Golden Ratio in the Human Body

THE GOLDEN RATIO IN THE HUMAN BODY GABRIELLE NAHAS IBDP MATH STUDIES THURSDAY, FEBRUARY 23rd 2012 WORD COUNT: 2,839 INTRODUCTION: The Golden Ratio, also known as The Divine Proportion, The Golden Mean, or Phi, is a constant that can be seen all throughout the mathematical world. This irrational number, Phi (? ) is equal to 1. 618 when rounded. It is described as â€Å"dividing a line in the extreme and mean ratio†. This means that when you divide segments of a line that always have a same quotient. When lines like these are divided, Phi is the quotient: When the black line is 1. 18 (Phi) times larger than the blue line and the blue line is 1. 618 times larger than the red line, you are able to find Phi. What makes Phi such a mathematical phenomenon is how often it can be found in many different places and situations all over the world. It is seen in architecture, nature, Fibonacci numbers, and even more amazingly,the human body. Fibonacci Numbers have proven to be closely rela ted to the Golden Ratio. They are a series of numbers discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, every number is the sum of the two before it.The term number is known as ‘n’. The first term is ‘Un’ so, in order to find the next term in the sequence, the last two Un and Un+1 are added. (Knott). Formula: Un + Un+1 = Un+2 Example: The second term (U2) is 1; the third term (U3) is 2. The fourth term is going to be 1+2, making U3 equal 3. Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144†¦ When each term in the Fibonacci Series is divided by the term before it, the quotient is Phi, with the exception of the first 9 terms, which are still very close to equaling Phi. Term (n)| First Term Un| SecondTerm Un+1| Second Term/First Term (Un+1 /Un)| 1| 0| 1| n/a| 2| 1| 1| 1| 3| 1| 2| 2| 4| 2| 3| 1. 5| 5| 3| 5| 1. 667| 6| 5| 8| 1. 6| 7| 8| 13| 1. 625| 8| 13| 21| 1. 615| 9| 21| 34| 1. 619| 10| 34| 55| 1. 618| 11| 55| 89| 1. 618| 12| 89| 144| 1. 618| Lines that follow the Fibonacci Series are found all over the world and are lines that can be divided to find Phi. One interesting place they are found is in the human body. Many examples of Phi can be seen in the hands, face and body. For example, when the length of a person’s forearm is divided by the length of that person’s hand, the quotient is Phi.The distance from a person’s head to their fingertips divided by the distance from that person’s head to their elbows equals Phi. (Jovanovic). Because Phi is found in so many natural places, it is called the Divine ratio. It can be tested in a number of ways, and has been by various scientists and mathematicians. I have chosen to investigate the Phi constant and its appearance in the human body, to find the ratio in different sized people and see if my results match what is expected. The aim of this investigation is to find examples of the number 1. 