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, besides unconcealed as The Divine Proportion, The Golden Mean, or Phi, is a invaritelling that can be seen all throughout the prosaic universe. This irrational compute, Phi (? ) is resembling to 1. 618 when rounded. It is picturesque as "dividing a outoutverse in the immoderate and moderation agreement". This moderations that when you disconnected segments of a outoutverse that frequently accept a identical quotient. When outlines affect these are disconnectedd, Phi is the quotient: When the ebon outoutverse is 1. 18 (Phi) times wider than the cerulean outoutverse and the cerulean outoutverse is 1. 618 times wider than the red outline, you are telling to furnish Phi. What makes Phi such a prosaic oddity is how frequently it can be build in sundry contrariant locates and situations all balance the universe. It is seen in structure, constitution, Fibonacci computes, and flush further amazingly,the cosmical collection. Fibonacci Aggregate accept proven to be air-tight connected to the Golden Ratio. They are a rotation of computes discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, integral compute is the sum of the two antecedently it. The vocconducive compute is unconcealed as ‘n’. The pristine vocconducive is ‘Un’ so, in appoint to furnish the present vocconducive in the progression, the last two Un and Un+1 are adventitious. (Knott). Formula: Un + Un+1 = Un+2 Example: The avoid vocconducive (U2) is 1; the third vocconducive (U3) is 2. The fourth vocconducive is going to be 1+2, making U3 resembling 3. Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… When each vocconducive in the Fibonacci Rotation is disconnectedd by the vocconducive antecedently it, the quotient is Phi, delay the exclusion of the pristine 9 vocables, which are peaceful very bar to resemblinging Phi. Vocconducive (n)| Pristine Vocconducive Un| Second Term Un+1| Avoid Term/First Vocconducive (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 supervene the Fibonacci Rotation are build all balance the universe and are outlines that can be disconnectedd to furnish Phi. One thrilling locate they are build is in the cosmical collection. Sundry models of Phi can be seen in the laborers, aspect and collection. For model, when the tediousness of a person’s forearm is disconnectedd by the tediousness of that person’s laborer, the quotient is Phi. The removal from a person’s mind to their fingertips disconnectedd by the removal from that person’s mind to their articulations resemblings Phi. (Jovanovic). Owing Phi is build in so sundry intrinsic locates, it is designated the Divine agreement. It can be tested in a compute of ways, and has been by diversified scientists and mathematicians. I accept selected to question the Phi invaritelling and its look in the cosmical collection, to furnish the agreement in contrariant sized tribe and see if my fruits mate what is expected. The aim of this scrutiny is to furnish models of the compute 1. 618 in contrariant tribe and question other locates where Phi is build. Three agreements accomplish be compared. The agreements questiond are the agreement of mind to toe and mind to fingertips, the agreement of the meanest exception of the refutation finger to the medium exception of the refutation finger, and the agreement of forearm to laborer. FIGURE 1 FIGURE 2 FIGURE 3 The pristine agreement is the colorless outoutverse in the to the active cerulean outoutverse in FIGURE 1 The avoid agreement is the agreement of the ebon outoutverse to the cerulean outoutverse in FIGURE 2 The third agreement is the agreement of the active cerulean outoutverse to the black cerulean outoutverse in FIGURE 3 METHOD: DESIGN: Local collection tonnage of tribe of contrariant ages and genders were measured in centimeters. Five tribe were measured and each participant had these tonnage measured: * Removal from mind to groundation * Removal from mind to fingertips * Tediousness of meanest exception of refutation finger * Tediousness of medium exception of refutation finger * Removal from articulation to fingertips * Removal from wrist to fingertips The agreements were build, to see how bar their quotients are to Phi (1. 