Golden Ratio in the Human Body

THE florid RATIO IN THE HUMAN BODY GABRIELLE NAHAS IBDP MATH STUDIES THURSDAY, FEBRUARY 23rd 2012 WORD COUNT 2,839 INTRODUCTION The fortunate Ratio, also known as The Divine Proportion, The Golden Mean, or Phi, is a unalterable that can be chinkn all throughout the mathematical world. This irrational number, Phi (? ) is equal to 1. 618 when rounded. It is described as dividing a railway in the extreme and mean ratio. This means that when you divide segments of a delineate that always have a same quotient. When lines like these argon divided, Phi is the quotient When the black line is 1. 18 (Phi) times larger than the sick line and the zesty 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 engraft in many distinct places and situations all everyplace the world. It is seen in architecture, nature, Fibonacci numbers, and even more amazingly,the human eubstance. Fibonacci Numbers have proven to be closely tie in 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 dickens before it.The term number is known as n. The front 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 3rd 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, receipts 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 here and nowTerm 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 Fibona cci Series are found all over the world and are lines that can be divided to find Phi. One enkindle place they are found is in the human embody. Many examples of Phi can be seen in the hands, typeface and body. For example, when the length of a persons forearm is divided by the length of that persons hand, the quotient is Phi.The distance from a persons head to their fingertips divided by the distance from that persons 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 tried 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 expect. The aim of this investigation is to find examples of the number 1. 618 in different people and investigate some other places where Phi is found.Three ratios will be compared. The ra tios investigated are the ratio of head to toe and head to fingertips, the ratio of the utmost prick of the indicant finger to the middle section of the index finger, and the ratio of forearm to hand. dactyl 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 part of people of different ages and genders were measured in centimeters.Five people were measured and each participant had these parts measured * duration from head to foot * Distance from head to fingertips * duration 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 going was found for each re sult. PARTICIPANTS The people were of different ages and genders. For variety, a 4- category-old fe manful, 8-year-old virile, 18-year-old womanly, 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/ young-bearing(prenominal) 8/male 18/female 18/male 45/male Distance from head to foot 105. 5 124. 5 167 1 hundred eighty 185 Distance from head to fingertips 72. 5 84 97 110 cxv 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 50Distance from wrist to fingertips 15 18. 5 25 28 31 RATIO 1 RATIO OF HEAD TO TOE AND HEAD TO FINGERTIPS Measurements Participant Distance from head to foot (0. 5 cm) Distance from hea d 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 pass judgment 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. 50. 5 cm = 1. 455 1. 2% 2. 8-year-old male 124. 50. 5 cm/ 850. 5 cm = 1. 465 1. 0% 3. 18-year-old female 1670. 5 cm/ 970. 5 cm = 1. 722 5. 2% 4. 18-year-old male 1800. 5 cm/ 1100. 5 cm = 1. 636 1. 0% 5. 45-year-old male 1850. 5 cm/ 1150. 5 cm = 1. 609 0. 7% How close each result is to Phi This shows the remainder mingled with the actual quotient, what was measured, and the pass judgment quotient (1. 618). This is found by subtracting the actual quotient from Phi and using the absolute regard as 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% come 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 diaphragm 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. 617. 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