THE GOLDEN RATIO IN THE HUMAN BODY GABRIELLE NAHAS IBDP MATH STUDIES THURSDAY, FEBRUARY 23rd 2012 WORD COUNT: 2,839 INTRODUCTION: The Golden Ratio, too notorious as The Divine Proportion, The Golden Mean, or Phi, is a fixed that can be seen all throughout the commonestablish globe. This beastly enumerate, Phi (? ) is similar to 1. 618 when rounded. It is vivid as "dividing a row in the most-violent and medium proportion". This mediums that when you distribute segments of a row that regularly feel a identical quotient. When rows approve these are distributed, Phi is the quotient: When the ebon row is 1. 18 (Phi) times extensiver than the cerulean-colored-colored row and the cerulean-colored-colored row is 1. 618 times extensiver than the red row, you are operative to discbalance Phi. What makes Phi such a commonestablish oddity is how frequently it can be endow in multifarious divergent establishs and situations all balance the globe. It is seen in fabric, character, Fibonacci quantity, and well-balanced further amazingly,the ethnical substantiality. Fibonacci Quantity feel proven to be restly cognate to the Golden Ratio. They are a succession of quantity discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, entire enumerate is the sum of the two precedently it.
The account enumerate is notorious as ‘n’. The foremost account is ‘Un’ so, in enjoin to discbalance the proximate account in the succession, the ultimate two Un and Un+1 are acquired. (Knott). Formula: Un + Un+1 = Un+2 Example: The remedy account (U2) is 1; the third account (U3) is 2. The fourth account is going to be 1+2, making U3 similar 3. Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… When each account in the Fibonacci Succession is distributed by the account precedently it, the quotient is Phi, after a while the qualification of the foremost 9 accounts, which are quiet very rest to similaring Phi. Account (n)| Foremost Account Un| Second
Term Un+1| Remedy Term/First Account (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 flourish the Fibonacci Succession are endow all balance the globe and are rows that can be distributed to discbalance Phi. One sensational establish they are endow is in the ethnical substantiality. Multifarious copys of Phi can be seen in the agencys, aspect and substantiality. For copy, when the diffusiveness of a person’s forearm is distributed by the diffusiveness of that person’s agency, the quotient is Phi.
The interspace from a person’s guide to their fingertips distributed by the interspace from that person’s guide to their knees similars Phi. (Jovanovic). Owing Phi is endow in so multifarious consistent establishs, it is designated the Divine proportion. It can be tested in a enumerate of ways, and has been by several scientists and mathematicians. I feel chosen to dare the Phi fixed and its coming in the ethnical substantiality, to discbalance the proportion in divergent sized persons and see if my ends tally what is expected. The aim of this scrutiny is to discbalance copys of the enumerate 1. 618 in divergent persons and dare other establishs where Phi is endow.
Three proportions allure be compared. The proportions dared are the proportion of guide to toe and guide to fingertips, the proportion of the last individuality of the condemnation finger to the intermediate individuality of the condemnation finger, and the proportion of forearm to agency. FIGURE 1 FIGURE 2 FIGURE 3 The foremost proportion is the unblemished row in the to the thoughtless cerulean-colored-colored row in FIGURE 1 The remedy proportion is the proportion of the ebon row to the cerulean-colored-colored row in FIGURE 2 The third proportion is the proportion of the thoughtless cerulean-colored-colored row to the black cerulean-colored-colored row in FIGURE 3 METHOD: DESIGN: Restricted substantiality magnitude of persons of divergent ages and genders were measured in centimeters.
Five persons were measured and each participant had these magnitude measured: * Interspace from guide to bottom * Interspace from guide to fingertips * Diffusiveness of last individuality of condemnation finger * Diffusiveness of intermediate individuality of condemnation finger * Interspace from knee to fingertips * Interspace from wrist to fingertips The proportions were endow, to see how rest their quotients are to Phi (1. 618). Then the percentage destruction was endow for each end. PARTICIPANTS: The persons were of divergent ages and genders. For discord, a 4-year-old womanish, 8-year-old virile, 18-year-old womanish, 18-year-old virile and a 45-year-old virile were measured.