618 in different people and investiga te other places where Phi is found.Three ratios will be compared. The ratios investigated are the ratio of head to toe and head to fingertips, the ratio of the lowest section of the index finger to the middle section of the index finger, and the ratio of forearm to hand. FIGURE 1 FIGURE 2 FIGURE 3 The first ratio is the white line in the to the light blue line in FIGURE 1 The second ratio is the ratio of the black line to the blue line in FIGURE 2 The third ratio is the ratio of the light blue line to the dark blue line in FIGURE 3 METHOD: DESIGN: Specific body parts of people of different ages and genders were measured in centimeters.Five people were measured and each participant had these parts measured: * Distance from head to foot * Distance from head to fingertips * Length of lowest section of index finger * Length of middle section of index finger * Distance from elbow to fingertips * Distance from wrist to fingertips The ratios were found, to see how close their quotients are to Phi (1. 618). Then the percentage difference was found for each result. PARTICIPANTS: The people were of different ages and genders. For variety, a 4-year-old female, 8-year-old male, 18-year-old female, 18-year-old male and a 45-year-old male were measured.All of the measurements are in this investigation with the ratios found and how close they are to the constant Phi are analyzed. The results were put into tables by each set of measurements and the ratios were found. DATA: | Participant Measurement ( ± 0. 5 cm)| Measurement| 4/female| 8/male| 18/female| 18/male| 45/male| Distance from head to foot| 105. 5| 124. 5| 167| 180| 185| Distance from head to fingertips| 72. 5| 84| 97| 110| 115| Length of lowest section of index finger| 2| 3| 3| 3| 3| Length of middle section of index finger| 1. 2| 2| 2. 5| 2| 2| Distance from elbow to fingertips| 27| 30| 40| 48| 50|Distance from wrist to fingertips| 15| 18. 5| 25| 28| 31| RATIO 1: RATIO OF HEAD TO TOE AND HEAD TO FINGERTIPS Measure ments Participant| Distance from head to foot ( ±0. 5 cm)| Distance from head to fingertips ( ±0. 5 cm)| 4-year-old female| 105. 5| 72. 5| 8-year-old male| 124. 5| 85| 18-year-old female| 167| 97| 18-year-male| 180| 110| 45-year-old male| 185| 115| Ratios: These are the original quotients that were found from the measurements. According to the Golden Ratio, the expected quotients will all equal Phi (1. 618). Distance from head to footDistance from head to fingertips 1. 4-year-old female: 105.  ±0. 5 cm/ 72. 5 ±0. 5 cm = 1. 455  ± 1. 2% 2. 8-year-old male: 124. 5 ±0. 5 cm/ 85 ±0. 5 cm = 1. 465  ± 1. 0% 3. 18-year-old female: 167 ±0. 5 cm/ 97 ±0. 5 cm = 1. 722  ± 5. 2% 4. 18-year-old male: 180 ±0. 5 cm/ 110 ±0. 5 cm = 1. 636  ± 1. 0% 5. 45-year-old male: 185 ±0. 5 cm/ 115 ±0. 5 cm = 1. 609  ± 0. 7% How close each result is to Phi: This shows the difference between the actual quotient, what was measured, and the expected quotient (1. 618). This is foun d by subtracting the actual quotient from Phi and using the absolute value to get the difference so it does not give a negative answer. |1. 18-Actual Quotient|=difference between result and Phi The difference between each quotient and 1. 618: 1. 4-year-old female: |1. 618- 1. 455  ± 1. 2%| = 0. 163  ± 1. 2% 2. 8-year-old male: |1. 618- 1. 465  ± 1. 0%| = 0. 153  ± 1. 0% 3. 18-year-old female: |1. 618- 1. 722  ± 5. 2%| = 0. 1  ± 5. 2% 4. 18-year-old male: |1. 618- 1. 636  ± 1. 0%| = 0. 018 5. 45-year-old male: |1. 