Fibonacci and the Golden Section

by Patrick Marsolek (4/2013)

In 2011, the American Museum of Natural History awarded a young 7th grader named Aiden Dwyer its Young Naturalist Award for proposing an arrangement of solar panels based on the Fibonacci series. Aiden noticed that the natural pattern that tree leaves were arranged around a stem, which correlated to the Fibonacci series, would be more efficient than traditional arrays of solar panels. Similarly, in the field of biomimicry, a company called PAX Scientific out of San Rafael, California has been developing air and fluid movement technologies also based on the Fibonacci series. Their "Streamlining Principle" is being applied to fans, mixers, impellers and such that move air and liquids more efficiently in systems. Tapping into the wonder of the Fibonacci series, and its correlate, the Golden Ratio, has been happening for thousands of years. Whether consciously recognizing it or not, humans have an amazing relationship with the Fibonacci series and the Golden Ratio.

What is the Fibonacci series? Although Leonardo of Pisa, aka Fibonacci, didn’t discover the series, he became famous for it. Though this numerical series was recorded by Indian mathematicians at least as early as the 6th century, Fibonacci made it well known around the year 1202, when he published his mathematical treatise, Liber Abaci. In what is almost a footnote in his book, he proposed a problem of calculating the number of pairs of reproductive rabbits that would result from month to month from one starting pair of rabbits. The series proceeds in this way: 1,1,2,3,5,8,13,21,34... and on into infinity. Each successive number in the series is the sum of the two previous numbers. The Fibonacci series is a self-generative series, growing out of itself like a plant.

The Fibonacci series is related to the Golden Ratio because, as the numbers in the series increase, the ratio of any two neighboring numbers becomes closer and closer to the Golden Ratio, also referred to as the Golden Section or the Greek letter Phi (f). We say that a line segment is divided at the Phi point, if the larger part divided by the smaller part is equal to the whole line divided by the larger part. The Phi ratio is the ratio of the larger part to the smaller, which is approximately 1.6180339887..., and is more commonly approximated as 1.618.

This number is not a rational number, meaning the decimals never repeat and it can not be written as a fraction of two integers. Some mathematicians have claimed Phi may be the most irrational number known. It is written algebraically as f = (1 + ÷5)/2. It is interesting to note that f2 = 2.61803... and 1/f  = .61803... Each exponent has the same digits in the decimal places. The number Phi truly is self-similar and rises out of itself like the Fibonacci series.

A rectangle that is drawn with the sides in the Phi proportion is a Golden Rectangle. If we cut off a square from the rectangle we’re left with another Golden Rectangle. The dimensions of the smaller rectangle are smaller than those of the ‘parent’ rectangle by precisely a factor f.  Draw two diagonals of any mother-daughter pair of rectangles in the series and they will all intersect at the same point. The series of continuously diminishing rectangle converges to that never-reachable point. Clifford A. Pickover suggested that we should refer to that point as the “Eye of God.”

Many researchers and historians have claimed that the Golden Rectangle and Golden Proportion are incorporated into many sacred structures such as Stonehenge, ancient Indian Temples, the Egyptian pyramids, pyramids in the Americas, and the Parthenon in Greece. For example, researchers recently looked for Phi in the Great Mosque at Kairouan in Tunisia, which was begun in 670 AD and modified over several hundred years. The plan of the mosque is not perfectly square due to site conditions. Taking this into consideration, researchers still concluded that the overall rectangle of the building plan, the relationship of the inner exposed courtyard to the whole building, the placement of the tower, and the height and proportions of the different stages of the tower all utilize the Phi proportion.

Skeptics of claims about the presence of the Golden Ratio in most architectural structures state that it is very easy to juggle numbers to find ratios that are close, and that people often overlook inaccuracies in the measurements. In one critical look at the appearance of the Golden Ratio in the Parthenon, the Golden Ratio is often drawn over the image of the front of the temple. The researchers pointed out that invariably parts of the structure fall outside of the sketched Golden Rectangles and that the dimensions vary from source to source. Skeptics also claim that it is possible to draw all sorts of geometric figures over any site plan or elevation of a monument, but if the major vertices of these do not fall on an actual physical point, intersection or corner, the conclusions drawn from such a figure are “at best arbitrary, and at worst, nonsense”.

