The Golden Ratio
A Golden Spiral created from a Golden Rectangle expands in dimension by the Golden Ratio with every quarter, or 90 degree, turn of the spiral. This can be constructed by starting with a golden rectangle with a height to width ratio of 1.618 (or Phi Ф). The rectangle is then divided to create a square and a smaller golden rectangle. It appears everywhere in nature if you know what to look for. This particular spiral is a graphical representation of the Fibonacci sequence which is a series of numbers that continues infinitely following a specific pattern. The Fibonacci numbers were first introduced in 1202 by Leonardo of Pisa. He is better known today as Fibonacci. The Fibonacci sequence appeared as a result in the following problem: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?”
The resulting sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and so on. The nth number of the sequence was the sum of the two previous numbers. Phi (Ф) is the Greek letter that represents the value of the Golden Ratio and is known as the Golden Number. It can also be written as φ which is the lower case form of the same letter. It is the 21st letter of the Greek alphabet. Only when representing the golden ratio, phi has the value of 1.618 and it can be reached using the equation: φ = (1 + √5) / 2
Perhaps the most fascinating of the occurrences of this pattern in nature is the Fibonacci spiral and how this sequence can identify the particular way in which plants grow. But first it will be necessary to correct a common misconception about the Fibonacci spiral. It is often believed that the spiral is represented by a nautilus shell when in fact the nautilus shell is a representation of a slightly different kind of curve. The mathematics behind the nautilus spiral differ slightly from the Fibonacci spiral. “The center/vortex of the spiral increases to a width of 1 at point A. The half rotation of 180 degrees to point B expands the width of the spiral to 1.618, the golden ratio. Continue another half turn of 180 degrees to point C to complete the full rotation of 360 degrees. The width of the spiral from the center is now 2.618, which is the golden ratio (phi) squared. Another full rotation expands the length from the vortex by phi squared, from phi to phi cubed. And so the pattern of expansion continues” (www.goldennumber.net/ nautilus-spiral-golden-ratio/). The Golden Ratio still plays a key role in the nautilus spiral, but it is important to give each spiral its recognition.
Now we can get back to how plants have this ingenious way of making sure that they are able to soak up the sun’s rays as efficiently as possible. This occurs by following a set of rules as to when the plant is determining where to grow each of its leaves. This is a brilliant occurrence of the Fibonacci sequence in nature. For example, one can observe the bottom of a pine cone and the curve of the seed pods on the pine cone’s base. Lo and behold the presence of adjoining numbers of the Fibonacci sequence on the base 8 and 13. Drawing lines on any number of different plants will produce a similar effect and one will find that each subsequently longer series of curves is a number within the Fibonacci sequence. This is fun to do with pineapples and to measure how many spirals curve around its body. Like the pine cone 8 and 13 are present, but there are 21 curves that can be counted around its body which is of course the next number in the sequence after 13. Of course, the Golden Spiral makes an appearance in many other instances all throughout nature besides plants. It can be found within the eye of a storm or within the swirling interior shapes of spiral galaxies.
The Golden Spiral also plays a role in what can be found attractive to the eye. Ears tend to be a good example of the Fibonacci spiral on humans and whether or not they are found attractive. There is also a set of proportions related to the Golden Ratio. The Golden Ratio also has a lot to do with composition within a drawing, or painting, or photograph. The rule of thirds is derived from the makeup of the Fibonacci spiral as well as the focus of interest or the subject of the image. For example, the rose to the left commands most of its attention in the center of the photograph while the left and right thirds remain sparsely detailed. If the Fibonacci spiral would be overlayed on an image, one may likely find the subject, when not perfectly centered, will tend to favor a particular corner over any other part of the image and as one travels away from the subject, the imagery becomes less descript in order to draw focus to the subject. Of course the point of interest on the Fibonacci spiral may not necessarily be the subject of the painting or image, but of something that is particularly important to what the image is trying to convey. The Golden Spiral may also play a role in the composition of the painting and the placement of the subjects within it. The image to the right is an example of the composition relying on the Golden Spiral.