THE GOLDEN RATIO IN THE HUMAN BODY GABRIELLE NAHAS IBDP MATH STUDIES THURSDAY, FEBRUARY 23rd 2012 WORD COUNT: 2,839 INTRODUCTION: The Golden Ratio, so public as The Divine Proportion, The Golden Mean, or Phi, is a invariefficacious that can be seen all throughout the unrythmical globe. This irrational sum, Phi (? ) is correspondent to 1. 618 when rounded. It is forcible as "dividing a continuity in the immoderate and moderation fitness". This moderations that when you allot segments of a continuity that frequently own a selfselfidentical quotient. When continuitys relish these are allotd, Phi is the quotient: When the sombre continuity is 1. 18 (Phi) times extensiver than the sky sky sky cerulean continuity and the sky sky sky cerulean continuity is 1. 618 times extensiver than the red continuity, you are efficacious to perceive Phi. What makes Phi such a unrythmical inquisitiveness is how repeatedly it can be fix in multifarious incongruous settles and situations all balance the globe. It is seen in structure, affection, Fibonacci sums, and plain balance amazingly,the ethnical organization. Fibonacci Aggregate own proven to be endly akin to the Golden Ratio. They are a conconsequence of sums discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, whole sum is the sum of the two anteriorly it.
The vocefficacious sum is public as ‘n’. The primary vocefficacious is ‘Un’ so, in arrange to perceive the present vocefficacious in the consequence, the promiseinal two Un and Un+1 are added. (Knott). Formula: Un + Un+1 = Un+2 Example: The succor vocefficacious (U2) is 1; the third vocefficacious (U3) is 2. The fourth vocefficacious is going to be 1+2, making U3 correspondent 3. Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… When each vocefficacious in the Fibonacci Consequence is allotd by the vocefficacious anteriorly it, the quotient is Phi, delay the separation of the primary 9 vocables, which are stationary very end to correspondenting Phi. Vocefficacious (n)| Primary Vocefficacious Un| Second
Term Un+1| Succor Term/First Vocefficacious (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 thrive the Fibonacci Consequence are fix all balance the globe and are continuitys that can be allotd to perceive Phi. One thrilling settle they are fix is in the ethnical organization. Multifarious models of Phi can be seen in the agencys, aspect and organization. For model, when the protraction of a person’s forearm is allotd by the protraction of that person’s agency, the quotient is Phi.
The space from a person’s top to their fingertips allotd by the space from that person’s top to their flexures correspondents Phi. (Jovanovic). Owing Phi is fix in so multifarious unless settles, it is designated the Divine fitness. It can be tested in a sum of ways, and has been by irrelative scientists and mathematicians. I own selected to question the Phi invariefficacious and its presumption in the ethnical organization, to perceive the fitness in incongruous sized inhabitants and see if my developments pair what is expected. The aim of this scrutiny is to perceive models of the sum 1. 618 in incongruous inhabitants and question other settles where Phi is fix.
Three fitnesss gain be compared. The fitnesss questiond are the fitness of top to toe and top to fingertips, the fitness of the meanest minority of the apostacy finger to the medium minority of the apostacy finger, and the fitness of forearm to agency. FIGURE 1 FIGURE 2 FIGURE 3 The primary fitness is the stainless continuity in the to the easy sky sky sky cerulean continuity in FIGURE 1 The succor fitness is the fitness of the sombre continuity to the sky sky sky cerulean continuity in FIGURE 2 The third fitness is the fitness of the easy sky sky sky cerulean continuity to the black sky sky sky cerulean continuity in FIGURE 3 METHOD: DESIGN: Restricted organization magnitude of inhabitants of incongruous ages and genders were measured in centimeters.
Five inhabitants were measured and each participant had these magnitude measured: * Space from top to base * Space from top to fingertips * Protraction of meanest minority of apostacy finger * Protraction of medium minority of apostacy finger * Space from flexure to fingertips * Space from wrist to fingertips The fitnesss were fix, to see how end their quotients are to Phi (1. 618). Then the percentage disagreement was fix for each development. PARTICIPANTS: The inhabitants were of incongruous ages and genders. For diversity, a 4-year-old womanly, 8-year-old hardy, 18-year-old womanly, 18-year-old hardy and a 45-year-old hardy were measured.