Similarly, whether the Great Pyramid in Egypt was built using the Phi ratio has also been the subject of much contention. Some researchers claim that the slope of the sides of the Great Pyramid are extremely close to the slope of a Golden Pyramid, which has the f ratio as an integral part of its geometry. Since there’s no historical evidence that the Egyptians knew about Phi or even had the mathematics to calculate the angle of the slope of the Great Pyramid many believe its presence in their architecture is a coincidence.

Schwaller de Lubicz, the French occultist who was interested in sacred geometry believed the Egyptians did know about f. He studied the Temple of Luxor in Egypt for many years. He wrote an extensive volume entitled The Temple of Man sharing his research. In his writings, de Lubicz claims that Phi is encoded throughout the temple, and that the temple displays a kind of gnomonic growth that is similar to the geometrical progression in the Fibonacci series and is expressed in the human body.

A gnomon is a figure which, when added to an original figure, results in a larger figure of similar shape. This is a quality of the Fibonacci series and Golden Ratio spirals and rectangles. (image) This is also the way hard tissues in animal bodies, like bones, teeth, shells and horns develop. In The Temple of Man, DeLubicz demonstrates how the sections of the Temple of Luxor display this Gnomonic progression, much like some Hindu temples, which corresponds to the growth of a human body. Each section of the temple builds on the previous section directed by the Phi proportions. He believes the entire temple represents a complete human.

Though skeptics again claim that DeLubicz has juggled the numbers to illustrate his theories, DeLubicz points out that though he could measure the existence of Phi in the proportions of the temple, Phi also has a universal symbolic quality that is not physically present. The analogy he suggests, is to imagine a revolving sphere which presents us with the notion of an axis. We imagine this axis, yet it has no objective existence. We use this imaginary axis to calculate the properties, movement and mass of the sphere. De Lubicz claims the stones of the Temple of Luxor are a record of the numerical expression of Phi as with the measurable properties of the sphere. Phi, in it’s universal aspect is unmanifest, yet very much pervades the temple, as does the axis of the sphere.

DeLubicz isn’t the first to see Phi in this way. The 17th century German mathematician and astronomer Johannes Kepler believed the Golden Ratio served as a fundamental tool for God in creating the universe. The Swiss architect Le Corbusier focused on systems of harmony and proportion. He described the properties of the Golden Ratio as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

As with the Golden Rectangle, Phi is an integral part of the mathematics of all pentagons. Unlike three, four and six sided regular polygons, the pentagon does not show up in crystalline structures, because its shape can’t be tiled together tightly. The pentagon fits better with asymmetric forms and living growth. Because of this, some people relate the pentagon and Phi with living things. Many plants seem to have a preference for pentagonal symmetry.

Fibonacci numbers are found in many spiral arrangements of leaves on the twigs of plants and trees.(images) From any leaf on a branch, if you count up the number of leaves around a stalk until you reach the leaf directly above it, the number of leaves is often a Fibonacci number, typically 3,5 or 8. The number of turns from the starting leaf to the terminal leaf is also usually a Fibonacci number. The ratio of the number of turns to the number of leaves is called the phyllotactic ratio of the tree. Though the phyllotactic ratio doesn’t always follow the Fibonacci series, it frequently does: in basswood and elm, it is 1/2; for beech and hazel, it is 1/3; for apricot, cherry, and oak, it is 2/5; for pear and poplar, it is 3/8; and for almond and willow, it is 5/13.  These are all alternating fibonacci numbers. This is the arrangement the 7th grader, Aiden Dwyer  suggested to make his solar panels more efficient.

Similarly, the spiral patterns on pine cones, artichokes, and pineapples provide excellent examples of Fibonacci numbers. If you count the spirals going both directions on these plants, they are often adjacent Fibonacci numbers. Some cones have three clockwise spirals and five counterclockwise spirals, some have 5 and 8, and some have 8 and 13. Some pine cones and pineapples show three different spiral patterns which are all adjacent Fibonacci numbers such as 5, 8, and 13. The spirals in sunflowers similarly show Fibonacci pairs of 21 and 34, 34 and 55, 55 and 89, or even 89 and 144. If you measure the angles between each leaf, bud or scale in these plants, you will find Phi.