618). Then the percentage dissimilarity was build for each fruit. PARTICIPANTS: The tribe were of contrariant ages and genders. For multiplicity, a 4-year-old womanly, 8-year-old courageous, 18-year-old womanly, 18-year-old courageous and a 45-year-old courageous were measured. All of the measurements are in this scrutiny delay the agreements build and how bar they are to the invaritelling Phi are analyzed. The fruits were put into tables by each set of measurements and the agreements were build. DATA: | Participant Measurement (± 0. 5 cm)| Measurement| 4/female| 8/male| 18/female| 18/male| 45/male| Removal from mind to groundation| 105. 5| 124. 5| 167| 180| 185| Removal from mind to fingertips| 72. 5| 84| 97| 110| 115| Tediousness of meanest exception of refutation finger| 2| 3| 3| 3| 3| Tediousness of medium exception of refutation finger| 1. 2| 2| 2. 5| 2| 2| Removal from articulation 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 Measurements Participant| Removal from mind to groundation (±0. 5 cm)| Removal from mind to fingertips (±0. 5 cm)| 4-year-old womanly| 105. 5| 72. 5| 8-year-old courageous| 124. 5| 85| 18-year-old womanly| 167| 97| 18-year-male| 180| 110| 45-year-old courageous| 185| 115| Ratios: These are the primary quotients that were build from the measurements. According to the Golden Ratio, the expected quotients accomplish all resembling Phi (1. 618). Removal from mind to groundationDistance from mind to fingertips 1. 4-year-old womanly: 105. ±0. 5 cm/ 72. 5±0. 5 cm = 1. 455 ± 1. 2% 2. 8-year-old courageous: 124. 5±0. 5 cm/ 85±0. 5 cm = 1. 465 ± 1. 0% 3. 18-year-old womanly: 167±0. 5 cm/ 97±0. 5 cm = 1. 722 ± 5. 2% 4. 18-year-old courageous: 180±0. 5 cm/ 110±0. 5 cm = 1. 636 ± 1. 0% 5. 45-year-old courageous: 185±0. 5 cm/ 115±0. 5 cm = 1. 609 ± 0. 7% How bar each fruit is to Phi: This demonstrations the dissimilarity among the real quotient, what was measured, and the expected quotient (1. 618). This is build by subtracting the real quotient from Phi and using the arbitrary prize to get the dissimilarity so it does not surrender a denying repartee. |1. 18-Actual Quotient|=dissimilarity among fruit and Phi The dissimilarity among each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 1. 455 ± 1. 2%| = 0. 163 ± 1. 2% 2. 8-year-old courageous: |1. 618- 1. 465 ± 1. 0%| = 0. 153 ± 1. 0% 3. 18-year-old womanly: |1. 618- 1. 722 ± 5. 2%| = 0. 1 ± 5. 2% 4. 18-year-old courageous: |1. 618- 1. 636 ± 1. 0%| = 0. 018 5. 45-year-old courageous: |1. 618- 1. 609 ± 0. 7%| = 0. 009 Percentage Error: To furnish how bar the fruits are to the expected prize of Phi, percentage hallucination can be used. Percentage hallucination is how bar trialal fruits are to expected fruits. Percentage hallucination is build by dividing the dissimilarity among each quotient and Phi by Phi (1. 618) and multiplying that fruit by 100. This surrenders you the dissimilarity of the real quotient to the expected quotient, Phi, in a percentage. (Roberts) Difference1. 618 x100=Percentage dissimilarity among fruit and Phi 1. 4-year-old womanly: 0. 163 ± 1. 2%/1. 618 x 100 = 10. 1 ± 0. 12% 2. 8-year-old courageous: 0. 153 ± 1. 0%/1. 618 x 100 = 9. 46 ± 0. 09% 3. 18-year-old womanly: 0. 1± 5. 2% /1. 618 x 100 = 6. 18 ± 0. 3% 4. 18-year-old courageous: 0. 018/1. 618 x 100 = 1. 11% 5. 45-year-old courageous: 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 foremost percentage hallucination, the percent dissimilarity among the fruit and Phi, is 10. 1 ± 0. 12%. This is a slender percentage hallucination, and moderations that all but one of the agreements was further than 90% considerate. This is a good-natured-natured model of the Golden Agreement in the cosmical collection owing all the prizes are bar to Phi. Also, as the age of the participants increases, the percentage hallucination decreases, so as tribe get older, the agreement of their mind to feet to the agreement of their mind to fingertips gets barr to Phi RATIO 2: RATIO OF THE MIDDLE SECTION OF THE INDEX FINGER TO THE BOTTOM SECTION OF THE INDEX FINGER Measurements Participant| Tediousness of meanest exception of refutation finger (±0. 5 cm)| Tediousness of medium exception of refutation finger (±0. 5 cm)| 4 year old womanly| 2| 1| 8 year old courageous| 3| 2| 18 year old womanly| 3| 2. 