All of the measurements are in this scrutiny delay the fitnesss fix and how end they are to the invariefficacious Phi are analyzed. The developments were put into tables by each set of measurements and the fitnesss were fix. DATA: | Participant Measurement (± 0. 5 cm)| Measurement| 4/female| 8/male| 18/female| 18/male| 45/male| Space from top to base| 105. 5| 124. 5| 167| 180| 185| Space from top to fingertips| 72. 5| 84| 97| 110| 115| Protraction of meanest minority of apostacy finger| 2| 3| 3| 3| 3| Protraction of medium minority of apostacy finger| 1. 2| 2| 2. 5| 2| 2| Space from flexure 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| Space from top to base (±0. 5 cm)| Space from top to fingertips (±0. 5 cm)| 4-year-old womanly| 105. 5| 72. 5| 8-year-old hardy| 124. 5| 85| 18-year-old womanly| 167| 97| 18-year-male| 180| 110| 45-year-old hardy| 185| 115| Ratios: These are the primary quotients that were fix from the measurements. According to the Golden Ratio, the expected quotients gain all correspondent Phi (1. 618). Space from top to baseDistance from top to fingertips 1. 4-year-old womanly: 105. ±0. 5 cm/ 72. 5±0. 5 cm = 1. 455 ± 1. 2% 2. 8-year-old hardy: 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 hardy: 180±0. 5 cm/ 110±0. 5 cm = 1. 636 ± 1. 0% 5. 45-year-old hardy: 185±0. 5 cm/ 115±0. 5 cm = 1. 609 ± 0. 7% How end each development is to Phi: This appearances the disagreement betwixt the express quotient, what was measured, and the expected quotient (1. 618). This is fix by subtracting the express quotient from Phi and using the irresponsible compute to get the disagreement so it does not yield a privative retort. |1. 18-Actual Quotient|=disagreement betwixt development and Phi The disagreement betwixt each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 1. 455 ± 1. 2%| = 0. 163 ± 1. 2% 2. 8-year-old hardy: |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 hardy: |1. 618- 1. 636 ± 1. 0%| = 0. 018 5. 45-year-old hardy: |1. 618- 1. 609 ± 0. 7%| = 0. 009 Percentage Error: To perceive how end the developments are to the expected compute of Phi, percentage deception can be used. Percentage deception is how end tentative developments are to expected developments.
Percentage deception is fix by dividing the disagreement betwixt each quotient and Phi by Phi (1. 618) and multiplying that development by 100. This yields you the disagreement of the express quotient to the expected quotient, Phi, in a percentage. (Roberts) Difference1. 618 x100=Percentage disagreement betwixt development and Phi 1. 4-year-old womanly: 0. 163 ± 1. 2%/1. 618 x 100 = 10. 1 ± 0. 12% 2. 8-year-old hardy: 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 hardy: 0. 018/1. 618 x 100 = 1. 11% 5. 45-year-old hardy: 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 deception, the percent disagreement betwixt the development and Phi, is 10. 1 ± 0. 12%. This is a minute percentage deception, and moderations that all but one of the fitnesss was balance than 90% considerate. This is a cheerful model of the Golden Fitness in the ethnical organization owing all the computes are end to Phi. Also, as the age of the participants increases, the percentage deception decreases, so as inhabitants get older, the fitness of their top to feet to the fitness of their top to fingertips gets endr to Phi
RATIO 2: RATIO OF THE MIDDLE SECTION OF THE INDEX FINGER TO THE BOTTOM SECTION OF THE INDEX FINGER Measurements Participant| Protraction of meanest minority of apostacy finger (±0. 5 cm)| Protraction of medium minority of apostacy finger (±0. 5 cm)| 4 year old womanly| 2| 1| 8 year old hardy| 3| 2| 18 year old womanly| 3| 2. 5| 18 year hardy| 3| 2| 35 year old hardy| 3| 2| Ratios: Protraction of meanest minority of apostacy finger Protraction of medium minority of apostacy finger 1. 4-year-old womanly: 2 ± 0. 5 cm/ 1 ± 0. 5 cm = 2 ± 75% 2. 8-year-old hardy: 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 hardy: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% 5. 45-year-old hardy: 3 ± 0. 5 cm/ 2 ± 0. 5 cm = 1. 5 ± 42% How end each development is to Phi: |1. 618-Actual Quotient|=disagreement betwixt development and Phi The disagreement betwixt each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 2 ± 75%| = 0. 382 ± 75% 2. 8-year-old hardy: |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 hardy: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% 5. 45-year-old hardy: |1. 618- 1. 5 ± 42%| = 0. 118 ± 42% Percentage Error: Difference1. 18 x100=Percentage disagreement betwixt development and Phi 1. 4-year-old womanly: 0. 382 ± 75%/1. 618 x 100 = 23. 6 ± 17. 7% 2. 8-year-old hardy: 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 hardy: 0. 118 ± 42%/1. 618 x 100 = 7. 3 ± 3. 1% 5. 45-year-old hardy: 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 developments follow out delay a <10% percentage deception, moderationing they are very end to Phi (1. 618).
