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, too disclosed as The Divine Proportion, The Golden Mean, or Phi, is a perpetual that can be seen all throughout the veracious globe. This irrational calculate, Phi (? ) is similar to 1. 618 when rounded. It is picturesquely as "dividing a course in the remote and balance fitness". This balances that when you deal-out segments of a course that frequently entertain a selfselfidentical quotient. When courses enjoy these are deal-outd, Phi is the quotient: When the sombre course is 1. 18 (Phi) times vastr than the sky bluish-colored-colored course and the sky bluish-colored-colored course is 1. 618 times vastr than the red course, you are conducive to meet Phi. What makes Phi such a veracious rarity is how frequently it can be rest in manifold irrelative fixs and situations all aggravate the globe. It is seen in edifice, constitution, Fibonacci calculates, and flush further amazingly,the civilized assemblage. Fibonacci Aggregate entertain proven to be air-tight kindred to the Golden Ratio. They are a train of calculates discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, integral calculate is the sum of the two precedently it. The message calculate is disclosed as ‘n’. The primitive message is ‘Un’ so, in classify to meet the present message in the regulate, the latest two Un and Un+1 are assumed. (Knott). Formula: Un + Un+1 = Un+2 Example: The promote message (U2) is 1; the third message (U3) is 2. The fourth message 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 message in the Fibonacci Train is deal-outd by the message precedently it, the quotient is Phi, delay the exclusion of the primitive 9 messages, which are peaceful very plug to similaring Phi. Message (n)| Primitive Message Un| Second Term Un+1| Promote Term/First Message (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 prosper the Fibonacci Train are rest all aggravate the globe and are courses that can be deal-outd to meet Phi. One sensational fix they are rest is in the civilized assemblage. Manifold illustrations of Phi can be seen in the influences, visage and assemblage. For illustration, when the prolixity of a person’s forearm is deal-outd by the prolixity of that person’s influence, the quotient is Phi. The interspace from a person’s leader to their fingertips deal-outd by the interspace from that person’s leader to their junctions similars Phi. (Jovanovic). Accordingly Phi is rest in so manifold primordial fixs, it is designated the Divine fitness. It can be tested in a calculate of ways, and has been by diverse scientists and mathematicians. I entertain clarified to summon the Phi perpetual and its air in the civilized assemblage, to meet the fitness in irrelative sized commonalty and see if my fruits companion what is expected. The aim of this ventilation is to meet illustrations of the calculate 1. 618 in irrelative commonalty and summon other fixs where Phi is rest. Three fitnesss earn be compared. The fitnesss summond are the fitness of leader to toe and leader to fingertips, the fitness of the weakest minority of the abjuration finger to the intermediate minority of the abjuration finger, and the fitness of forearm to influence. FIGURE 1 FIGURE 2 FIGURE 3 The primitive fitness is the clear course in the to the active sky bluish-colored-colored course in FIGURE 1 The promote fitness is the fitness of the sombre course to the sky bluish-colored-colored course in FIGURE 2 The third fitness is the fitness of the active sky bluish-colored-colored course to the sombre sky bluish-colored-colored course in FIGURE 3 METHOD: DESIGN: Biased assemblage compressiveness of commonalty of irrelative ages and genders were measured in centimeters. Five commonalty were measured and each participant had these compressiveness measured: * Interspace from leader to pavement * Interspace from leader to fingertips * Prolixity of weakest minority of abjuration finger * Prolixity of intermediate minority of abjuration finger * Interspace from junction to fingertips * Interspace from wrist to fingertips The fitnesss were rest, to see how plug their quotients are to Phi (1. 618). Then the percentage distinction was rest for each fruit. PARTICIPANTS: The commonalty were of irrelative ages and genders. For dissimilarity, 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 ventilation delay the fitnesss rest and how plug they are to the perpetual Phi are analyzed. The fruits were put into tables by each set of measurements and the fitnesss were rest. DATA: | Participant Measurement (± 0. 5 cm)| Measurement| 4/female| 8/male| 18/female| 18/male| 45/male| Interspace from leader to pavement| 105. 5| 124. 5| 167| 180| 185| Interspace from leader to fingertips| 72. 5| 84| 97| 110| 115| Prolixity of weakest minority of abjuration finger| 2| 3| 3| 3| 3| Prolixity of intermediate minority of abjuration finger| 1. 2| 2| 2. 5| 2| 2| Interspace from junction 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 leader to pavement (±0. 5 cm)| Interspace from leader 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 primordial quotients that were rest from the measurements. According to the Golden Ratio, the expected quotients earn all similar Phi (1. 618). Interspace from leader to pavementDistance from leader 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 plug each fruit is to Phi: This pretences the distinction betwixt the express quotient, what was measured, and the expected quotient (1. 618). This is rest by subtracting the express quotient from Phi and using the despotic rate to get the distinction so it does not yield a denying exculpation. |1. 18-Actual Quotient|=distinction betwixt fruit and Phi The distinction betwixt 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 meet how plug the fruits are to the expected rate of Phi, percentage untruth can be used. Percentage untruth is how plug tentative fruits are to expected fruits. Percentage untruth is rest by dividing the distinction betwixt each quotient and Phi by Phi (1. 618) and multiplying that fruit by 100. This yields you the distinction of the express quotient to the expected quotient, Phi, in a percentage. (Roberts) Difference1. 618 x100=Percentage distinction betwixt fruit 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 primary percentage untruth, the percent distinction betwixt the fruit and Phi, is 10. 