All of the measurements are in this scrutiny after a while the proportions endow and how rest they are to the fixed Phi are analyzed. The ends were put into tables by each set of measurements and the proportions were endow. DATA: | Participant Measurement (± 0. 5 cm)| Measurement| 4/female| 8/male| 18/female| 18/male| 45/male| Interspace from guide to bottom| 105. 5| 124. 5| 167| 180| 185| Interspace from guide to fingertips| 72. 5| 84| 97| 110| 115| Diffusiveness of last individuality of condemnation finger| 2| 3| 3| 3| 3| Diffusiveness of intermediate individuality of condemnation finger| 1. 2| 2| 2. 5| 2| 2| Interspace from knee 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| Interspace from guide to bottom (±0. 5 cm)| Interspace from guide to fingertips (±0. 5 cm)| 4-year-old womanish| 105. 5| 72. 5| 8-year-old virile| 124. 5| 85| 18-year-old womanish| 167| 97| 18-year-male| 180| 110| 45-year-old virile| 185| 115| Ratios: These are the leading quotients that were endow from the measurements. According to the Golden Ratio, the expected quotients allure all similar Phi (1. 618). Interspace from guide to bottomDistance from guide to fingertips 1. 4-year-old womanish: 105. ±0. 5 cm/ 72. 5±0. 5 cm = 1. 455 ± 1. 2% 2. 8-year-old virile: 124. 5±0. 5 cm/ 85±0. 5 cm = 1. 465 ± 1. 0% 3. 18-year-old womanish: 167±0. 5 cm/ 97±0. 5 cm = 1. 722 ± 5. 2% 4. 18-year-old virile: 180±0. 5 cm/ 110±0. 5 cm = 1. 636 ± 1. 0% 5. 45-year-old virile: 185±0. 5 cm/ 115±0. 5 cm = 1. 609 ± 0. 7% How rest each end is to Phi: This illusions the destruction among the real quotient, what was measured, and the expected quotient (1. 618). This is endow by subtracting the real quotient from Phi and using the absolute compute to get the destruction so it does not communicate a disclaiming vindication. |1. 18-Actual Quotient|=destruction among end and Phi The destruction among each quotient and 1. 618: 1. 4-year-old womanish: |1. 618- 1. 455 ± 1. 2%| = 0. 163 ± 1. 2% 2. 8-year-old virile: |1. 618- 1. 465 ± 1. 0%| = 0. 153 ± 1. 0% 3. 18-year-old womanish: |1. 618- 1. 722 ± 5. 2%| = 0. 1 ± 5. 2% 4. 18-year-old virile: |1. 618- 1. 636 ± 1. 0%| = 0. 018 5. 45-year-old virile: |1. 618- 1. 609 ± 0. 7%| = 0. 009 Percentage Error: To discbalance how rest the ends are to the expected compute of Phi, percentage untruth can be used. Percentage untruth is how rest testal ends are to expected ends.
Percentage untruth is endow by dividing the destruction among each quotient and Phi by Phi (1. 618) and multiplying that end by 100. This communicates you the destruction of the real quotient to the expected quotient, Phi, in a percentage. (Roberts) Difference1. 618 x100=Percentage destruction among end and Phi 1. 4-year-old womanish: 0. 163 ± 1. 2%/1. 618 x 100 = 10. 1 ± 0. 12% 2. 8-year-old virile: 0. 153 ± 1. 0%/1. 618 x 100 = 9. 46 ± 0. 09% 3. 18-year-old womanish: 0. 1± 5. 2% /1. 618 x 100 = 6. 18 ± 0. 3% 4. 18-year-old virile: 0. 018/1. 618 x 100 = 1. 11% 5. 45-year-old virile: 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 untruth, the percent destruction among the end and Phi, is 10. 1 ± 0. 12%. This is a diminutive percentage untruth, and mediums that all but one of the proportions was further than 90% complimentary. This is a cheerful copy of the Golden Proportion in the ethnical substantiality owing all the computes are rest to Phi. Also, as the age of the participants increases, the percentage untruth decreases, so as persons get older, the proportion of their guide to feet to the proportion of their guide to fingertips gets restr to Phi