618- 1. 609  ± 0. 7%| = 0. 009 Percentage Error: To find how close the results are to the expected value of Phi, percentage error can be used. Percentage error is how close experimental results are to expected results.Percentage error is found by dividing the difference between each quotient and Phi by Phi (1. 618) and multiplying that result by 100. This gives you the difference of the actual quotient to the expected quotient, Phi, in a percentage. (Roberts ) Difference1. 618 x100=Percentage difference between result and Phi 1. 4-year-old female: 0. 163  ± 1. 2%/1. 618 x 100 = 10. 1  ± 0. 12% 2. 8-year-old male: 0. 153  ± 1. 0%/1. 618 x 100 = 9. 46  ± 0. 09% 3. 18-year-old female: 0. 1 ± 5. 2% /1. 618 x 100 = 6. 18  ± 0. 3% 4. 18-year-old male: 0. 018/1. 618 x 100 = 1. 11% 5. 45-year-old male: 0. 009/1. 618 x 100 = 0. 5% AVERAGE: 10. 1  ± 0. 12% + 9. 46  ± 0. 09% + 6. 18  ± 0. 3% + 1. 11% + 0. 55% / 5 = 5. 48  ± 0. 5% ANALYSIS: The highest percentage error, the percent difference between the result and Phi, is 10. 1  ± 0. 12%. This is a small percentage error, and means that all but one of the ratios was more than 90% accurate. This is a good example of the Golden Ratio in the human body because all the values are close to Phi. Also, as the age of the participants increases, the percentage error decreases, so as people get older, the ratio of their head to feet to the ratio of their head to fingertips gets closer to PhiRATIO 2: RATIO OF THE MIDDLE SECTION OF THE INDEX FINGER TO THE BOTTOM SECTION OF THE INDEX FINGER Measurements Participant| Length of lowest section of index finger ( ±0. 5 cm)| Length of middle section of index finger ( ±0. 5 cm)| 4 year old female| 2| 1| 8 year old male| 3| 2| 18 year old female| 3| 2. 5| 18 year male| 3| 2| 35 year old male| 3| 2| Ratios: Length of lowest section of index finger Length of middle section of index finger 1. 4-year-old female: 2  ± 0. 5 cm/ 1  ± 0. 5 cm = 2  ± 75% 2. 8-year-old male: 3  ± 0. 5 cm/ 2  ± 0. 5 cm = 1. 5  ± 42% 3. 18-year-old female: 3  ± 0. 5 cm/ 2.  ± 0. 5 cm = 1. 2  ± 37% 4. 18-year-old male: 3  ± 0. 5 cm/ 2  ± 0. 5 cm = 1. 5  ± 42% 5. 45-year-old male: 3  ± 0. 5 cm/ 2  ± 0. 5 cm = 1. 5  ± 42% How close each result is to Phi: |1. 618-Actual Quotient|=difference between result and Phi The difference between each quotient and 1. 618: 1. 4-year-old female: |1. 618- 2  ± 75%| = 0. 382  ± 75% 2 . 8-year-old male: |1. 618- 1. 5  ± 42%| = 0. 118  ± 42% 3. 18-year-old female: |1. 618- 1. 2  ± 37%| = 0. 418  ± 37% 4. 18-year-old male: |1. 618- 1. 5  ± 42%| = 0. 118  ± 42% 5. 45-year-old male: |1. 618- 1. 5  ± 42%| = 0. 118  ± 42% Percentage Error: Difference1. 18 x100=Percentage difference between result and Phi 1. 4-year-old female: 0. 382  ± 75%/1. 618 x 100 = 23. 6  ± 17. 7% 2. 8-year-old male: 0. 118  ± 42%/1. 618 x 100 = 7. 3  ± 3. 1% 3. 18-year-old female: 0. 418  ± 37%/1. 618 x 100 = 25. 8  ± 9. 5% 4. 18-year-old male: 0. 118  ± 42%/1. 618 x 100 = 7. 3  ± 3. 1% 5. 45-year-old male: 0. 118  ± 42%/1. 618 x 100 = 7. 3  ± 3. 1% AVERAGE: 23. 6 ±17. 7% + 7. 3  ±3. 1% + 25. 8  ±9. 5% + 7. 3  ±3. 1% + 7. 3  ±3. 1%/5= 14. 3  ± 36. 5% ANALYSIS: With this ratio, 3 of the results come out with a