1 ± 0. 12%. This is a weak percentage untruth, and balances that all but one of the fitnesss was further than 90% servile. This is a good-natured-natured illustration of the Golden Fitness in the civilized assemblage accordingly all the rates are plug to Phi. Also, as the age of the participants increases, the percentage untruth decreases, so as commonalty get older, the fitness of their leader to feet to the fitness of their leader to fingertips gets plugr to Phi RATIO 2: RATIO OF THE MIDDLE SECTION OF THE INDEX FINGER TO THE BOTTOM SECTION OF THE INDEX FINGER Measurements Participant| Prolixity of weakest minority of abjuration finger (±0. 5 cm)| Prolixity of intermediate minority of abjuration 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: Prolixity of weakest minority of abjuration finger Prolixity of intermediate minority of abjuration 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 plug each fruit is to Phi: |1. 618-Actual Quotient|=distinction betwixt fruit and Phi The distinction betwixt 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 distinction betwixt fruit 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: Delay this fitness, 3 of the fruits after out delay a <10% percentage untruth, balanceing they are very plug to Phi (1. 618). In the measurements, 3 of the participants had the selfselfidentical fitness of 3:2. This fruit is altogether sensational accordingly 3 and 2 are rest in the Fibonacci Series. This proves that the Fibonacci train is kindred to the Golden Ratio. The primitive fruit rest was 2:1; these are too Fibonacci calculates. Both Fibonacci calculates and the Golden Fitness were seen in the fitness. RATIO 3: RATIO OF THE LENGTH OF THE FOREARM TO THE LENGTH OF THE HAND Measurements Participant| Prolixity of forearm (±0. 5 cm)| Prolixity of influence (±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: Prolixity of forearm Prolixity of influence 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 plug each fruit is to Phi: |1. 618-Actual Quotient|=distinction betwixt fruit and Phi The distinction betwixt 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 distinction betwixt fruit 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% far from Phi, not including the untruth. The barely fruit that differed was the four-year-old womanish participant’s fruit, which could be accordingly she is peaceful developing. The other 4 fruits were very plug to Phi and pretence the Golden Fitness in the civilized assemblage obstructly correspondently. CONCLUSION AND VALIDITY: The fruits of this ventilation pretence that commonalty of irrelative sizes all entertain assemblage fitnesss that after very plug to similaring the Golden Ratio. When the average percentage untruths were rest for each of the three tested fitnesss, none of them were superior than 14. 3 ± 36. 5%. This balances that all of the percentage untruths were low, thus, all the average fitnesss rest were very plug to the expected rate of 1. 618 (Phi). The third fitness, the fitness of the prolixity of the forearm to the prolixity of the influence, was the fitness plugst to the Golden Fitness delay a percentage untruth of barely 2. 6± 7. 0%. On average, the fitnesss were barely about 2. 6% far from 1. 618. Within the participants, the fitness of forearm to influence was remotely plug to similaring Phi. This proves the fancy that the Golden Fitness can be rest in this minority of the civilized assemblage. Looking at each of the participants partially, the 4-year-old womanish had the primary percentage untruth in two of the three fitnesss that were tested. In two of the fitnesss, the 45-year-old virile had the weakest percentage untruth. In integral fitness, the 45-year-old virile had a significantly inferior percentage untruth than the 4-year-old womanish, and it was too clear that as age went up, the percentage untruth decreased. This suggests that as commonalty develop, their assemblage fitnesss develop plugr to the Golden Ratio. The fitness delay the primary average percent untruth was the fitness of the intermediate minority of the finger to the inferior minority of the finger. Flush though it had the primary percentage untruths, it did entertain the most kindred to the Fibonacci Series, which has proven to be air-tight kindred to the Golden Ratio. Three of the participants had 3cm and 2cm for their measurements; 3 and 2 are disclosed as Fibonacci calculates. Another participant had 1 and 2, which are too Fibonacci calculates. In this ventilation, it was rest that the Golden Fitness is very plug to the measurements of fitnesss rest in the civilized assemblage. There was space for untruth in this ventilation. The participants had a vast whole of change and all of them differed in age and gender. If replicated, this exemplification would boon from further participants of the selfselfidentical age and gender so their fruits can be compared and can too be considered further availconducive and conducive to be generalized. This ventilation tested three assemblage fitnesss for the Golden Ratio, the fitness of leader to pavement and leader to fingertips, intermediate minority of abjuration finger to weakest minority of abjuration finger and forearm to influence. There are manifold other Phi fitnesss that can be summond in the civilized assemblage. In classify to get allure a construction of the Golden Fitness in the civilized assemblage, other fitness’s should be testes, such as the ones rest in the civilized visage. In this ventilation unconnected fitnesss were rest. When looking at the quittance and partition of the fitnesss tested delay the biased participants, it is clear that adults entertain assemblage fitnesss plugr to the Golden Ratio, making consequence entertain a possibility to be considered outliers and yield unconnected fruits. If this were to be conducted frequently, the superiority of participants would be aggravate the age of 18 years, or consequence could be measured in a entirely detached exemplification. REFERENCES: Jovanovic, Radoslav. "The Golden Minority and The Civilized Body. " Rasko Jovanovic's Globe 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 Fitness / Divine Proportion and Fibonacci Numbers. PhiPoint Solutions, LLC. , 1997. Web. 22 Feb. 2012. PhiPoint Solutions, LLC. "The Civilized Body. " Civilized Assemblage 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>.