The golden ratio is derived from the Fibonacci sequence, and is seen universally in varied natural elements. It is a part of the natural dimensions of most biological as well as non-biological entities on this planet.

*“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”*

―Johannes Kepler

The golden ratio is referred to by many diverse terms, such as golden mean, golden section, medial section, divine proportion, golden cut, and extreme and mean ratio. All these names point to the fact that, it is a ratio of dimensions of a given entity, but this description seems vague. A more accurate way to describe it would be, to call it a ratio of line segments when a line is divided into two parts (a and b), such that the ratio of ‘a’ to ‘b’ is the same as the ratio of (a+b) to ‘a’. This ratio is called the golden ratio, and is signified by the Greek letter phi (Φ). Its mathematical value is 1.61803398… For general purposes, the value is assumed to be 1.618. This value can be derived using basic quadratic equations, geometry, or by analyzing the Fibonacci sequence. This sequence is a series of numbers, where each number is the sum of its two preceding numbers. The initial sequence is as follows – 0, 1, 1, 2, 3, 5, 8. 13. 21, 34, 55, 89, 144, and so on. The interesting aspect of this series is that, after the first four to five numbers, if each number is divided by its immediate predecessor, it yields a value close to 1.618. This value approaches closer to the golden ratio as the series progresses.

## Golden Ratio Representation

The Fibonacci series is often visually represented as given above. Each number is represented as a square, whose side measures the same as the value of the number. These squares are then placed adjacently as the series progresses, to yield what is known as the Fibonacci rectangle. If a spiral is drawn through the corners of each square, one obtains the Fibonacci spiral. Just as how the ratio of the numbers of the series yields the golden ration, so is the case with this spiral. The ratio of each turn of the spiral, or the ratio of its increasing radii, yields the golden ratio. This is the common form of manifestation of the divine ratio in natural elements

The elucidation of the relationship between the golden section and the Fibonacci sequence is vital in order to detect and identify the presentation of this particular ratio in nature.

## Real-life Examples of Golden Ratio

### Flower Petals

In almost all flowering plants, the number of petals on the flower is a Fibonacci number. It is extremely rare for the number of petals not to be so. Examples of this phenomenon are: Corn marigold, cineraria, and daisies have 13 petals; asters and chicory have 21 petals; plantain and pyrethum flowers have 34 petals, etc. The golden ratio is seen in these flowers in terms of petal arrangement. All the petals exhibit a twisting of about 1.618034°, in order to optimize exposure to sunlight.

Also, flowers with multiple layers of petals exhibit the Fibonacci sequence per layer, and the top view of the flower presents the Fibonacci spiral. The ratio of petals between each layer is the golden ratio. The same is also true for the leaf arrangement of most plants

### Seed Heads

Spiraling patterns of seed heads, as seen in case of sunflowers, are a great example of the Fibonaccian process and the divine ratio. In a seed head, typically, new seeds are formed at the center, and they migrate outwards in a radial fashion as they grow older. Since each whorl of the seed heads follow the sequence, it logically follows that the ratio of any two adjacent whorls is the golden ratio. The seed heads also exhibit two distinct radial orientations. If he number of total seed heads oriented in the two directions are compared, they yield the divine proportion.

### Pine Cones

Similar to the spiral patterns of the seed heads, the pods of the pine cone are also arranged in a Fibonaccian spiral. Each cone consists of pairs of alternating whorls, each oriented in the opposite direction to the other whorl. The ratio of the turn of each pod and the ratio between the number of pods in successive whorls is the golden ratio, i.e., 1.618.

### Fruits and Vegetables

The same pattern is observed in the case of fractal-like fruits and vegetables. The most common examples are pineapple, red cabbage, artichokes, and Romanian cauliflower (image). In these fruits and vegetables, it is easy to visualize the spiral patterns along their surface.

### Branching Pattern in Trees

When the main trunk of a tree branches out, it gives rise to a side-branch, which will further go on to divide and produce two more branches. One of these branches will split and form two new growth points, while the other branch remains dormant. This occurs at each branching event along the length of the tree over the course of its lifetime. This gives rise to branches, whose number follow the Fibonacci progression. This implies that, at each branching node, the ratio of new branches to old is 1.618.