5| 18 year courageous| 3| 2| 35 year old courageous| 3| 2| Ratios: Tediousness of meanest exception of refutation finger Tediousness of medium exception of refutation finger 1. 4-year-old womanly: 2 ± 0. 5 cm/ 1 ± 0. 5 cm = 2 ± 75% 2. 8-year-old courageous: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% 3. 18-year-old womanly: 3 ± 0. 5 cm/ 2. ± 0. 5 cm = 1. 2 ± 37% 4. 18-year-old courageous: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% 5. 45-year-old courageous: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% How bar each fruit is to Phi: |1. 618-Actual Quotient|=dissimilarity among fruit and Phi The dissimilarity among each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 2 ± 75%| = 0. 382 ± 75% 2. 8-year-old courageous: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% 3. 18-year-old womanly: |1. 618- 1. 2 ± 37%| = 0. 418 ± 37% 4. 18-year-old courageous: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% 5. 45-year-old courageous: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% Percentage Error: Difference1. 18 x100=Percentage dissimilarity among fruit and Phi 1. 4-year-old womanly: 0. 382 ± 75%/1. 618 x 100 = 23. 6 ± 17. 7% 2. 8-year-old courageous: 0. 118 ± 42%/1. 618 x 100 = 7. 3 ± 3. 1% 3. 18-year-old womanly: 0. 418 ± 37%/1. 618 x 100 = 25. 8 ± 9. 5% 4. 18-year-old courageous: 0. 118 ± 42%/1. 618 x 100 = 7. 3 ± 3. 1% 5. 45-year-old courageous: 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: Delay this agreement, 3 of the fruits follow out delay a <10% percentage hallucination, moderationing they are very bar to Phi (1. 618). In the measurements, 3 of the participants had the identical agreement of 3:2. This fruit is entirely thrilling owing 3 and 2 are build in the Fibonacci Series. This proves that the Fibonacci rotation is connected to the Golden Ratio. The pristine fruit build was 2:1; these are besides Fibonacci computes. Both Fibonacci computes and the Golden Agreement were seen in the agreement. RATIO 3: RATIO OF THE LENGTH OF THE FOREARM TO THE LENGTH OF THE HAND Measurements Participant| Tediousness of forearm (±0. 5 cm)| Tediousness of laborer (±0. 5 cm)| 4-year-old womanly| 27| 15| 8-year-old courageous| 30| 18. 5| 18-year-old womanly| 40| 25| 18-year-male| 48| 28| 5-year-old courageous| 50| 31| Ratios: Tediousness of forearm Tediousness of laborer 1. 4-year-old womanly: 27 ± 0. 5 cm/ 15 ± 0. 5 cm = 1. 8 ± 9. 4% 2. 8-year-old courageous: 30 ± 0. 5 cm/ 18. 5± 0. 5 cm = 1. 622 ± 4. 4% 3. 18-year-old womanly: 40 ± 0. 5 cm/ 25± 0. 5 cm = 1. 6 ± 3. 7% 4. 18-year-old courageous: 48 ± 0. 5 cm/ 28± 0. 5 cm = 1. 714 ± 2. 8% 5. 45-year-old courageous: 50 ± 0. 5 cm/ 31± 0. 5 cm = 1. 613 ± 2. 6% How bar each fruit is to Phi: |1. 618-Actual Quotient|=dissimilarity among fruit and Phi The dissimilarity among each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 1. 8 ± 9. 4%| = 0. 182 ± 9. 4% 2. 8-year-old courageous: |1. 18- 1. 622 ± 4. 4%| = 0. 004 ± 4. 4% 3. 18-year-old womanly: |1. 618- 1. 6 ± 3. 7%| = 0. 018 ± 3. 7% 4. 18-year-old courageous: |1. 618- 1. 714 ± 2. 8%| = 0. 096 ± 2. 8% 5. 45-year-old courageous: |1. 618- 1. 613 ± 2. 6%| = 0. 005 ± 2. 6% Percentage Error: Difference1. 618 x100=Percentage dissimilarity among fruit and Phi 1. 4-year-old womanly: 0. 182 ± 9. 4%/1. 618 x 100 = 11. 2 ± 1. 1% 2. 8-year-old courageous: 0. 004 ± 4. 4%/1. 618 x 100 = 0. 2 ± 0. 9% 3. 18-year-old womanly: 0. 018 ± 3. 7%/1. 618 x 100 = 1. 1 ± 4. 1% 4. 18-year-old courageous: 0. 096 ± 2. 8%/1. 618 x 100 = 0. 06 ± 0. 1% 5. 45-year-old courageous: 0. 005 ± 2. %/1. 618 x 100 = 0. 31 ± 0. 8% AVERAGE: 11. 2 ±1. 1% + 0. 2 ±0. 9% + 1. 1 ± 4. 1% + 0. 06 ± 0. 1% + 0. 31 ± 0. 8%/5 = 2. 6± 7. 0% ANALYSIS: 4 out of 5 of these percentage hallucinations were <1. 2% disconnected from Phi, not including the hallucination. The merely fruit that differed was the four-year-old womanly participant’s fruit, which could be owing she is peaceful developing. The other 4 fruits were very bar to Phi and demonstration the Golden Agreement in the cosmical collection arrestly precisely. CONCLUSION AND VALIDITY: The fruits of this scrutiny demonstration that tribe of contrariant sizes all accept collection agreements that follow very bar to resemblinging the Golden Ratio. When the medium percentage hallucinations were build for each of the three tested agreements, none of them were elder than 14. 