A Guide to Discovering Phi By Marc Matteson
Is it possible for certain numbers to be inherently sacred? Gematria, or Hebrew numerology, is a well known way to study the Torah. This practice gives each letter of the Hebrew alphabet a numerical value and the values of significant words and phrases are analyzed. Most Christians are aware of the reoccurring presence of the several numbers in the Bible such as seven, twelve and especially three. Since the time of the Greeks, mathematicians have revered a very specific number as the “Creator’s fundamental building block”. This number can be found everywhere in nature. It is present in plants, animals, and even you. This number is so essential to our existence that our eyes have learned to see this number as aesthetically pleasing. By the time you finish reading this article, you will be aware of this number’s presence all around you.
What is this number? This mystical number has many names: the golden ratio, golden section, golden mean, and divine proportion. The mathematical name is Phi (pronounced “fye”). Phi is represented as and is approximately 1.618. This value is a specific ratio that can be defined in many ways. One of the first definitions of Phi is to use a line of any length and divide it into two segments as shown below.
The position of the vertical bar B C divides the line A so that the relationship between the total length A A and the first segment B is the same Phi equals the ratio of A divided by B, and as the relationship between the two also equals the ratio of B divided by C. sections B and C. This is represented as the equation to the right. A B 1 B C Phi has an interesting ability to be defined in many mathematically unique ways.
1 1 2 1 2 (1 5) 2cos 5 2 sec 5 2 csc10
1 1 1 1 1... 1 1 1 1 1 1... Any of these equations gives the exact value for Phi to an infinite number of decimal places. There are other unique mathematical traits of this number that set it apart from the rest. Phi has a unique relationship to itself. If you add one to Phi and then divide the new value by Phi, the answer is again, Phi. This only works with our mystic number.
Where have I seen Phi? Phi can be seen everywhere from simple geometry, to some of the most revered, complex artwork of the world. The most recognizable form of Phi in our world is the golden spiral. The golden spiral is Golden spiral inscribed in golden rectangle constructed by making a rectangle with This spiral is seen in nature the dimensions of Phi and (Phi + 1). as nautilus shells, galaxies, etc. (Source: Wolfram1) This is known as the “golden rectangle.” It is shown to the right as the largest rectangle. Then, the longer side of the rectangle is sectioned as shown in the first diagram. This makes a very large square with sides equal to Phi and another, smaller rectangle. This rectangle is another golden rectangle since it has dimensions identical to the original only on a smaller scale. The Nautilus and Galaxy (Source: Golden1) same procedure is repeated to this rectangle making another square and smaller rectangle. This process can be repeated infinitely. When the corners of the squares are connected, they make a gracious spiral. This spiral is seen often in nature. The nautilus shell is a perfect example. The ratio of each spiral’s diameter to the next approaches Phi as well. The same spiral is found on sunflowers, pineapples, pinecones, and daisies. The seeds or petals form spirals in both directions. The shape of our galaxy is also a spiral. Again, the spiral diameter ratios approach Phi.
A more personal example of the presence of Phi in our lives is the human body itself. The ratios of different measurements on our bodies approach Phi. Your full height divided by the distance from your bellybutton to the ground should be close to Phi. The same is true of the distance from your shoulder to fingertip divided by elbow to fingertip. This can be applied to the location of your knees, wrists, knuckle, etc. This same strategy can be applied to other animals as well; the placement of eyes on mammals, the location of fins on fish, even the segments of an ant’s body are guided by this ratio. Since humankind lives in a world abound with examples of Phi, it makes sense that it is reiterated in art. See for yourself. (Source: Golden2)
The colored bars represent the golden ratio in the following images.(Source: Golden2)
Notre Dame Cathedral The Last Supper (Source: Golden2)
As shown above, the proportions of different elements are very similar to the golden ratio. In fact, Leonardo DaVinci intentionally used this ratio to paint The Last Supper. Why do I care? Everyone else is using it, why not you? Most songs you listen to reach their climax at a point related to Phi, approximately 61.8% into the song. Credit card companies are using Phi to shape their credit cards. They are roughly golden rectangles. The most impressive use of Phi is in the stock market. Analysts have found that many stock value increases and decreases can be shown to have time intervals at increments divisible by Phi. This is just the beginning of Phi’s relevance to our lives. Phi is so integrated into the world around us, it would seem wasteful to ignore its presence and not utilize its effects.
How can I learn more? The information presented here (including pictures) was collected from these websites. They go into great detail about the golden ratio and its relationship to our lives. They also describe an amazing relationship between Phi and the Fibonacci sequence. This sequence is just as integrated into our lives as Phi.
http://goldennumber.net/ http://mathworld.wolfram.com/GoldenRatio.html http://mathforum.org/dr.math/faq/faq.golden.ratio.html
Resources:
Wolfram1 - Weisstein, Eric W. Golden Rectangle. From MathWorld--A Wolfram Web Resource. Retrieved January 28, 2005, from http://mathworld.wolfram.com/GoldenRectangle.html
Meisner1 - Meisner, Gary. “Fibonacci Spirals.” Retrieved January 28, 2005, from http://goldennumber.net/spirals.htm
Meisner2 - Meisner, Gary. Golden Section. Retrieved January 28, 2005, from http://goldennumber.net/goldsect.htm