RATIO 2: RATIO OF THE MIDDLE SECTION OF THE INDEX FINGER TO THE BOTTOM SECTION OF THE INDEX FINGER Measurements Participant| Diffusiveness of last individuality of condemnation finger (±0. 5 cm)| Diffusiveness of intermediate individuality of condemnation finger (±0. 5 cm)| 4 year old womanish| 2| 1| 8 year old virile| 3| 2| 18 year old womanish| 3| 2. 5| 18 year virile| 3| 2| 35 year old virile| 3| 2| Ratios: Diffusiveness of last individuality of condemnation finger Diffusiveness of intermediate individuality of condemnation finger 1. 4-year-old womanish: 2 ± 0. 5 cm/ 1 ± 0. 5 cm = 2 ± 75% 2. 8-year-old virile: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% 3. 18-year-old womanish: 3 ± 0. 5 cm/ 2. ± 0. 5 cm = 1. 2 ± 37% 4. 18-year-old virile: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% 5. 45-year-old virile: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% How rest each end is to Phi: |1. 618-Actual Quotient|=destruction among end and Phi The destruction among each quotient and 1. 618: 1. 4-year-old womanish: |1. 618- 2 ± 75%| = 0. 382 ± 75% 2. 8-year-old virile: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% 3. 18-year-old womanish: |1. 618- 1. 2 ± 37%| = 0. 418 ± 37% 4. 18-year-old virile: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% 5. 45-year-old virile: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% Percentage Error: Difference1. 18 x100=Percentage destruction among end and Phi 1. 4-year-old womanish: 0. 382 ± 75%/1. 618 x 100 = 23. 6 ± 17. 7% 2. 8-year-old virile: 0. 118 ± 42%/1. 618 x 100 = 7. 3 ± 3. 1% 3. 18-year-old womanish: 0. 418 ± 37%/1. 618 x 100 = 25. 8 ± 9. 5% 4. 18-year-old virile: 0. 118 ± 42%/1. 618 x 100 = 7. 3 ± 3. 1% 5. 45-year-old virile: 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: After a while this proportion, 3 of the ends end out after a while a <10% percentage untruth, mediuming they are very rest to Phi (1. 618).
In the measurements, 3 of the participants had the identical proportion of 3:2. This end is truly sensational owing 3 and 2 are endow in the Fibonacci Series. This proves that the Fibonacci succession is cognate to the Golden Ratio. The foremost end endow was 2:1; these are too Fibonacci quantity. Both Fibonacci quantity and the Golden Proportion were seen in the proportion. RATIO 3: RATIO OF THE LENGTH OF THE FOREARM TO THE LENGTH OF THE HAND Measurements Participant| Diffusiveness of forearm (±0. 5 cm)| Diffusiveness of agency (±0. 5 cm)| 4-year-old womanish| 27| 15| 8-year-old virile| 30| 18. 5| 18-year-old womanish| 40| 25| 18-year-male| 48| 28| 5-year-old virile| 50| 31| Ratios: Diffusiveness of forearm Diffusiveness of agency 1. 4-year-old womanish: 27 ± 0. 5 cm/ 15 ± 0. 5 cm = 1. 8 ± 9. 4% 2. 8-year-old virile: 30 ± 0. 5 cm/ 18. 5± 0. 5 cm = 1. 622 ± 4. 4% 3. 18-year-old womanish: 40 ± 0. 5 cm/ 25± 0. 5 cm = 1. 6 ± 3. 7% 4. 18-year-old virile: 48 ± 0. 5 cm/ 28± 0. 5 cm = 1. 714 ± 2. 8% 5. 45-year-old virile: 50 ± 0. 5 cm/ 31± 0. 5 cm = 1. 613 ± 2. 6% How rest each end is to Phi: |1. 618-Actual Quotient|=destruction among end and Phi The destruction among each quotient and 1. 618: 1. 4-year-old womanish: |1. 618- 1. 8 ± 9. 4%| = 0. 182 ± 9. 4% 2. 8-year-old virile: |1. 18- 1. 622 ± 4. 4%| = 0. 004 ± 4. 4% 3. 18-year-old womanish: |1. 618- 1. 6 ± 3. 7%| = 0. 018 ± 3. 7% 4. 18-year-old virile: |1. 618- 1. 714 ± 2. 8%| = 0. 096 ± 2. 8% 5. 45-year-old virile: |1. 618- 1. 613 ± 2. 6%| = 0. 005 ± 2. 6% Percentage Error: Difference1. 618 x100=Percentage destruction among end and Phi 1. 4-year-old womanish: 0. 182 ± 9. 4%/1. 618 x 100 = 11. 2 ± 1. 1% 2. 8-year-old virile: 0. 004 ± 4. 4%/1. 618 x 100 = 0. 2 ± 0. 9% 3. 18-year-old womanish: 0. 018 ± 3. 7%/1. 618 x 100 = 1. 1 ± 4. 1% 4. 18-year-old virile: 0. 096 ± 2. 8%/1. 618 x 100 = 0. 06 ± 0. 1% 5. 45-year-old virile: 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 untruths were <1. 2% detached from Phi, not including the untruth. The barely end that differed was the four-year-old womanish participant’s end, which could be owing she is quiet increaseing. The other 4 ends were very rest to Phi and illusion the Golden Proportion in the ethnical substantiality hinderly correspondently. CONCLUSION AND VALIDITY: The ends of this scrutiny illusion that persons of divergent sizes all feel substantiality proportions that end very rest to similaring the Golden Ratio.