Better Major: Computer Science or Software Engineering?

Technology powers our world and is deeply embedded into nearly every facet of our lives. The prospect of working with computers as a career is becoming increasingly popular with high school and college students; the number of computer science majors is rising every year, and even younger students are getting involved with programming and building computer hardware. There’s a good reason for this rise in interest. Demand is high for qualified employees in technology fields , and some of those positions offer particularly high pay and other perks. Someone has to invent, build, and maintain all that technology we use on a regular basis, and in the future, you might be the one filling that role. If you’re interested in pursuing a college major involving computers, however, you might be confused by the variety of different programs with different names that are available at colleges across the United States. Mainly, you’ll find a difference drawn between programs that go by the name â€Å"computer science,† and those that are called â€Å"computer engineering† or similar names. As a high school student with minimal experience in the field, you’re likely wondering what distinguishes these two fields, and which major it’s better to pursue. Read on for ’s take on this popular field and its variations. Variations exist in how different colleges and organizations use these terms, so it’s always a good idea to research what a particular major involves at a particular school. You might also find additional variations on this theme; MIT, for example, offers four different undergraduate majors that include computer science. Variations aside, however, here’s how computer science and computer engineering are generally used as specific titles for undergraduate majors. In some contexts, computer science can be used as an umbrella term that covers many different intellectual and academic endeavors related to computers. When it’s used as a specific field, computer science is typically more theoretical than computer engineering. It addresses the ways in which computers think on a fundamental level, requiring you to get to the root of how technology solves problems and meets human needs. If you want to study computer science, you’re going to have to take a lot of math courses — you might be surprised at the amount of overlap between the two fields. Your math studies will be higher-level and more abstract than what’s covered in typical high school math classes. An undergraduate computer science degree can qualify you to become a programmer or software developer, which is a job you can perform in almost any field, from healthcare to government to cutting-edge technology. Eventually, some positions or career paths may require you to get a master’s degree, but even with a bachelor’s degree alone, your career and income prospects are strong. You’re more likely to find computer science as an option at a wide range of top-tier colleges than computer engineering. For example, all the schools in the Ivy League offer majors in computer science, but not all offer the ability to specifically major in computer engineering. Computer engineering is generally considered to be a more practical, less theoretical major than computer science. While you’ll still need to build a strong math background, if you study computer engineering, you’ll spend more time working with actual computer hardware and focusing on practical, hands-on skills for working with technology and solving real-world technical problems. Some computer engineering programs are specifically intended to teach you what you’ll need to know for a career as an information technology professional. IT professionals work in every industry to administer and maintain computer networks, manage an organization’s computer resources, and troubleshoot everyday computer issues. Generally, computer engineering programs fall under a university’s engineering department, and may be grouped with electrical engineering. You’re more likely to find computer engineering (or a similar title) available as a major at engineering-heavy colleges and institutes of technology, or at public colleges that offer a particularly long list of majors. Check with your college for more information. There’s no definitive answer as to whether computer science or computer engineering is the better major. Each has its pros and cons, and each approaches the subject of computer technology in a different way. Each major will prepare you for a slightly different career path, but neither path is â€Å"better.† Practically speaking, you’re likely to find a lot of overlap between these two majors when it comes to what you’ll study. The basics of programming will be the same, and though computer engineering tends to involve somewhat less theoretical math, you’ll still need to build a strong understanding of how the subject relates to technology and programming. Desirable companies like Google are notorious for including difficult, math-heavy programming tasks in their hiring process, even if you won’t have to do that kind of problem-solving on a daily basis in that position. At the same time, practical skills in working with computers are always beneficial when you’re planning to go into a technical field. Rather than asking which of these majors is â€Å"better,† ask which of these majors appears to be the better fit for you — your needs, your talents, and your career plans. If one of these approaches to technology is particularly appealing to you, gives you opportunities to use your talents in a satisfying way, and allows you to achieve your personal goals for your adult life, it’s likely the better choice for you as an individual. Finally, your choice between computer science and computer engineering as undergraduate majors may be less important than your overall choice of college. Choosing a college that’s a good fit for you across the board is tremendously important, and the presence of your preferred major may not outweigh the absence of other features you find important. If you’re worried about your future career prospects, keep in mind that many jobs aren’t tied to one particular college major, and you’ll have time to amass work experience and other qualifications that are equally important. Also, remember that if you attend a competitive and well-regarded college (especially in the field of technology), having that recognizable name on your diploma will reflect well upon you, regardless of the specific name of your major. You’ll also need to take into account the differences between specific computer-related undergraduate programs at specific colleges. Not all computer science or computer engineering programs are the same, and while a certain type of program might seem preferable for you in general, there may well be exceptions. (For example, you might be especially excited to work with a certain professor, and that might influence your college choice.) Besides the importance of fit, you’ll need to consider that your future plans may change after you arrive at college. This is incredibly common — multiple studies have shown that a strong majority of students, perhaps as much as 80%, change their major at least once. If you choose a school that’s an all-around solid match to you, you’ll be better equipped to handle these future changes if they come to pass. Estimating your chance of getting into a college is not easy in today’s competitive environment. Thankfully, with our state-of-the-art software and data, we can analyze your academic and extracurricular profile and estimate your chances. Our profile analysis tool can also help you identify the improvement you need to make to enter your dream school. You won’t always have a choice between computer science and computer engineering at a given college; many colleges only offer one or the other. However, as you incorporate your preferences regarding your major into your college choices, it’s wise to think about which path is a better fit. Here are some questions you can ask yourself to help assess whether computer science or computer engineering is a better choice for you, generally speaking: Again, there’s a lot more to choosing a college than whether the school offers the best possible major for you. Don’t focus too much on your major — it will only be one facet of your college experience and preparation for the working world, and it may very well change later on. Make sure you do your research about what else each college has to offer, inside and outside the classroom. Teaching style, campus facilities, financial aid availability, and the general â€Å"feel† of the campus are all important factors. Visiting the school is a good idea if it’s possible for you to do so, and talking to current and former students provides valuable personal perspectives. Technology is a burgeoning industry, and many high school students today are interested in building a strong background in the computer field. If you’re one of these students, it pays to get started early in exploring this field. Check out these posts from the blog for more information about options you might find interesting: Looking for more help navigating the often-confusing world of college admissions? ’s experienced advisors are here to help you with every aspect of the process. We can even match you up with a current college student who’s interested in computers and can offer specialized advice.

