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