A Study of Phi and its Importance in Human Choices Concerning Beauty

By: Anthony McCabe

Abstract
This paper aims to answer what Phi is, where it is found in nature, and how it affects humans concerning our search for beauty. This is done through graphically and mathematically finding Phi, and identifying its unique properties. The history of Phi is explored, and its usage in the past is covered. Phi is then applied to nature, through its presence in the Golden Angle, nature, and architecture. Phi is then explored in human nature, when it comes to our physique and psychological choices. This leads to a conducted survey showing human wants in facial appearance, relevant to Phi. The results show a significant amount of people prefer the face closest related to Phi, supporting the hypothesis that Phi plays an important role in human beauty. Phi is found to be a mathematical phenomenon that predates even math itself, and has always been useful to societies and to nature itself. Phi is found everywhere in our world and makes objects and patterns seem more elegant because of its presence. This is relevant to humans as well, as, concerning beauty, Phi is a powerful measurement that psychologically attracts us at our most basic and primitive levels.

A Study of Phi and its Importance in Human Choices Concerning Beauty

One object, one thing, can be viewed in many different ways by many different types of people. For example, a piece of wood is a tool, or a building block for a carpenter. It’s a way to make a profit for the lumberjack. It’s a chain of glucose molecules for a biologist. We have our ways to view things in life depending on who we are, and how we function. Beauty, like all other things, is therefore a relative subject. A mask with strange face painting on may be incredibly beautiful to an artist, yet strange, unfamiliar, and maybe even scary to someone who doesn’t study that type of artistry.

As humans we look for beauty not only in things, but in each other. We look at what attracts us, the symmetry on a face, and the way a person’s eyes look. The amount of small tidbits in people that can possibly attract us is nearly endless. Therefore the constant search for a quantitative way to measure beauty has been a long and arduous task, one which is still debated by many. But one thing has come out of it, one mathematical ratio that has been viewed as the ratio of beauty, the ratio of good, or more commonly, The Golden Ratio, Phi. The golden ratio is not just a mathematical formula for structural beauty throughout history, but also a psychological attraction for humans that helps steer us towards other people. What is Phi, where and why is it in nature, and how does it affect human choices?

II. The Basics

What is Phi? This question is harder to answer than most may believe. Like Pi, or 3.14… Phi is an irrational number that never ends. Its first seven digits are 1.618033, but it keeps on going (Knott). This mathematical ratio is used in numerous ways, for many different purposes. One of the more common uses is the idea of the Golden Section. This is used with line segments interacting with each other. The figure on the right shows the idea of the Golden Section. It is showed by the ratio of the whole line (A) to the large segment (B), which is equal to the ratio of the large segment (B) to the smaller segment (C). That may be confusing, but a simpler explanation is A is to B as B is to C. This seems pretty strange, that randomly these segments will have the same proportions to each other. This phenomenon can only happen if A is 1.618… times larger than B, and B is 1.618… times larger than C. (The Golden Number) Here Phi shows some of its incredible power. Because the segments are ratios of each other, they must have the Golden Ratio in them.
The above was an illustrated description; now let’s look at the mathematics behind it. To find Phi, you can use a simple equation using exponents. The equation is x2-x-1=0.

This simple equation can be graphically representated using a Graphing Calculator, and then find its roots or zeros. The left side of the Y-Axis will show -.618034… while the right side will show 1.618034… This is Phi and its inverse. In fraction form, 1.618034 is 5+12. This is the most common equation used when calculation with Phi (Knott).
This equation can be used with the previous example of the ratios of A, B and C. If ordinary number are plugged in, with the prerequisites in mind (A=B+C and AB=BC ) Then a numbers like A=3, B=2, and C=1 can be used for A=B+C, but it doesn’t follow the rule of the ratio. In order for the ratio to work, B must =C(1.618…) and A must=B(1.618…). Using a calculator, and the arbitrary number C=2, you can get: B=2(1.618…), B=3.24… and A=B(1.618…), A=5.24… Rounding to the nearest hundredths. Therefore, 5.24=3.24+2, which is correct, and 5.243.24=3.242, which works out if the number for B and A are stored in their decimal format. (To find 1.618 in this exercise, enter 512+12 for phi and use the radical in order to find B and A)

III. Origins of Phi
There is no real way to say when we first found Phi. The ratio has been there forever, and has appeared in nature throughout time. However, the ratio itself has been discovered, forgotten, rediscovered, and forgotten once again over and over in history, and that is probably why it goes by so many different names.

