Research Work

 Structural harmony always elevates the aesthetic value of anything, including architecture. The book "De Architectura", a book by a Roman architect Marcus Vitruvius solidifies this statement. He also stated that architecture consists of order, arrangement, proportion, symmetry, décor, and distribution. Theorists, mathematicians, astronomers and architects such as Cartwright, Sartwell, Johannes Kepler, Akhtaruzzuman and Shafie. (Stakhov 42) Examples can be found scattered throughout history, dating back to Ancient Greece and Egypt.  It is speculated that in Gothic castles were made more durable as builders formed bricks in a specific rectangular "parallelepiped" shapes, and were referred to as "Golden Bricks". According to Akharuzzuman and Shafie pointed out that even in the Renaissance architecture contained hints of it.  Posamentier and Lehmann 231 states that many architects throughout the ages deliberately used the Golden Section "in their sketche...

  Mathematical harmony has, over the years, been used deliberately and unknowingly in musical composition. Mathematician Gottfried Leibnez wrote that "music is the pleasure the human mind experiences from counting without being aware that it is counting." Italian composer and lutenist, Vincenzo Galelie also wrote a discourse on string theory, which states that "the universe is constructed by tiny vibrating particles", which are assumed to be even smaller than subatomic particles, and these interact with each other by twisting, folding and vibrating. They create phenomena like electromagnetism and gravity. This string theory may also be linked to sound and hence music.   Composers often rely on the golden ratio and Fibonacci sequence to enhance their compositions, and construct balanced, melodic, dynamic movements and rhythms. Claude Debussy's music, for example, use the golden ratio, prominently the "dramatic" climax of "Cloches a travers les feui...

 The Golden Ratio And Fibonacci Sequence In Art And Architecture  Although humans are blessed with the gift of nature, we tend to enjoy aesthetically pleasing order and symmetry. According to evidence, humans are likely to detect symmetry in as little as 0.05 of a second! It is evident that humans do not only like seeing symmetry and Divine Proportion, but also working to create and display it. It can be seen in art and architecture.   In paintings, portraits and drawings, artists like following it.  Vitruvian Man (Leonardo Da Vinci)  The Great Wave (Katsushika Hokusai)   The Last Supper (Leonardo Da Vinci) Mona Lisa (Leonardo da Vinci) Rembrandt's Self Portrait Mondrian, Tableau 1. For example, the Vitruvian Man exhibits a pentagonal symmetry, and the square side to circle radius ratio is the golden ratio. The dimensions of the table and the proportions of the wall and background in Da Vinci's "The Last Supper" display an attempt of following the gold...

 According to theorists Boeyens and Thackeray, The Golden Ratio is reflected in birds and in animal kingdoms, from large scale to minor entities.  For example, according to Boeyens and Thackeray, The Golden Spiral is visible even in elephant tusks, and as I was observing it, I realized that even its gathered trunk reflects the Golden Spiral.  Moreover, the Golden Spiral is also reflected in the horns of a ram. Matila C. Ghyka, a novelist and theorist identified a golden ratio between the leg of the horse and the vertical thickness of the body.  According to Akhtaruzzuman and Shafie, even a penguin's body reflects the proportions of Golden Ratio; with the eyes, beak, wing, and significant markings falling at the top golden sections.  The peacock's feathers are one of the most beautiful example of The Golden Ratio in nature.  The Golden Ratio and Fibonacci Sequence in more distinctly minute features includes "microtubules" of animal cells. In mammals, ea...

Bee Families: If a unmated female bee lays an unfertilized egg, it hatches a male bee, whereas a female bee is hatched if the egg is fertilized by a male.  Therefore, a male bee has 1 parent, 2 grandparents, 3 great grand parents, and 5 great great grandparents. This phenomenon is observed in the family tree of drone bees (lineage, forebears).  The number of ancestors at each level,  F n , is the number of female ancestors, which is  F n −1 , plus the number of male ancestors, which is  F n −2. Some insects happen to possess the ability to create harmonious structures and distinguish between mathematical shapes.  Moreover, the Fibonacci Sequence can be observed in body sections such as the position of eye-like spots, and the golden ratio can be observed on the shapes on butterfly wings. 

   Fibonacci sequence might as well be the overlooked backbone of patterns in nature. In the branches of trees, arrangements of leaves, the lineage of bees, the fruitlets of pineapples, in chamomiles,  and daisies.  Botany:  Pentagonal symmetry is one of the structural symmetries. It includes 5 parts around a  central axis (72 degrees apart). It is seen in phyla, in the likes of green algae and the water  net,  "Hydrodyctyaceae Hydrodictyon"  Moreover, the Fibonacci Sequence may also be observed in families of vascular plants,  compositae. It includes the likes of sunflowers, daisies and asters. The have a disc- shape arrangement, composed of pentamerous (arranged in groups of five) florets  centripetal to an involucrate head, surrounded by fanning petals on the outside.  The sunflower displaying an equiangular spirals running clockwise and counterclockwise It is observed that these bidirectional spirals intersect each oth...

  Fibonacci Sequence  It is said that the golden ratio itself springs from the Fibonacci sequence, so it is important to first understand the importance and the meaning of the Fibonacci Sequence.  What is the Fibonacci Sequence?  In simplest words, it is a naturally occurring sequence or series of numbers, in which each new number is added to the one that precedes it. It can also be described as the sum of two numbers before it. For example, if it begins with 0, and 1 is added, we get 1, then when 1 is added again, we get two. And the sum of 1 and 2 is 3, the sum of 2 and 3 is 5, and the sum of 3 and 5 is 8.  So we can say that it goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and so on. It doesn't necessarily have a limit and may go on till infinity.  There are many debates regarding the origin of the Fibonacci sequence, as some sources associate it with the Greeks, some claim it was an Indian discovery, and some argue that it is an Italian d...

  Research Method and Design and Data Type: Secondary Data Analysis with mainly qualitative data used for analysis.  The access to internet has made it much easier to conduct research, and so most of the data I will collect, store on my blog, and use to draw conclusions will be from multiple sources online. The following sources are the most significant contributors to my potential data: www.google.com fibonacci.com https://en.wikipedia.org/ The book: Master Fibonacci - The Man Who Changed Mathematics I will add further sources if I use any.  More sources:  https://www.invisionapp.com/ https://www.sciencedirect.com/

  Directional Hypothesis:  The Golden Ratio and Fibonacci Sequence has practical effects, it is generously scattered throughout nature, and can successfully flourish the aesthetic aspect of man-made creations, such as forms of art. 

Have the Golden Ratio and Fibonacci Sequence been observed in nature and history and can they be applied to various works and projects to add value and enhance the product/results?