### Shells

The outer calcareous shell in the case of snails, seashells, and other such examples, also exhibit the Fibonacci spiral. Snail and nautilus shells are obvious examples, where the spiral is plainly observable. Each chamber of the nautilus, when compared to its immediate successor, reveals the golden ratio. The same is true in case of snails. In case of bivalve type clams, which exhibit grooves on their shells, the ratio of the grooves to the ridges equals the golden mean. The same phenomenon is also seen in the case of horns of rams and goats, the shape of certain spider webs, and the inner cochlea of the ear.

### Spiral Galaxies

The Fibonaccian spiral is also observed in case of a spiral galaxy. Our own galaxy―the Milky Way―is one such celestial entity. Certain other entities within the galaxy also exhibit the golden ratio. It is found in the ratio of the diameters of Saturn and its rings. It is also the ratio of the distances of Venus and the Earth from the Sun. Interestingly, the ratio of the revolutions of these two planets also yields the golden ratio.

### Hurricanes

As in the case of shells and spiral galaxies, the movement of air and wind in hurricanes also follows the Fibonaccian spiral, revealing the golden ratio. The spiral nature of a hurricane is largely due to the simultaneous movement of the air and atmospheric elements between a low pressure area (epicenter of the hurricane) and the surrounding high pressure area.

### Faces

Various features of the human face exhibit the divine proportion. It is seen when one compares the relative position of facial features. Some common examples of such ratios are:

► Center of the pupil ● Bottom of the teeth ● Bottom of the chin

► Outer and inner edge of the eye ● Center of the nose

► Outer edges of the lips ● Upper ridges of the lips

► Width of the center tooth ● Width of the second tooth

► Width of the eye ● Width of the iris

Along with these, the human face also shows the golden ratio in terms of vertical and horizontal ratios. In addition, the shape of the ear resembles the shape of a Fibonaccian spiral. Numerous studies have concluded that faces with facial features that exhibit a precise golden ratio are deemed to be highly attractive and regarded as extremely beautiful.

### Reproduction Dynamics

In honeybee populations, the ratio of females to males is 1.618. Also, according to bee reproduction, fertilized eggs become female bees, whereas the unfertilized ones become males. Therefore. The females possess two parents, while the male only possesses one parent. Hence, if one were to examine the family tree of individual bees, the number of parents would progress from the newest to the oldest in a Fibonacci sequence.

### DNA

A DNA molecule is 34Å in length and 21Å in width. The ratio is approximately equal to the golden ratio. The same is true for the ratio of the two grooves of the helical DNA molecule, i.e., the major (21Å) and the minor (13Å) groove.

### Animal Bodies

Animals show a wide range of body structures. Despite this vast range, they still exhibit the divine proportion in various parts of their bodies. Some examples include:

► Dolphins: Dimensions (length:breadth) of eyes, fins, as well as tail section.

► Penguins: The ratio of the position of the body markings at the eyes, beak, and wings, in contrast with its total height.

► Tiger: Almost all the facial features and their positions show golden sections, including the ratio of the length and breadth of the face.

► Insects: The ratios of the body segments (head, thorax, and abdomen) to each other are golden sections.

### Human Bodies

Golden ratios that are observed in the human body are as follows:

► Head to toe ● Head to navel

► Ratio of the length of each digit of a finger

► Shoulder to fingertip ● Shoulder to elbow

► Hip bone to heel ● Hip bone to knee

► Chest length ● Waist length

In addition to these examples, the divine proportions is also seen in various architectural wonders, like the Greek Parthenon, paintings like the Last Supper, in musical symphonies and instruments, and even in biblical texts (dimensions of Noah’s Ark). This ratio has been revered as divine, and called God’s fingerprint due to its presentation in numerous living as well as non-living entities. Some claim that this is evidence of God’s presence and his intelligent design of the universe, whereas, at the same time, others point out that these are mere statistical manipulations. Whatever the case may be, it is interesting to note the presence of this ratio in so many varied forms in nature.