The Golden Ratio

The golden ratio is a mathematical law known by many names, which include the: golden proportion, golden mean, golden cut, divine proportion, extreme and mean ratio, phi or Ф (the 21st letter of the Greek alphabet).

It can be expressed in many ways, including numerically, geometrically, in length, area, volume, distribution, beauty, consciousness and creation itself. It has the ability to create infinite diversity on all levels of creation.

The golden ratio, phi, is essentially a relationship between two or more components of a whole; where the whole is divided into a small and large segment. Any whole can be divided into two segments. However, there is only one way that something can be divided so that the two parts are unified proportionally to each other in the same way as they are unified to the original undivided whole.

The simplest expression of this can be demonstrated by the division of a line (c), into a small (a) and a large (b) segment, producing three unequal parts (small segment, large segment and original whole line), all of which are unified through proportional similarity.

Mathematically, these ratios are such that the longer segment is 1.618034... times the length of the shorter segment (if the shorter segment equals 1.00), while the shorter is 0.618034… times the length of the whole line; and the whole line is 2.618034… (phi squared) times the length of the larger segment.

(N.B. The three dots after the numbers indicate an unending ratio, meaning the numbers stream off to infinity and are usually rounded off to three decimal places).

In the average human hand, the length of the bone segments in the fingers of a hand measure in the following way. If the length of the smallest segment of a finger (any finger) is given the value of one unit of measurement, the length of the second segment of that finger will measure approximately 1.618 times longer than that; the third segment approximately 1.618 times longer than the second; and the fourth segment in the palm 1.618 times longer than that

Phi in the ideal human face: Length of face… width of face Lips and eyebrows…length of nose Length of face… tip of jaw and eyebrows Length of mouth… width of nose Width of nose… distance between nostrils Distance between pupils… between eyebrows

 

 

 

The following comparisons of sections of the body are all in phi proportion: finger tip to elbow…wrist to elbow shoulder line/top of head… head length navel/top of head… shoulder line/top of head navel to knee… knee to end of foot.

Not only does the human form, on average, measure in this way, but the bone structures of all creatures are similarly based on the golden ratio. From microbes to the largest creatures upon the planet, phi is the prevalent ratio that carries through all species.

The very program of life itself—the DNA molecule—contains the golden ratio. One revolution of the double helix measures 34 angstroms while the width is 21 angstroms. The ratio 34/21 reflects phi 34 divided by 21 equals 1.619… a close approximation of phi’s 1.618.

A cross-sectional view of the DNA shows a double pentagon (decagon), which is itself an embodiment of the golden ratio (as demonstrated further on).

The orbits of certain planets are phi related: the distance from Mercury to Venus is approximately 1.618 times the distance from the Sun to Mercury. The distance from Earth to Mars is approximately 1.618 times the distance from Venus to Earth.

The orbital relationships between certain planets are also golden proportional. For example, the relationship between Jupiter and Saturn is such that over a sixty year period Jupiter makes five revolutions around the Sun. By marking the position of Saturn at the end of each of Jupiter’s orbital cycles, a golden-proportioned pentagon and pentagram are traced out in space.

The Fibonacci Series

In the 12th century, Leonardo Fibonacci discovered a mathematical series that is found throughout nature. Starting with 1, each new number in the series is simply the sum of the two before it.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

The series is generated by adding each set of numbers to make the next number. However, if instead of adding the numbers together, they are divided, an interesting phenomenon occurs:

As the series moves into the higher ranges the numbers converge on phi. After the 40th number in the series, the ratio is accurate to 15 decimal places: 1.618033988749895 . . .

The Fibonacci series is frequently found in nature in various modalities such as the leaf arrangements in plants, the pattern of florets in flowers, the logarithmic spirals of shells, the bracts of pinecones.

The golden angle provides the ideal positioning of leaf distribution around a stem (phyllotaxis). It is based on an angle of approximately 222.5 degrees rotated from one leaf to the next, which provides the maximum space for sunlight and rainfall for all the leaves.

It is called the golden angle because it divides the 360 degrees circle into a phi ratio: 222.5/360 = 0.618055… and 360/222.5 = 1.617977…

One of the most prominent forms through which the golden ratio expresses itself is the golden spiral, where the arc of the spiral is determined by phi. The scale can range from microscopic to super-galactic, but the arc remains constant at every stage of progression and level of existence. Not all spirals in nature are golden proportioned, but the golden spiral is prominently featured.

The golden spiral can be found in the tiniest sub-atomic particles of the microcosm to spiral galaxies of the macrocosm. At the center of a daisy, golden spirals unfold in clockwise and counter-clockwise directions – the number of the spirals determined by Fibonacci numbers. Swarms of insects, schools of fish and flocks of birds regroup into golden spirals after scattering from a disturbance. The golden ratio is present in the spiral of a fly approaching an object, or a falcon circling its prey.

One of the most exquisite examples of the golden ratio in nature is the chambered nautilus shell. The nautilus begins its journey of evolution constructing a shell to protect itself from the outside elements. As it continues to grow, it builds another chamber at the shell mouth bigger than the one before it, and when it moves into this larger area it seals the old chamber off behind it. It continues this process of building larger and larger chambers along a logarithmic spiral, always closing off the previous chamber to increase buoyancy.

The effect is remarkable. The linear spiral that the chambers are constructed along is a golden proportioned spiral; and three-dimensionally the space of each successive chamber (starting a few out from the center) has approximately 1.618 times more volume than the chamber preceding it.

The Golden Rectangle

The golden rectangle, expounded by the ancient Egyptians and Greeks, has the property that when a square is removed (square f in left diagram below) a smaller rectangle of the same shape remains. Then a smaller square can be removed (e), and so on with similar results. The squares are all in phi relationship. They can also generate a phi spiral by linking the quadrants (quarter circles) of each square. Each quadrant is 1.618 times longer than the one preceding it.

 

 

The golden rectangle featured in a famous experiment by Gustav Fechner in 1876 in a survey asking people to choose from a set of ten shapes, ranging from a square to a double square. There was a marked preference for choosing the golden rectangle. He conducted further research by measuring the proportions of hundreds of windows and doors and found that a lot of them approximated the golden ratio.