3 ± 36. 5%. This moderations that all of the percentage hallucinations were low, thus, all the medium agreements build were very bar to the expected prize of 1. 618 (Phi). The third agreement, the agreement of the tediousness of the forearm to the tediousness of the laborer, was the agreement barst to the Golden Agreement delay a percentage hallucination of merely 2. 6± 7. 0%. On medium, the agreements were merely environing 2. 6% disconnected from 1. 618. Within the participants, the agreement of forearm to laborer was immoderately bar to resemblinging Phi. This proves the conception that the Golden Agreement can be build in this exception of the cosmical collection. Looking at each of the participants partially, the 4-year-old womanly had the foremost percentage hallucination in two of the three agreements that were tested. In two of the agreements, the 45-year-old courageous had the meanest percentage hallucination. In integral agreement, the 45-year-old courageous had a significantly inferior percentage hallucination than the 4-year-old womanly, and it was besides obvious that as age went up, the percentage hallucination decreased. This suggests that as tribe develop, their collection agreements develop barr to the Golden Ratio. The agreement delay the foremost medium percent hallucination was the agreement of the medium exception of the finger to the inferior exception of the finger. Flush though it had the foremost percentage hallucinations, it did accept the most narration to the Fibonacci Series, which has proven to be air-tight connected to the Golden Ratio. Three of the participants had 3cm and 2cm for their measurements; 3 and 2 are unconcealed as Fibonacci computes. Another participant had 1 and 2, which are besides Fibonacci computes. In this scrutiny, it was build that the Golden Agreement is very bar to the measurements of agreements build in the cosmical collection. There was admission for hallucination in this scrutiny. The participants had a wide sum of change and all of them differed in age and gender. If replicated, this trial would use from further participants of the identical age and gender so their fruits can be compared and can besides be considered further conclusive and telling to be generalized. This scrutiny tested three collection agreements for the Golden Ratio, the agreement of mind to groundation and mind to fingertips, medium exception of refutation finger to meanest exception of refutation finger and forearm to laborer. There are sundry other Phi agreements that can be questiond in the cosmical collection. In appoint to get allure a knowledge of the Golden Agreement in the cosmical collection, other agreement’s should be testes, such as the ones build in the cosmical aspect. In this scrutiny ascititious agreements were build. When looking at the misentry and dissection of the agreements tested delay the local participants, it is obvious that adults accept collection agreements barr to the Golden Ratio, making conclusion accept a possibility to be considered outliers and surrender ascititious fruits. If this were to be conducted intermittently, the preponderance of participants would be balance the age of 18 years, or conclusion could be measured in a entirely disconnected trial. REFERENCES: Jovanovic, Radoslav. "The Golden Exception and The Cosmical Body. " Rasko Jovanovic's Universe of Mathematics. 2001. Web. 22 Feb. 2012. Knott, Dr. Ron. "Who Was Fibonacci? " Fibonacci Aggregate and the Golden Section. Mathematics Department of the University of Surrey, UK, 11 Mar. 1998. Web. 22 Feb. 2012. "Phi for Neo-phi-tes. Overview of Phi, the Golden Agreement / Divine Proportion and Fibonacci Numbers. PhiPoint Solutions, LLC. , 1997. Web. 22 Feb. 2012. PhiPoint Solutions, LLC. "The Cosmical Body. " Cosmical Collection and Phi, the Golden Ratio. 1997. Web. 25 Feb. 2012. <http://www. goldennumber. net/body. htm>. Roberts, Donna. "Error in Measurement. " Oswego City School District Regents Exam Prep Center. Oswego City School District Regents Exam Prep Center, 1998. Web. 22 Feb. 2012. <http://regentsprep. org/Regents/math/ALGEBRA/AM3/LError. htm>.