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, also known as The Divine Proportion, The Golden Mean, or Phi, is a constant that can be seen all throughout the mathematical world. This irrational number, Phi (? ) is equal to 1. 618 when rounded. It is described as “dividing a line in the extreme and mean ratio”. This means that when you divide segments of a line that always have a same quotient. When lines like these are divided, Phi is the quotient: When the black line is 1. 18 (Phi) times larger than the blue line and the blue 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 found in many different places and situations all over the world. It is seen in architecture, nature, Fibonacci numbers, and even more amazingly,the human body. Fibonacci Numbers have proven to be closely related 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 two before it.

The term number is known as ‘n’. The first 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 third 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, 144… 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| Second

Term 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 Fibonacci Series are found all over the world and are lines that can be divided to find Phi. One interesting place they are found is in the human body. Many examples of Phi can be seen in the hands, face and body. For example, when the length of a person’s forearm is divided by the length of that person’s hand, the quotient is Phi.

The distance from a person’s head to their fingertips divided by the distance from that person’s 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 tested 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 expected. The aim of this investigation is to find examples of the number 1. 618 in different people and investigate other places where Phi is found.

Three ratios will be compared. The ratios investigated are the ratio of head to toe and head to fingertips, the ratio of the lowest section of the index finger to the middle section of the index finger, and the ratio of forearm to hand. FIGURE 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 parts of people of different ages and genders were measured in centimeters.

Five people were measured and each participant had these parts measured: * Distance from head to foot * Distance from head to fingertips * Length 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 difference was found for each result. PARTICIPANTS: The people were of different ages and genders. For variety, a 4-year-old female, 8-year-old male, 18-year-old female, 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/female| 8/male| 18/female| 18/male| 45/male| Distance from head to foot| 105. 5| 124. 5| 167| 180| 185| Distance from head to fingertips| 72. 5| 84| 97| 110| 115| 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| 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| Distance from head to foot (±0. 5 cm)| Distance from head 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 expected 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. 5±0. 5 cm = 1. 455 ± 1. 2% 2. 8-year-old male: 124. 5±0. 5 cm/ 85±0. 5 cm = 1. 465 ± 1. 0% 3. 18-year-old female: 167±0. 5 cm/ 97±0. 5 cm = 1. 722 ± 5. 2% 4. 18-year-old male: 180±0. 5 cm/ 110±0. 5 cm = 1. 636 ± 1. 0% 5. 45-year-old male: 185±0. 5 cm/ 115±0. 5 cm = 1. 609 ± 0. 7% How close each result is to Phi: This shows the difference between the actual quotient, what was measured, and the expected quotient (1. 618). This is found by subtracting the actual quotient from Phi and using the absolute value 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% AVERAGE: 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 Phi

RATIO 2: RATIO OF THE MIDDLE 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. 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: With this ratio, 3 of the results come out with a <10% percentage error, meaning they are very close to Phi (1. 618).