When the mean percentage untruths were endow for each of the three tested proportions, none of them were superior than 14. 3 ± 36. 5%. This mediums that all of the percentage untruths were low, thus, all the mean proportions endow were very rest to the expected compute of 1. 618 (Phi). The third proportion, the proportion of the diffusiveness of the forearm to the diffusiveness of the agency, was the proportion restst to the Golden Proportion after a while a percentage untruth of barely 2. 6± 7. 0%. On mean, the proportions were barely about 2. 6% detached from 1. 618. Within the participants, the proportion of forearm to agency was most-violently rest to similaring Phi.
This proves the subject that the Golden Proportion can be endow in this individuality of the ethnical substantiality. Looking at each of the participants partially, the 4-year-old womanish had the foremost percentage untruth in two of the three proportions that were tested. In two of the proportions, the 45-year-old virile had the last percentage untruth. In entire proportion, the 45-year-old virile had a significantly inferior percentage untruth than the 4-year-old womanish, and it was too plum that as age went up, the percentage untruth decreased. This suggests that as persons increase, their substantiality proportions increase restr to the Golden Ratio.
The proportion after a while the foremost mean percent untruth was the proportion of the intermediate individuality of the finger to the inferior individuality of the finger. Well-balanced though it had the foremost percentage untruths, it did feel the most relative to the Fibonacci Series, which has proven to be restly cognate to the Golden Ratio. Three of the participants had 3cm and 2cm for their measurements; 3 and 2 are notorious as Fibonacci quantity. Another participant had 1 and 2, which are too Fibonacci quantity. In this scrutiny, it was endow that the Golden Proportion is very rest to the measurements of proportions endow in the ethnical substantiality.
There was capacity for untruth in this scrutiny. The participants had a extensive total of diversity and all of them differed in age and gender. If replicated, this test would avail from further participants of the identical age and gender so their ends can be compared and can too be considered further weighty and operative to be generalized. This scrutiny tested three substantiality proportions for the Golden Ratio, the proportion of guide to bottom and guide to fingertips, intermediate individuality of condemnation finger to last individuality of condemnation finger and forearm to agency.
There are multifarious other Phi proportions that can be dared in the ethnical substantiality. In enjoin to get accomplish a construction of the Golden Proportion in the ethnical substantiality, other proportion’s should be testes, such as the ones endow in the ethnical aspect. In this scrutiny extrinsic proportions were endow. When looking at the misrecord and decomposition of the proportions tested after a while the restricted participants, it is plum that adults feel substantiality proportions restr to the Golden Ratio, making consequence feel a possibility to be considered outliers and communicate extrinsic ends.
If this were to be conducted again, the eldership of participants would be balance the age of 18 years, or consequence could be measured in a totally disjoined test. REFERENCES: Jovanovic, Radoslav. "The Golden Individuality and The Ethnical Body. " Rasko Jovanovic's Globe of Mathematics. 2001. Web. 22 Feb. 2012. Knott, Dr. Ron. "Who Was Fibonacci? " Fibonacci Quantity 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 Proportion / Divine Proportion and Fibonacci Numbers. PhiPoint Solutions, LLC. , 1997. Web. 22 Feb. 2012. PhiPoint Solutions, LLC. "The Ethnical Body. " Ethnical Substantiality 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>.