The Fibonacci Sequence

Of all the mathematical patterns discovered throughout history, none have had the profound impact as the Fibonacci Sequence. Originally discovered in ancient India, the sequence was not formally known in Europe until Leonardo of Pisa (aka Fibonacci) published his ground breaking book Liber Abaci in 1202. By studying the reproduction of rabbits and analyzing the growth of the population over time, Fibonacci was able to observe a breeding pattern unrealistic in most other species. The unique characteristics of the sequence set it apart from all other recursive mathematical series. The first two numbers of the sequence are 0 and 1. All numbers following are calculated by taking the sum of the two previous numbers. Thus the first fifteen numbers of this infinite sequence are the following: {0,1,1,2,3,5,8,13,21,34,55,89,144,233,377}.
How the sequence corresponds to the Golden Ratio is seen by dividing any number in the sequence by the previous number. By doing this you obtain numbers very close to one another and that oscillate around Golden Ratio. In fact after thirteen iterations the numbers becomes fixed (≈1.618) and converge to more and more precise approximations of the Golden Ratio. Next post I will touch on some mind-blowing examples of this pattern in nature.

The Golden Ratio

Dating back to over 2,400 years ago, famous mathematicians including Pythagoras and Euclid have dedicated countless hours into the study of what is now referred to as the Golden Ratio. This perfect ratio has been a topic of interest for mathematicians, philosophers, architects, and even artist. Ancient Greeks used mathematics to develop a relationship between beauty and truth. Aristotle believed in the existence of an ideal median that divided two extremes of a single entity. This perfect balance between excess and deficiency must satisfy properties such as symmetry, proportionality, and harmony. The Golden Ratio encompasses all of these characteristics and has been the cornerstone for architects and artists due to its aesthetically pleasing visual representations. The Parthenon is believed to have been designed using approximations of the ratio. Renowned artists such as Leonardo DaVinci and Salvador Dali are believed to have incorporated the ratio into some of their most famous artwork. The Golden Ratio, represented by the Greek letter Φ, is approximately 1.61803...or [ (1+√5)/2 ]. This is an irrational constant, meaning that it can’t be plotted on a number line because its decimal places go on infinitely, never converging to a finite value. By many, the Golden Ratio is thought to be a preexistent model for the natural balance of equilibrium that is a part of any changing life form. Below are several geometric representations of the ratio, and how it is found in the human body. Next post I am going to touch on how this ratio is found in countless mathematical sets including the Fibonacci Sequence. Because some of these pictures are difficult to understand I have included the verbal proportions of the body as well.
*Note all of the following ratios are equivalent, but the perfect ratio can only be approximated based on how close your body is to having the ideal proportions.
Length of face: width of face
Distance between the lips and where the eyebrows meet: length of nose
Length of face: distance between tip of jaw and where the eyebrows meet
Length of mouth: width of nose
Width of nose: distance between nostrils
Distance between pupils: distance between eyebrows

Human Evolution

I have recently read a blog post by Susan Hawks. I found the blog to be very interesting although she does makes some rather big stretches in her argument. Based on Susan’s profile, one can assume she is a very free thinking person. She has a unique philosophy of spirituality and governments' role in society. I agree with most of her ideas and liked her connection between the metamorphosis of a caterpillar into a butterfly and our current state in human existence. I agree that in order for human evolution to continue and new institutions to emerge; older systems must die off to allow for these new organizations to take hold. She believes that a radical rebirth of society is necessary and inevitable. Susan views government as an immune system that’s role is to protect the organism (the human race). However, during times of such rapid growth the government does not recognize this process as necessary, and fights the transformation to the point of actually self-destructing. Susan is quoted as saying, “When the caterpillar is in a chrysalis stage, its physical body goes through such a rapid transformation that its own immune system recognizes the change as an outside attack. It then begins to attack its own organism because it doesn’t recognize that the process it is undergoing is natural and necessary.” If interested one can read her entire argument and other posts by going to . Also, here is a short clip from NOVA summarizing its series on fractal geometry. I would recommend anyone interested to check out the site because it offers very good videos on math and science.