# Golden Ratio Research Paper

Name: Marina Thomas Date: 10/17/14 Research Paper Draft Topic:The Golden Ratio So why is it that Angelina Jolie is so gorgeous or why is the Mona Lisa a perfect work of art? Art and beauty is just one of the many pleasures of life. However, it is unlikely for someone to ponder on the detail of this beauty which in fact relates to mathematics. The Golden Ratio or Rectangle is believed to make the most pleasing shape. The golden ratio is approximately equal to 1. 618. It is actually an irrational number that continues on called phi. Phi can be shown as y or 0.

The golden ratio is expressed as a rectangle that can be applied in many areas of life today. This includes architecture, art, and geometry. The golden ratio can be applied to art whereas a face can be made very appealing due to correct proportion which connects with the Golden Ratio. For example, magazines may use the Golden Ratio in their art. The Golden Ratio is also named in other ways as well as Divine Proportion, Golden Number, Golden Mean and Golden Section. In 300 B. C. , a mathematician called Euclid created the first definition of the Golden Ratio.

He said, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. ” An approximate idea of phi is 1. 16180339887. Phidias being the father of this mathematical idea was the effect of making the numerical value of the Golden Ratio “phi”. He had used the Golden Ratio as a sculptor for architecture, specifically the Parthenon. Figure #1: Phi can be represented geometrically through ratios of line segments and lengths. -(illustrated by http://www. mathsisfun. om/numbers/goldenratio. html)

Phi was derived based on a line segment. It was simply divided. The large piece can be divided by the smaller proportioned piece and still be equal to the overall length divided by the large piece. In other words Phi+1=Phi2 (squared) or longer unit divided by short unit=whole unit divided by longer unit. The reciprocal with one added can also be equal to this because by dividing both sides by Phi you would result with Phi=1+1/Phi. A pentagram or more commonly known as a five pointed star was associated with Pythagoras.

He was just one of the many influences toward the Golden Ratio and its extensive proportions and helped develop the proportions to the Golden Rectangle. It can actually form itself into a continuous or infinite number of golden rectangles which is just one of the amazing features of this astounding shape. Take the following for example:Figure #2: The figure has an inscribed pentagram with the regular pentagram.

When examined, one can find that all the line segments found are equivalent to one of the five line segments in length. – (illustrated by http://www. ontracosta. edu/ legacycontent/math/pentagrm. htm) a+b=1. 618… b+c=1. 618… c+d=1. 618… Pythagoras was able to derive within a pentagram that the ratios of the lines shown above, all were found equal to the golden ratio. If the pentagram is correctly drawn, many golden rectangles can be found. A regular pentagon can be identified with equivalent sides, angles as well as diagonals. The diagonal lines within a regular pentagon creates a star with five points, also known as a pentagram. Five overlapping triangles can be found in the pentagram.

If a circle were to be drawn around the pentagon, the golden ratio would be found as a result. The Golden Ratio has a special relationship with the Fibonacci Sequence. The sequence, 0,1,1,2,3,5,8,13,21,34.. , is a representation of the Fibonacci Sequence . It may be noticeable that each pair of numbers starting from O will be equal to the third. If you were to take the ratio of two successive numbers, you would find it getting closer to the Golden Ratio: 1/1=1 2/1=2 3/2=1. 5 5/3=1. 666.. 8/5=1. 6 13/8=1. 625 ….. This would continue until it was reaching a particular value being the Golden Ratio.

In another case, if you had drawn the Golden Ratio as a rectangle and cut a perfect square into it you would remain with another rectangle within the main one. This rectangle would have the same ratio as the main rectangle. It is possible to continue this process as repetition or a pattern and result with a spiral. Figure # 3:-The golden spiral is found in the complexity of all things appealing to the human eye. It is developed through a series of golden ratios all adding up to create the sum of one overall golden ratio. (illustrated by http:// tinyrottenpeanuts. om/golden-ratio-for-kids/) You could then draw a spiral starting from the left corner and making your way around the opposite corners of the squares. This is known as the Golden Spiral. This spiral is supposedly pleasing to the eye and is actually applied in many forms of nature. Some examples include pinecones, seashells, flowers and even the human face. A major example is the sunflower.

A spiral is formed in the center where seeds are formed or are growing. Once a new cell has formed, the spiral turns. As many seeds possible is squeezed with the absence of any possible gaps. In order to optimize the filling, it’s necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. ” (http://www. popmath. org. uk/rpamaths/rpampages/ sunflower. html) The Golden Ratio also in fact, has a connection with proportion, presumably the human body. Leonardo Da Vinci referred to it as the Divine Proportion. He had used this mathematical form in his art, for example, the Vitruvian man. Three Golden Rectangles are found upon the head, torso and legs.