The Shawshank Redemption and its Relation to Greek Mythology Term Paper

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The Principles of Corporate Social Responsibility Concept Assignment

The Principles of Corporate Social Responsibility Concept - Assignment Example Currently, it has become a focus for many companies due to its continuing commitment to help businesses contribute to economic development by being responsible to their respective stakeholders (Hancock, 2005, 12). The greater part of the CSR has helped manage or improve environmental and social impacts of the company. The increased use of CSR has encouraged the corporate sector to be more environmentally and socially responsible particularly at a time when the crisis has damaged consumer’s level of trust in the operations of the company. This coupled with the rise in transparency and social media has facilitated companies to begin taking CSR more seriously in order to help reduce their environmental impacts (Louche et al, 2010, 209). Corporate social responsibility has helped enterprises to contribute to the highly competitive social market economy and sustainable development. Responsible business conduct has helped many private sector operators to provide public services. For instance, they have provided employment opportunities, social amenities as well as sustainable, smart and inclusive growth (Mullerat & Brennan, 2011, 29).   CSR has facilitated engagement with both external and internal stakeholders thus enabling businesses to take advantage of societal fast-changing expectations as well as operating conditions. The enterprises have acted as drivers for the emergence of other markets and the development of real opportunities for growth. Companies normally commit responsible investment in various sectors of the economy to enhance the social welfare of their stakeholders. This results in economic growth thus improving the living standards of the stakeholders (Mullerat & Brennan, 2011, 89). Corporate social responsibility planning has come to the forefront for major companies’ future growth opportunities. Their increased investment in CSR initiatives, for instance, environmental initiative, has been very attractive in inducing economic stimulus of a country while at the same  time achieving solid CRS goals.

Strategic Management Assignment Example | Topics and Well Written Essays - 1500 words

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FDA policy Essay Example | Topics and Well Written Essays - 1250 words

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Nestle Infant Formula Article Example | Topics and Well Written Essays - 500 words

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