‘Golden Ratio’ and ‘Fibonacci Numbers’: The Ultimate Toolkit for Defining the Geometry of Universe
The study of mathematical constructs and their implications on Nature helps us to apply that knowledge during designing and engineering man-made structures (such as buildings, robots, machines, websites etc). This makes our engineering more Naturalistic and robust to the changes of the surrounding environment.
This article is going to describe about a single numerical constant that governs the shape and form of most of the known entities in Nature and the entire universe. In other words, what if I tell you that the structure of all natural shapes and forms of this universe is based on a single number? Believe it or Not, it is true and this article is going to describe this “golden number”, which is called the “golden ratio” and its amazing mathematical implications behind formation of shapes and forms in the universe.
As the first part of this article, let’s review the definition of “golden ratio” and various geometrical shapes formed based on the value of “golden ratio”.
Golden Ratio & Fibonacci Numbers
Suppose we divide any single line into 2 non-equal parts, and the ratio of lengths between the longer and shorter part is x:1, the golden ratio (denoted by greek letter φ)is defined as the value of x in which the longer part length divided by shorter part length (a/b in Figure 1) is also equal to the total line length divided by longer part length ([a+b]/a in Figure 1).
The value of the golden ratio (φ) is a constant which is approximately equal to 1.618.
“Fibonacci numbers” is a self-generating series in which; the first two numbers are 1, 1 and each number after that is the sum of the previous two numbers. As you can see in Figure 3, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8 = 21 and so on…
But is there a connection between the “Fibonacci numbers” and the “golden ratio”? As in Figure 4; when you divide any number in the Fibonacci series with the previous number (except during the first few cases), the answer coincides with the value of the golden ratio. (e.g. 34/21, 21/13, 13/8 and so on…)
Golden Angle
Another interesting concept derived from golden ratio is the “golden angle”, which is the value of the angle formed when a circle is divided into 2 arcs in which the arch lengths are split as per “golden ratio”, φ (see Figure 5). Hence the value of the “golden angle” can be expressed either as an obtuse angle of 137.5 degrees approximately (or as a reflexive angle of [360–137.5] degrees).
Golden Spiral
Figure 6 illustrates an animation of a Nautilus shell. The proportion between the blue and white areas observed when traversing the spiral is aligned with 1:1.618 (1:φ), all the time (which is the definition for the “golden spiral”).
A golden spiral can be constructed using a set of shapes formed with proportions based on “Fibonacci numbers” (Figure 7).
Golden Triangles and Golden Pentagon
Figure 8, shows triangles constructed based on the golden ratio, in which 2 to 1 sides having a ratio φ:1 between side lengths. The triangle on the left (Figure 8) is called the “golden acute triangle” (since all 3 angles are less than 90 degrees). The triangle on the right (Figure 8) is called the “golden obtuse triangle” (since one of the angles is greater than 90 degrees).
A regular pentagon (Figure 9), also called the “golden pentagon” is formed by combining 2 golden obtuse triangles and 1 golden acute triangle.
Now that we know on how to form various geometrical shapes based on the value of “golden ratio”, the next step of this article will discuss about the amazing correlation of above shapes with natural constructions of the universe.
Golden Ratio and Plants
“Golden ratio” is observed in tree branching. As you can see in Figure 10, when a tree trunk grows wide while splitting into branches; the branches tend to split in a pattern that the total branch count at a given height level with the immediate below/above level falls for a ratio between immediate “Fibonacci numbers” (which we know is the value of “golden ratio”).
Usually the nearby leaves in a tree branch are oriented at different directions. In this way when we count the number of leaves along a spiral until finding a leaf with the same orientation in which we started counting (see Figure 11), the number of leaves counted along the spiral tends to align with a “Fibonacci number”. As shown in Figure 11, the number of leaves associated for this characteristic differs from tree species to species, but always fall for a number in the “Fibonacci Series”.
If we number the leaves of a tree branch from top to bottom (see Figure 12); the angle in which leaves of any 2 consecutive indices align (e.g. the angular difference between leaf 3 & leaf 4, or leaf 8 & leaf 9 etc), approximates to value of the “golden angle”(approximately 137.5 degrees).
Trees follow the above patterns related with “golden ratio” during growth, in order to make sure that their leaves absorb the maximum amount of sunlight to enable optimal photosynthesis. These “golden ratio” oriented growth patterns were first observed by Charles Bonnet, a Swiss naturalist during 1754 and was termed as Phyllotaxis.
So it propagates all the way to flowers starting from branches and leaves. Figure 13, illustrates how the seeds are formed in the middle of a sunflower (on the right), aligned with a “group of concentric golden spirals” (on the left).
Figure 14, shows how the formation of flower petals (in this case roses) is based on the multiple “golden spiral” setup, similar to seed formation pattern in Figure 13.
Golden Ratio and Space
Some of the galaxies in universe (such as Milky Way, Andromeda) are of spiral shape. Their photographs show the resemblance of the “golden spiral” as the basic shape of construction (see Figure 15).
Planet Saturn in the Solar System, has proportions related to “golden ratio” among its diameter and rings as illustrated by Figure 16.