A person’s eyes are Phi of a distance apart from the human’s chin. You can even find the Golden spiral upon the ears of a person, furthermore, making our ears proportional. Based on the original idea of the golden ratio a comparison of any two two aspects will be led to proportion in an ideal way. Examining A and B algebraically, it has to be such that A + B divided by A=A divided by B. It is then identifiable that a comparison will result in the ratio 1:1. 618. It is able to be found all over the human body.

For example, if the hand has the value of 1, then naturally, 1. 18 will become the result for the combination of both the forearm and the hand. Going down to the lower body, the same situation is found. If the foot is found with the value 1 then the combination of both the foot and the shin will be found to equal 1. 618. The face is probably the strongest point of examination toward the golden ratio. The human head forms a golden rectangle where the eyes would be the midpoint. The Golden Sections of the lower area of the chin as well as eyes become the placement for the mouth and nose. Adjacent teeth are in proportion to Phi. A mating factor for example, would be broad shoulders.

They introduce an idea of strength and are therefore attractive to the opposite gender. Looking into the definition of “broad”; something can be defined as “broad” only when what it is being compared to is smaller. Many men then find themselves into the knowledge of a shoulder to waist measurement which is where the golden ratio is put into place. Leonardo Da Vinci had placed himself in extensive study of the Golden Ratio toward the human body. Corpses were examined and then derived for ratios. What is interesting about the relationship to of the human body to the Golden Ratio is the effect on physiology.

Our brains are found to look for anything attracting. “The attracting factor can be considered as anything with clear symmetry and balance. This means that what most humans consider to be a beautiful body is one with symmetry which is a result of the Golden Ratio. It has been argued in past times and today if what we know as beauty is based on close proportion. Another famous figure that had worked with Phi was Plato. He has applied it to his design of Platonic Solids. Illustrations of Platonic solids and parts of polyhedra were draw by Albrecht Durer, a German artist.

In the way it was drawn it was portrayed where a single piece could be cut of figures and folded. After folding it could then be formed to make three dimensional solids. “The Golden Ratio plays a crucial role in the dimensions and symmetry properties of some platonic solids. In particular, a dodecahedron with an edge length (the segment along which two faces join) of one unit has a total surface of 156/13-4 and a volume of 543/6-20. While not all the properties were known in antiquity, neither Plato nor his followers failed to see their sheer beauty. (Livio 70-71) Salvador Dali had illustrated the Golden Rectangle in his art, specifically the “Last Supper”.

The Golden Ratio is found through accurate proportion whereas Jesus and his twelve disciples are all proportional. This includes the table and the windows as well. The Golden Ratio is applied to many forms of beauty. In Ancient Egypt as well as the Renaissance Era this ratio has been used among many famous artists as well as their upholding pieces of art. It is believed that the Ancient Egyptians were the first to use the Golden Ratio when applying it to their construction of pyramids.

When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt” (http://www. goldennumber. net/architecture/) A major example includes the Great Pyramd. Another famous structure of the Golden Ratio is the Parthenon created by Phidias in Athens Greece. The Golden Rectangle is continuously seen in the Parthenon. Each area of the overall Golden Rectangle will contain a square along with another Golden Rectangle. This pattern will continue as proportions are repeated. As the pattern proceeds a spiral will be formed.

Figure #4: Many artists and architects had applied the Golden Ratio to their art. One specifically being the Parthenon created by Phidias. When mathematically examined, the Golden Ratio is found through the identification of the Golden Spiral. (illustrated by http:// www. goldennumber. net/parthenon-phi-golden-ratio/) Another famous applier of the Golden Ratio is Mondrian. He had used it in much of his art, his most famous being “Composition in Red White and Blue. ” Other famous pieces of art include but are not limited to the Taj Mahal, the Mona Lisa, the Notre Dame.

Figure 5: One of the most painting worldwide can be found as the Mona Lisa. It’s superiority as well as appealing figure and beauty is an easy characteristic identifiable to the examiner. When mathematically examined, it is then found of the Golden Ratio which in fact is not a surprise due to the clear beauty in the piece. (illustrated by https://www. vismath. eu/en/blog/goldenratio-mean-art-nature) All these famous buildings are built in proportion with the Golden Ratio and are seen as beautiful to most people. Today the Golden Ratio is still applied to architecture as well as magazines.