One interesting side tangent here, a crop circle that looked like a sunflower appeared on Wooborough Hill, in Wiltshire England in 2000. This circle is composed of 22 clockwise and 22 counterclockwise circles, clearly not Fibonacci numbers, and not a representation of a natural flower with its Phi relationships.

As Schwaller de Lubicz suggested, the Phi ratio is also present in the human body. The navel (the top of the hips on a skeleton) divides the body according to the Golden Section. Taking the full height as 1, the body from the feet to the navel is equal to 1/ f, with the portion from the navel to the top of the head equal to 1/ f2. Yet this proportion is not static. At birth the navel is in the center, and its position on the body moves as we reach adulthood. In the adult female the navel is usually a little above the Golden Section and in males it is usually a little below. Similar relationships occur between the sections of the legs and the arms. You can look at your index finger and see the Phi ratio. The proportion of the first two segments to the full length of the finger is close to the Golden Ratio. Curl the index finger as if making a fist and you can see a rectangle that is close to Phi in your curled finger. Similar correspondences occur in the face and the head.

From beetles to butterflies, from horses to frogs researchers have noted that the relationships of their different body parts are often close to Phi. It must be noted that the measurements of these different animals and body parts do not exactly fit the Golden Section. However, when one averages the measurements, they often do come quite close. Living, growing forms dance within and around this perfect ratio.

The Golden Spiral, which can be formed from the Golden Triangle or the Golden Rectangle, also mirrors the most common spiral found in nature. This is the logarithmic spiral that we see in ferns, nautilus shells, rams’ horns, hurricanes and even in galaxies. Just like the Golden Section or the Golden Rectangle, these spirals have the same proportion regardless of scale, which some claim is an expression of the infinite.

Humans live in a body and a world inscribed with Phi. It seems natural we would seek to find its expression in our architecture, art, music and language. We’ve already seen how the Golden Proportion may be in some sacred buildings. Fibonacci numbers also showed up in the Matra-vittas of 7th century Sanskrit and Prakit poetry. One author  gives the rule specifically for the creation of meters by adding the two previous meters and calculates the series of meters, 1,2,3,5,8,13,21... which is the Fibonacci sequence.

Others have suggested that Leonardo Da Vinci used this proportion in some of his paintings, such as the Mona Lisa and The Last Supper. Paul Larson of Temple University claimed that he discovered the Golden Ratio in some of the earliest Gregorian chants known as Liber Usualis. The music seems to be arranged with phrases on some of the Golden Proportions. Others have analyzed some of Bela Bartok’s music, specifically his “Music for strings, percussion and celesta”. This music shows divisions at 89, 55, 34, 21, 13, and 8 measures, all Fibonacci numbers. It’s generally believed that Bartok didn’t consciously try to use Fibonacci numbers in his music, at least he made no record of it.

Gustav Fechner measured the dimensions of thousands of printed books, picture frames in galleries, windows and other common rectangular objects such as playing cards. The results varied diversely, yet the averages of many of these objects did come close to the Phi ratio. Interestingly, many books produced between 1550 and 1770 had the Phi ratio to within a millimeter.

Clearly, if one reaches far enough, one can find the Golden Ratio wherever one looks. We can see it in ancient architecture, art, poetry and jump to the conclusion that our ancestors knew about this proportion and used it. Yet, there is very little factual evidence that ancient peoples had the mathematics we have today to understand Phi, or that even historical artists and musicians consciously used it.

That being said, there is something curious going on with our relationship to this ratio, It does show up again and again in the things we create and may continue to provide us with new elegant solutions to modern problems. Perhaps it may be as Schwaller de Lubicz suggests, that the Phi ratio is in some way an underlying constant to the world we live in. When ancient peoples were in tune with the natural world and built a sacred space, they didn’t need to know the formula for Phi -- the ratio was built into that structure as a reflection of themselves and their connection with the natural world.

Patrick Marsolek is a writer, dancer, facilitator, clinical hypnotherapist and the director of Inner Workings Resources. He leads groups and teaches classes in extended human capacities, consciousness exploration, personal development, and  compassionate communication. He offers his services to businesses, individuals and families and in self-empowerment seminars. He is the author of Transform Yourself: A Self-hypnosis manual and A Joyful Intuition. See for more information.