The orbital years of planets; Mercury, Venus, Jupiter & Saturn relative to an Earth year approximates to different powers of “golden ratio” (φ) as shown in Figure 17. Hence all the 5 planets together including Earth are called the “golden planets”.
A team of scientists and mathematicians have calculated the overall shape of the universe based on microwaves received as cosmic background radiation dating back to the infancy of the universe. The answer they’ve come for the shape is a “Dodecahedron”; a 3D shape formed based on the “golden pentagon” (Figure 18).
Figure 19, illustrates on how the space in the universe would fit into a “3D Dodecahedral model”. Although this is still a theory, it can be proved based on cosmic background radiation collected via microwave space antennas.
Golden Ratio and Geography
Hurricanes get formed as spirals and tend to align with the “golden spiral” upon formation (Figure 20).
Wave formation happens similar to hurricanes, in which the wave is slowly shaped towards a “golden spiral” upon its peak (Figure 21).
The African continent is naturally shaped as a “golden spiral” (Figure 22).
Golden Ratio and Life
When an embryo develops inside a mother’s womb the overall shape tends to align with the “golden spiral” (Figure 23).
Figure 24 illustrates the relationship between a hen egg and the “golden spiral” shape(on the left).
Also certain proportions of the “egg shape” are closely associated with the “golden ratio” value as illustrated in Figure 24 (on the right).
How many “golden rectangles” (rectangles with length:width ratio being φ:1) can you figure out when looking at a Koala’s face? (Figure 25)
Figure 26 shows a detailed analysis of a butterfly in relation to its overall shape. The interesting observation to note here is the contribution of fundamental “golden ratio” based shapes; “golden rectangle” and “golden spiral”, for the construction of the overall butterfly shape.
Figure 27 is a collection of photographs containing different snail types. How many “golden spirals” can you find in each snail shell?
The proportions of lengths between head, thorax and abdomen of an insect approximately aligns with different powers of “golden ratio” (see Figure 28).
Figure 29, illustrates the length and width of different types beetles and how these dimensions correspond to “golden ratio” (φ).
When a fly approaches an object it follows a “spiral” path quite close to a “golden spiral” due to the structure of its eyes and the nervous system.
In any bee colony (i.e. a bee hive), there is one big female bee called the “queen” who lays eggs. If the eggs laid by the queen are fertilized by a male bee, then it leads to forming another female bee who would form a new colony as its queen with a subset of male bees in the existing colony. An egg laid by the queen, those are not fertilized by a male bee; forms a male worker bee. Hence every female bee will have both a male and a female parent, whereas every male bee will only have a female parent (see Figure 31). Hence for a particular bee’s ancestry, it will always have a “fibonacci number” of ancestors for a particular earlier generation.
Observing birds, we could see that the sub skeleton parts making a complete wing approximates to “golden ratio” between their lengths (see Figure 32).
The overall shape of the head and beak together forms a “golden spiral” in many bird species (Figure 33).
As a final refresher to this section, just think about your Cat looking very cute when lying on the floor confirming the shape of a “golden spiral”.
Golden Ratio and Human Body
Figure 35, illustrates the organization of different proportions in a human arm and their relationship with the “golden ratio”.
How many “golden ratio” values can you make by dividing different proportions marked in the hand and face illustrated in Figure 36?
Figure 37; a detailed analysis of “golden ratio” done for a model’s photo, illustrates how “golden ratio” accounts for forming the overall shape of the human body. Here comes a question for all girls, who have read up to this point. How would you decide the height of your shoe heels?
Figure 38, shows the contribution of the “golden spiral” when forming the overall shape of the human ear and even its little details.
Figure 39, explains the contribution of “fibonacci circles” for different aspects of the human eye.
Figure 40, illustrates how you could explain different curvatures of the human body based on the “golden spiral”.
How many “golden ratios” can you make by dividing different lengths measured in your feet? (see Figure 41)
Figure 42, illustrates the amazing involvement in “golden spiral” for defining a well-shaped body confirming with a healthy body mass index.
I hope these evidence prompt you to think that the beauty of a human face could be analyzed by using the “golden ratio”. In fact, it is possible and already done which is a topic for another day. Yet, the interested can read it here.
Putting It All Together
After a comprehensive study about the contribution of “golden ratio” for forming shapes in Nature and universe; this final section focuses on few ideas behind on how it has been applied to design man-made artifacts, demonstrating the usefulness of the study.
Figure 43, provides evidences pointing out the knowledge of “golden ratio” from the time of Leonardo da Vinci and its application in some arts produced during renaissance.
Figures 44 and 45; illustrate 2 naturalistic paintings (butterflies and a mollusk shell) drawn by a Venezuelan architect Rafael Araujo, in which the initial design sketch is based on proportions derived from “golden ratio”.
As stated during the beginning of this article the knowledge of “golden ratio” is universal and hence could be applied from man-made Arts to all disciplines of engineering. Figure 46, illustrates how “golden ratio” is applied in software engineering for designing logos of 3 popular trade brands.
In fact you could derive novel design concepts on your current corporate website making it more attractive, Naturalistic and user-friendly. Application of “golden ratio” for website design is a topic that needs to be discussed in separate. It will be an extremely pleasant learning experience, based on the knowledge gained from this article. For those who want to take up the challenge you can read it all here.