One of the earliest discoveries of Phi is in the great pyramids. It has been theorized that if you draw a straight line from the tip of the pyramid to the center of the base, and then one parallel to the ground out to one of the sides, the right angle triangle created has sides, which display Phi. This supports the idea that we instinctively follow Phi when we want beauty, even before any of these measurements could be carried out (Great Golden Pyramid).

Phidias, the creator of the Parthenon, studied Phi. His creation is said to show Phi in the measurements of rectangles (Patterns of Visual Math) If you look at the front of the Parthenon, there are three distinct sections: the left side, which is made up of three columns, the middle which is made up of two, and the right section, which is another three columns. However, if you split up the sides into rectangles, and then measure, you can see that the two extreme sides are not the same measurement, and actually show Phi when compared to the middle section, as described previously (Phi, 1.618).

One of the first mathematicians to show Phi was Fibonacci. His series displays Phi by relating the preceding number to the following. The law of the series is that the next number is the sum of the previous two. The series starts with 0, then 1, then follows the predetermined parameters. A few first digits are: 0,1,1,2,3,5,8,13, etc. If we take any two numbers besides the first 0 and 1, and relate it to the last via making it into a ratio, you will find something approximate to Phi. For example: 53=1.6667… and 85=1.6. This continues throughout the entire series, and actually gets closer and closer to Phi the closer the numbers get to infinity ().

The Golden Ratio has also been “found” and explored by many throughout history. For example, in Ancient Greece, Phi was discovered in the pentagram, or five-pointed star. The five-pointed star is famous for being the sign of the Pythagorean Brotherhood (The Golden Rectangle). Euclid referred to the Golden Ratio as “extreme and mean ratio” and he discovered a way to construct it by using a compass and a straightedge (The golden Section). During the Renaissance, Pacioli created The Divine Proportion, which included many geometric shapes that showed the Golden Ratio, and which was illustrated by Leonardo da Vinci. Pacioli believed that Phi was a message from God, and a source of secret knowledge about the inner beauty of things (A Brief History of Phi).

The Golden Ratio is a universal law “which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.” Adolf Zeising, late nineteenth century. Zeising was the first person to claim that the Partheon was the shape of the Golden Rectangle, which in turn inspired Mark Barr, an American mathematician, to name the ratio Phi after Phidas (Bellos 298).

IV. Phi in Life There are a couple of different applications of Phi that makes it even more amazing. One such application is the Logarithmic Spiral, or spira mirabilis (marvelous spiral). This is approximately found by slicing a paper into the Golden Rectangle, as described in The Basics. After the initial slice, take the smaller side of the paper, and slice that one into another Golden Rectangle. Keep doing so with each smaller side, and you will get smaller and smaller rectangles that circle in on themselves.
The next step is to take a compass, and creating an origin in the bottom right hand of the first square, you draw a curve to fit in the square. This connects to the second piece at the top right hand corner. In the next smaller square, you put the compass in the bottom left hand corner, and draw another curve to connect with the third. Then so on and so forth. The spiral created once they are all connected is very close to the Logarithmic Spiral (Bellos 293).

The spira mirabilis is everywhere in nature, in our world. The nautilus shell is in the shape of a spira mirabilis. The flight patterns of falcons, as Vance Tucker of Duke University realized, descend on their prey in the shape of the spira mirabilis (The Golden Number). Even a galaxy, for example the Whirlpool Galaxy, is also in the shape of the spira mirabilis. It’s amazing to think that this spiral is so perfect for so many different jobs. The nautilus shell grows each section larger than the last to accommodate for its host, and the shape is perfect using the logarithmic spiral. Falcons decent in Phi pattern because of the position of their eyes, on the side of their head, and how decent in the shape of the spira mirabilis gives them a view of their target at all times (The Golden Number). Naturally, evolution chooses Phi because of its amazing properties, and its perfect fit in nature.

My personal favorite example of Phi in nature is the growing of leaves on a plant. Leaves grow in a very specific way in order to capture the most amount of sunlight (Bellos 296). Picture for a moment a stem growing straight up, barren, without any leaves on it. If the first leaf grew out, where would the next leaf be placed in order to maximize the amount of sun absorbed by the plant? Most would assume a quarter turn, or a half turn, away. But, take for example the quarter turn; if you continue this pattern, then every fifth leaf would be on top of the first. The sixth leaf would be on top of the second, and so forth. What is the best positioning of the leaves in order to maximize sunlight? The answer: 137.5 degrees.