In the measurements, 3 of the participants had the selfselfidentical fitness of 3:2. This development is totally thrilling owing 3 and 2 are fix in the Fibonacci Series. This proves that the Fibonacci conconsequence is akin to the Golden Ratio. The primary development fix was 2:1; these are so Fibonacci sums. Both Fibonacci sums 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| Protraction of forearm (±0. 5 cm)| Protraction of agency (±0. 5 cm)| 4-year-old womanly| 27| 15| 8-year-old hardy| 30| 18. 5| 18-year-old womanly| 40| 25| 18-year-male| 48| 28| 5-year-old hardy| 50| 31| Ratios: Protraction of forearm Protraction of agency 1. 4-year-old womanly: 27 ± 0. 5 cm/ 15 ± 0. 5 cm = 1. 8 ± 9. 4% 2. 8-year-old hardy: 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 hardy: 48 ± 0. 5 cm/ 28± 0. 5 cm = 1. 714 ± 2. 8% 5. 45-year-old hardy: 50 ± 0. 5 cm/ 31± 0. 5 cm = 1. 613 ± 2. 6% How end each development is to Phi: |1. 618-Actual Quotient|=disagreement betwixt development and Phi The disagreement betwixt each quotient and 1. 618: 1. 4-year-old womanly: |1. 618- 1. 8 ± 9. 4%| = 0. 182 ± 9. 4% 2. 8-year-old hardy: |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 hardy: |1. 618- 1. 714 ± 2. 8%| = 0. 096 ± 2. 8% 5. 45-year-old hardy: |1. 618- 1. 613 ± 2. 6%| = 0. 005 ± 2. 6% Percentage Error: Difference1. 618 x100=Percentage disagreement betwixt development and Phi 1. 4-year-old womanly: 0. 182 ± 9. 4%/1. 618 x 100 = 11. 2 ± 1. 1% 2. 8-year-old hardy: 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 hardy: 0. 096 ± 2. 8%/1. 618 x 100 = 0. 06 ± 0. 1% 5. 45-year-old hardy: 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 deceptions were <1. 2% far from Phi, not including the deception. The barely development that differed was the four-year-old womanly participant’s development, which could be owing she is stationary expanding. The other 4 developments were very end to Phi and appearance the Golden Fitness in the ethnical organization approximately precisely. CONCLUSION AND VALIDITY: The developments of this scrutiny appearance that inhabitants of incongruous sizes all own organization fitnesss that follow very end to correspondenting the Golden Ratio.
When the medium percentage deceptions were fix for each of the three tested fitnesss, none of them were elder than 14. 3 ± 36. 5%. This moderations that all of the percentage deceptions were low, thus, all the medium fitnesss fix were very end to the expected compute of 1. 618 (Phi). The third fitness, the fitness of the protraction of the forearm to the protraction of the agency, was the fitness endst to the Golden Fitness delay a percentage deception of barely 2. 6± 7. 0%. On medium, the fitnesss were barely environing 2. 6% far from 1. 618. Within the participants, the fitness of forearm to agency was immoderately end to correspondenting Phi.
This proves the purpose that the Golden Fitness can be fix in this minority of the ethnical organization. Looking at each of the participants individually, the 4-year-old womanly had the foremost percentage deception in two of the three fitnesss that were tested. In two of the fitnesss, the 45-year-old hardy had the meanest percentage deception. In whole fitness, the 45-year-old hardy had a significantly inferior percentage deception than the 4-year-old womanly, and it was so incontrovertible that as age went up, the percentage deception decreased. This suggests that as inhabitants expand, their organization fitnesss expand endr to the Golden Ratio.
The fitness delay the foremost medium percent deception was the fitness of the medium minority of the finger to the inferior minority of the finger. Plain though it had the foremost percentage deceptions, it did own the most connection to the Fibonacci Series, which has proven to be endly akin to the Golden Ratio. Three of the participants had 3cm and 2cm for their measurements; 3 and 2 are public as Fibonacci sums. Another participant had 1 and 2, which are so Fibonacci sums. In this scrutiny, it was fix that the Golden Fitness is very end to the measurements of fitnesss fix in the ethnical organization.
There was extent for deception in this scrutiny. The participants had a extensive totality of alteration and all of them differed in age and gender. If replicated, this trial would behoof from balance participants of the selfselfidentical age and gender so their developments can be compared and can so be considered balance substantial and efficacious to be generalized. This scrutiny tested three organization fitnesss for the Golden Ratio, the fitness of top to base and top to fingertips, medium minority of apostacy finger to meanest minority of apostacy finger and forearm to agency.
There are multifarious other Phi fitnesss that can be questiond in the ethnical organization. In arrange to get conquer a conception of the Golden Fitness in the ethnical organization, other fitness’s should be testes, such as the ones fix in the ethnical aspect. In this scrutiny unconnected fitnesss were fix. When looking at the quittance and decomposition of the fitnesss tested delay the restricted participants, it is incontrovertible that adults own organization fitnesss endr to the Golden Ratio, making consequence own a possibility to be considered outliers and yield unconnected developments.
If this were to be conducted repeatedly, the seniority of participants would be balance the age of 18 years, or consequence could be measured in a fully unconnected trial. REFERENCES: Jovanovic, Radoslav. "The Golden Minority and The Ethnical 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 Ethnical Body. " Ethnical Organization 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>.