This paper reflects the research and thoughts of a student at the time the paper was written for a course at Bryn Mawr College. Like other materials on Serendip, it is not intended to be "authoritative" but rather to help others further develop their own explorations. Web links were active as of the time the paper was posted but are not updated.

## Be Careful What you Believe:A Study of the Number Phi and it's Effects on Beauty

### Malorie Garrett

"So if I don't fit this number, does that mean I am ugly?" Laura said as I calculated the ratio of each of my friends heights in relation to the distance between her navel and her feet. "Because if it does," she said "then that's dumb." The number she was referring to is the number F, Phi, which is an irrational number, approximately 1.61803398874989. A number is considered irrational if it has a decimal which never repeats and never ends. Phi is often called the golden ratio or the golden sections and it is attributed to many different aspects of the natural and human made world. It also has many interesting mathematical properties, which will be described in detail later. Some people believe that the closer one comes to this ratio, the more beautiful you are. So, being the mathematician that I am, I decided to gather some data to see how well me and my friends fit into this ratio. We got together and we measured each other from the belly button up and down to see how closely we fit this ratio ( For data see cart attached). As I started to crunch the numbers, all my friends were very interested to see how well they "measured up" to the "ideal number".

This is when Laura made the comment that one number could not possible determine whither or not someone is beautiful. And while I completely agree with that comment, I do not think we can dismiss Phi and it's relationship with many objects, including that of the human body. But as I started to explain that it did not mean you were ugly, but that does not mean it's "dumb", I realized I was surrounded by English and Anthropology majors. They all looked at me with squinty eyes as I recited some math jargon. Then there was a pause. Marcelina said "You cannot classify peoples beauty by a number!". I sighed heavily and shock my head. That's when I decided the purpose of this paper: to explain the mathematical significance of Phi and it's relation to beauty. Yes, Phi explains a lot of the world, but that does not mean if you don't fit within it, something is wrong with you. More importantly, is it even appropriate to classify the human body in terms of Phi? In order to answer this question and there discover Phi relation to beauty we first must understand Phi mathematically and historical.

Initially called division in extreme mean and ratio, Phi is the irrational number one gets from dividing a line segment in such a way that the ratio of the whole segment to the larger segment is equal to the ratio of the larger segment to the smaller segment. In other words, if you divide the whole line by the larger part it should be the same as dividing the larger segment by the smaller segment. If you are still confused, look at this image:

So AB/AC=AC/CB=F»1.6180339. And if you are feeling particularly mathematical, you can let AB equal 1, in which case you will get (1/X)=X/(1-X) or X² + X -1=0. If you solve this equation for X, you get two roots, the positive root is X=(1+Ö5)/2» 0.6180339, which you may notice is Phi minus one. The reciprocal of X is Phi, 1/X»1.6180339. The other root is X»-1.6180339, negative Phi.

Phi is a very unique number. When you square it, it is the same as adding one to Phi. Phi is also connected to other mathematical properties. One in particular that it is related to is the Fibonacci sequence, which is a string of numbers generated by adding the current number to the previous number, starting with zero and one. So you have 0+1=1, then 1+1=2, then 2+1=3, then 2+3=5, ect. Fibonaccis' numbers are another thing that seems to appear everywhere in nature. The number of most flowers petals are Fibonacci numbers. The reason for that is quite simple, it is "Nature being efficient" (Devlin). The leave or petals of pants arrange themselves in a way so as to obscure the least amount of each other. They are also really important when dealing with fractals. If you take one number in the Fibonacci sequence, starting at around five, and divide it by the previous number, then tend towards Phi. For example, the numbers 233/144» 1.6180. Phi also is closely related to geometry. An isosceles triangle, a triangle with two equal angles and two equal sides, with angles 72°,72°, and 36° contains the golden ratio between a side and the base. One can also create rectangles which fit the golden ratio as well. If you set up a series of rectangles next to each other involving the golden ratio and Fibonaccis' numbers, they will all have the same proportions. This sequence of rectangles creates a unique spiral.

So all that math is great, but what does it mean and how is related to beauty? Well, some believed that the Greeks thought that golden rectangular was the most aesthetically pleasing rectangular. It is also speculated that they used Phi to create the Parthenon and in paintings and sculptures. The uses of the golden ratio has not only been attributed to the Greeks, but the Egyptians and their Pyramids, DaVinci's Mona Lisa, the human body, and even the Washington Monument. And while it would be nice if all of this was true, it is just not very plausible. First off, we can never truly know what the ancients were thinking when they made the pyramids or the Parthenon. More importantly, most of these items mathematically don't add up to reach Phi. One problem is that "measurements of real objects can only be approximations. Surfaces of real objects are not perfectly flat" (Markowsky 5). This means that people who 'see' the golden ratio are working with inaccurate numbers. And while these numbers are only off by a fraction, they add up, and when you work with a number such as Phi with so many decimal places, that little but can through it off a lot. Also, sometimes where they measure of the golden rectangle is arbitrary or they place it in such a way that the entire object in question is not encompassed. The other problem also has to do with people 'seeing' it everywhere. As Martin Garderner explains, the problem here is to much information:

If you set about measuring a complicated structure like the Pyramid, you will quickly have on hand a great abundance of lengths to play with. If you have sufficient patience to juggle the amount in various ways, you are certain to come out with many figures which coincide with important historical dates or figures in the sciences. Since you are bound by no rules, it would be odd indeed if this for Pyramid 'truth' failed to meet with considerable success. Markowsky 5

Geometric shapes lead themselves to finding Phi because of their nature. It is easy, given time and some effort, to arbitrarily calculate the golden ratio. If you look hard enough, you can even see it in the human body.