137.5 degrees is what you get if you split a circle according to Phi. It is considered the Golden Angle, and the opposite angle is 222.5 degrees. To find this, the same a and b ratios can be used as in the beginning of this essay, however instead of being the two lengths of a line segment, it is the two arcs of a circle. A pictorial representation is as follows:

As with the line segment, if you take the whole of the circle (a+b) and divide it by the larger arc (a) such that it equals the ratio between the larger and smaller arcs (ab) then the only number that will make a+ba=ab is phi. The golden angle would be the angle subtending arc b. Mathematically, you can find this by:

The ratio of a to b is Phi. A is the same as the B is the same as the length Of circumfrance minus the arc, or theta times radius. The length of arc b

Eliminate the r’s to get:

Inversing in order to isolate Theta.

One of the amazing things about Phi, and most things associated with it, is that it is an irrational number. 1.618 doesn’t stop there, it continues out ad infinitum. 137.5 also continue forever and ever, and therefore won’t repeat, even if you turn it around and around over and over. That’s why each leaf is a 137.5 degree turn from the last, approximately. It allows each leaf to get sunlight even after the tenth rotation. Phi can also be found in the human body. To be more precise, it is found pretty much all over the human body. The proportions of different lengths compared to each other tend to come out approximately 1.618. Here are a couple of measurements: * The ratio of the length of the whole body as compared to the length from the head to the finger tips. * The ratio of the length from top of head to finger tips as compared to the length from the top of the head to the belly button/elbows. * The ratio of the length from the top of the head to the belly button as compared to the length of the forearms, the length of the shin bone, width of the shoulders, length from the inside top of one arm to the other, and the length from the head to the pectorals. * Length of the previous measurements, compared to the length of the top of the head to the base of the skull and the width of the abdomen.
(Phi, 1.618).
These lengths, on a normal, healthy human being, should be approximately 1.618. The amounts of ratios you can find are staggering, though not in themselves proof of anything but coincidence. The ratio of 1.618 is common, seeing as it’s just a bit more than 1:1.5. That is one thing being less than twice as long as something else, a useful ratio to have if you are thinking about, say, your upper torso to your whole body. However, the amount of times it pops up suggests that it may be something more than just coincidence (Patterns of Visual Math). Another instance of the Golden Ratio that may be interesting, though probably not as relevant, is the design of the human body. Our torso has five appendages (arms, legs and head), each of these appendages have five of their own (5 fingers on each hand, five toes on each foot, five openings on the face), and we have five senses (sight, sound, touch, taste and smell). The golden ratio can be found by the mathematical formula of fives: 5.5+.5=Phi, Using the four fives stated above. Once again, an interesting formula, though if it proves or states anything of importance is less prevalent.

V. Survey
Average
Attractive
Beautiful
I decided to test out a theory laid out by Marquardt Beauty Analysis (MBA). They propose that as humans we automatically attempt to find people with Golden Ratio features. Their webpage posts what are supposed to be standards of different degrees of beauty, ranging from unattractive to beautiful. For the sake of the survey I gave to the students, I used three standards; average, attractive and beautiful. The test was to see if students would gravitate towards the more attractive features, and if given a single choice between the three, would chose the one closest to Phi (which, according to MBA, is considered Beautiful). The results were incredibly favorable to the initial hypothesis, and supported the theory proposed by MBA. Each face was given a number, with Attractive being number one, Average being number two, and Beautiful being number three. The face that was the closest to phi ratio, the face labeled “Beautiful” got the most votes. The results are as follows: | Face | Number of Votes | Average | Attractive | Beautiful | | 1 | 10 | 50 |

This data can also be shown in a pie graph, which will show the percentage that chooses certain faces.
As shown with the 82% favour of the “Beautiful” face, angles and lines closer to Phi seems to be deemed more attractive by the public. This is one of the reasons why MBA does plastic surgery according to the Phi ratio. It is believed to make people more beautiful and add to their overall attractiveness. This survey follows the initial question of Phi’s relevance to human choices, and human nature overall. We instinctively choose Phi when we have a choice; we follow it and want it.

VI. Conclusion
Phi is a mysterious, powerful mathematical ratio that precedes mathematics itself. It is evident in its abundance and its usage in nature, with animals, and in our most primitive human instincts. Its universality and usage in such varied applications is one of its most fantastic qualities. According to a large percent of a school’s student body, the more pronounced Phi is in a person’s face, the more attractive they are. This is in immense support of my, and MBA’s theory of human’s progressive steering towards this mathematical formula for beauty. However, this ratio does not only have support in history, but also in future and present production and advertising. Companies are already using Phi in their advertisement campaigns, label creation, and product layout. This formula has by far been one of the most revolutionary techniques in marketing. We now have a mathematical calculation, a formula for human want and attraction. To this end, we can use mathematics to steer the will of people and society, to create true, calculable beauty and work towards a world where our daily life is filled with objects that please our eyes and make life that much more beautiful.

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