In the measurements, 3 of the participants had the same ratio of 3:2. This result is quite interesting because 3 and 2 are found in the Fibonacci Series. This proves that the Fibonacci series is related to the Golden Ratio. The first result found was 2:1; these are also Fibonacci numbers. Both Fibonacci numbers and the Golden Ratio were seen in the ratio. RATIO 3: RATIO OF THE LENGTH OF THE FOREARM TO THE LENGTH OF THE HAND Measurements Participant| Length of forearm (±0. 5 cm)| Length of hand (±0. 5 cm)| 4-year-old female| 27| 15| 8-year-old male| 30| 18. 5| 18-year-old female| 40| 25| 18-year-male| 48| 28| 5-year-old male| 50| 31| Ratios: Length of forearm Length of hand 1. 4-year-old female: 27 ± 0. 5 cm/ 15 ± 0. 5 cm = 1. 8 ± 9. 4% 2. 8-year-old male: 30 ± 0. 5 cm/ 18. 5± 0. 5 cm = 1. 622 ± 4. 4% 3. 18-year-old female: 40 ± 0. 5 cm/ 25± 0. 5 cm = 1. 6 ± 3. 7% 4. 18-year-old male: 48 ± 0. 5 cm/ 28± 0. 5 cm = 1. 714 ± 2. 8% 5. 45-year-old male: 50 ± 0. 5 cm/ 31± 0. 5 cm = 1. 613 ± 2. 6% 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- 1. 8 ± 9. 4%| = 0. 182 ± 9. 4% 2. 8-year-old male: |1. 18- 1. 622 ± 4. 4%| = 0. 004 ± 4. 4% 3. 18-year-old female: |1. 618- 1. 6 ± 3. 7%| = 0. 018 ± 3. 7% 4. 18-year-old male: |1. 618- 1. 714 ± 2. 8%| = 0. 096 ± 2. 8% 5. 45-year-old male: |1. 618- 1. 613 ± 2. 6%| = 0. 005 ± 2. 6% Percentage Error: Difference1. 618 x100=Percentage difference between result and Phi 1. 4-year-old female: 0. 182 ± 9. 4%/1. 618 x 100 = 11. 2 ± 1. 1% 2. 8-year-old male: 0. 004 ± 4. 4%/1. 618 x 100 = 0. 2 ± 0. 9% 3. 18-year-old female: 0. 018 ± 3. 7%/1. 618 x 100 = 1. 1 ± 4. 1% 4. 18-year-old male: 0. 096 ± 2. 8%/1. 618 x 100 = 0. 06 ± 0. 1% 5. 45-year-old male: 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 errors were <1. 2% away from Phi, not including the error. The only result that differed was the four-year-old female participant’s result, which could be because she is still growing. The other 4 results were very close to Phi and show the Golden Ratio in the human body almost exactly. CONCLUSION AND VALIDITY: The results of this investigation show that people of different sizes all have body ratios that come very close to equaling the Golden Ratio.

When the average percentage errors were found for each of the three tested ratios, none of them were greater than 14. 3 ± 36. 5%. This means that all of the percentage errors were low, thus, all the average ratios found were very close to the expected value of 1. 618 (Phi). The third ratio, the ratio of the length of the forearm to the length of the hand, was the ratio closest to the Golden Ratio with a percentage error of only 2. 6± 7. 0%. On average, the ratios were only around 2. 6% away from 1. 618. Within the participants, the ratio of forearm to hand was extremely close to equaling Phi.

This proves the idea that the Golden Ratio can be found in this section of the human body. Looking at each of the participants separately, the 4-year-old female had the highest percentage error in two of the three ratios that were tested. In two of the ratios, the 45-year-old male had the lowest percentage error. In every ratio, the 45-year-old male had a significantly lower percentage error than the 4-year-old female, and it was also evident that as age went up, the percentage error decreased. This suggests that as people grow, their body ratios grow closer to the Golden Ratio.

The ratio with the highest average percent error was the ratio of the middle section of the finger to the lower section of the finger. Even though it had the highest percentage errors, it did have the most relation to the Fibonacci Series, which has proven to be closely related to the Golden Ratio. Three of the participants had 3cm and 2cm for their measurements; 3 and 2 are known as Fibonacci numbers. Another participant had 1 and 2, which are also Fibonacci numbers. In this investigation, it was found that the Golden Ratio is very close to the measurements of ratios found in the human body.

There was room for error in this investigation. The participants had a large amount of variation and all of them differed in age and gender. If replicated, this experiment would benefit from more participants of the same age and gender so their results can be compared and can also be considered more valid and able to be generalized. This investigation tested three body ratios for the Golden Ratio, the ratio of head to foot and head to fingertips, middle section of index finger to lowest section of index finger and forearm to hand.

There are many other Phi ratios that can be investigated in the human body. In order to get obtain a understanding of the Golden Ratio in the human body, other ratio’s should be testes, such as the ones found in the human face. In this investigation extraneous ratios were found. When looking at the conclusion and analysis of the ratios tested with the specific participants, it is evident that adults have body ratios closer to the Golden Ratio, making children have a possibility to be considered outliers and give extraneous results.

If this were to be conducted again, the majority of participants would be over the age of 18 years, or children could be measured in a completely separate experiment. REFERENCES: Jovanovic, Radoslav. “The Golden Section and The Human Body. ” Rasko Jovanovic’s World of Mathematics. 2001. Web. 22 Feb. 2012. Knott, Dr. Ron. “Who Was Fibonacci? ” Fibonacci Numbers 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 Ratio / Divine Proportion and Fibonacci Numbers. PhiPoint Solutions, LLC. , 1997. Web. 22 Feb. 2012. PhiPoint Solutions, LLC. “The Human Body. ” Human Body 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>.