There are many sites on the internet that would want you to think that Phi can and should be applied to the human form as a way to describe beauty. Dr. Stephen Marquardt claims that the golden ratio can be applied to the human face. He has created a mask that one can place over the human face. Apparently, the closer one fit's the mask, the more beautiful ones face is. Once you apply the mask to your face, Dr. Marquardt believes that one can change ones face either with make-up or cosmetic surgery to become more beautiful. According to his site as well as some others I've found, including The Learning Channels homepage, confirm that he has in truth used phi to create a mask that according to him "radiates, it advertises and screams: 'human, human, human.'" tlc.discovery.com). He says that the most beautiful faces are also the most human looking, and that his mask emulates the most 'human' face. Another site says "Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented" (goldennumber.net). Though he may have done this research, it does not show on his site. Most of the faces are female and look similar, even those from different ethnicities. I was surprised to find some actual mathematical information on his site, so I also do not doubt that he actually calculated this mask from Phi and it's shapes. But what does this mean? That it describes the most beautiful face? He would say it does based on the idea that the golden ratio and the golden rectangle is the most ascetically pleasing shape and that it appears everywhere, which is not necessarily true.

But perhaps I am being biased because he is a plastic surgeon. Another site that advertise the "perfect face" also describes the length of the body has containing the golden ratio. Easier and less judgmentally than applying the mask, I decided to round up some test subjects to see how true it was for the average Bryn Mawr girl. This is what I got:

Name AB AC CB AB/AC AC/CB Ethnicity

Ideal 1.618034 1.618034
Tonda 61 35 25 1.742857 1.400000 Japanese and White
Marcelina 62.7 36.2 26.5 1.732044 1.366038 Mexican and White
Laura 63.3 36.8 26.5 1.720109 1.388679 Dutch and Canadian
Malorie 66.5 40 26.5 1.662500 1.509434 White
Emily 64 38.5 25.5 1.662338 1.509804 White (Greek heritage)
Sruti 62.5 37 25.5 1.689189 1.450980 Indian

As you can see, across ethnicities, most of my friends do not add up correctly to fit the ratio. Only half of us even got within the first decimal point of 1.6 in the first calculation of AB/AC. But still, the number are kind of close to Phi. So then I calculated AC/AB, because according to the formula AB/AC=AC/CB=F»1.6180339. While I was never expecting these numbers to work out perfectly, I was expecting AB/AC to roughly equal to AC/CB. But they are not. Me and Emily measurements are the closest, we are off only by a tenth, and Marcelina and Laura's measurements are off the most at four tenths. At this point I felt I did not need to calculate the percentage differences between AB/AC and AC/CB because the numbers are so off. And although my numbers and calculations are by no means error free, the numbers I got tell me that the assumption that human body's fit into the golden ratio is unfounded. All of the mathematical papers of the golden ratio I read confirm my assumption. Mathematician Keith Devlin says on the matter in an article entitled "Good stories, pity they're not true" :

First of all, you won't get exactly the number GR. You never can; GR is irrational, remember. But in the case of measuring the human body, there is a lot of variation. True, the answers will always be fairly close to 1.6. But there's nothing special about 1.6. Why not say the answer is 1.603? Besides, there's no reason to divide the human body by the navel. If you spend a half an hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6, or 1.5, or whatever you want. Devlin

Like the measuring of the Pyramids, the only people who see the golden ratio in the human forms are those who want to see it.

Astrophysicist Mario Levio says that "Mathematics is a human invention . . .but nature dictated to human beings what mathematics to invent" (Bramanti). This statement really explains the reasoning of math to me. It was created, like science, in order to explain the patterns humans where seeing in nature. In both fields, we do research, gather data, and then create hypotheses and sometimes even theorems to explain something. When we find that the conclusions we came to were wrong or misinformed, we change it or throw it out completely. Phi is no different than other mathematical equation in that sense. There are things we know for sure: it's unique mathematical properties, it's relation to Fibonaccis' numbers, and it's relation to plants in nature. But besides that, all the rest is not true or else up for speculation. To me, Phi is an exciting number and it makes me feel good that it has all of these properties. I like the relation it has to nature and the world. It makes me feel like we are justified in our creation and continued struggle with math. To me, that is what is beautiful about Phi. The number itself is not beautiful; in fact I find the repeating decimal both ugly and annoying. Also, to say that it creates the most beautiful shapes and that those shapes can be combined to the perfect face is unfounded and unethical. Dr. Marquardt misuses the math, and for what? Plastic surgery. Phi cannot tell you how beautiful you are or if you have the correct proportions because there is no such thing. It is closely related to nature, so we may in the future find evidence of Phi in the human body, but at time there is not sufficient evidence to confirm that there is a relation. What you have to remember is that information in the wrong hands can be misused and misrepresented, especially in this age of the world wide web. So as you search the world wide web, note who is publishing the information: a mathematician or a plastic surgeon?

### References

2) Bramanti, Matt. "U.Notre Dame: Author promotes 'Golden Ratio'". The America's Intelligence Wire. Feb. 5 2003. May 2005.

4) Maor, Eli. "Symbol of perfect proportions" Science. 299.5609 (2003): 1016(1).

5)Markowsky, George. "Misconceptions about the Golden Ratio". The College Mathmatics Journal. 23.1 (1992): 2-19.