor throughout the term. The paper should be no less than 10 pages, excluding the cover sheet and references page. Proper APA citations are expected. This milestone will be graded using the Final Project Rubric. Deliverable Milestones Milestone Deliverables Module Due Grading Paper will be graded (50 points total) with Final Project Topic Selection and Outline Rubric —feedback will be provided Paper will be graded (100 points total) with Strand Paper Rubric—feedback will be provided Paper will be graded (100 points total) with Strand Paper Rubric—feedback will be provided Paper will be graded (100 points total) with Strand Paper Rubric—feedback will be provided Graded separately; Final Project Rubric (200 points total) 1 Topic Approval & Paper Outline Two 2 Strand 1—Historical Significance Three 3 Strand 2—Mathematics Four 4 Strand 3—Real-World Applications Five 5 Final Product: Final Paper Seven Final Project Rubric Requirements of submission: Written components of projects must follow these formatting guidelines when applicable: double spacing, 12-point Times New Roman font, one-inch margins, and discipline-appropriate citations. Final paper should be no less than 10 pages, excluding coversheet and references page; proper APA formatting is expected. Critical Elements Communication Exemplary (100%) Demonstrates comprehensive communication of mathematical issues and ideas using accurate mathematical language and proper terminology Demonstrates knowledge of multiple mathematical issues through extensive collection and in-depth analysis of evidence to make informed conclusions All of the mathematical concepts are correctly applied and integrated with supporting evidence in a real-world context Proficient (85%) Demonstrates moderate communication of mathematical issues and ideas using accurate mathematical language and proper terminology Demonstrates knowledge of some mathematical issues through collection and in-depth analysis of evidence to make informed conclusions Needs Improvement (55%) Demonstrates minimal communication of mathematical issues and ideas using accurate mathematical language and proper terminology Demonstrates minimal knowledge of mathematical issues through collection and analysis of evidence to make informed conclusions Not Evident (0%) Does not demonstrate communication of mathematical issues and ideas using accurate mathematical language and proper terminology Does not demonstrate knowledge of mathematical issues through collection and analysis of evidence and does not make informed conclusions Value 20 Most of the mathematical concepts are correctly applied and integrated with supporting evidence in a real-world context Some of the mathematical concepts are correctly applied and integrated with supporting evidence in a real-world context Does not correctly apply or integrate mathematical concepts 20 Main Elements Includes almost all of the main elements and requirements and cites multiple examples to illustrate each element Includes most of the main elements and requirements and cites many examples to illustrate each element Includes some of the main elements and requirements Does not include any of the main elements and requirements 25 Writing (Mechanics/Citations) Student meets all requirements for submission. No errors related to organization, grammar and style, and citations Student meets most requirements for submission. Minor errors related to organization, grammar and style, and citations Student meets some requirements for submission. Some errors related to organization, grammar and style, and citations Student does not meet requirements for submission. Major errors related to organization, grammar and style, and citations 15 Knowledge of Fundamental Areas Integration and Application Total 20 100%

# The concepts of Fibonacci Sequence Project