...Central Luzon State University College of Arts and Sciences Department of Psychology The Golden Ratio: Reference of Beauty Submitted to: Nixon Agaser Submitted by: Jam Shimei I. Gamboa Jeanne R. Dadufalza Eden Leigh Pagoso Micah A. Montes The Golden Ratio: Reference of Beauty Beauty has been defined as a combination of qualities that give pleasure to the senses or to the mind. It is a philosophical concept, the aspects of which are studied under the term aesthetics, derived from the Greek word for perception (aisthesis). Aesthetics, therefore, is the study of beauty and, to a lesser extent, it’s opposite the ugly. It involves both the understanding and evaluation of beauty, proportions and symmetry. The assessment of facial beauty is immersed in subjectivity and therefore leans towards the world of art. Facial proportions and facial balance, however, can be measured and therefore fit somewhere between art and science. Aesthetics itself is now essentially a science in the formation, although obviously with a very strong philosophical and artistic background (Naini, et al. 2008.) According to Jang (2014) it is becoming clear that there is a science to beauty. Extensive research has shown that we have similar ideas about what constitutes a beautiful face. When British researchers asked women to rate pictures of various men, their choices were identical. When a group including English, Chinese, and Indians were asked to choose a “good looking” face from a selection of racially...
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...THE GOLDEN RATIO Enrico Freitag The golden ration can occur anywhere. The golden proportion is the ratio of the shorter length to the longer length which equals the ratio of the longer length to the sum of both lengths. The golden ratio is a term used to describe proportioning in a piece. In a work of art or architecture, if one maintained a ratio of small elements to larger elements that was the same as the ratio of larger elements to the whole, the end result was pleasing to the eye. The ratio for length to width of rectangles is 1.61803398874989484820. The numeric value is called “phi”. The Golden Ratio is also known as the golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern. The Golden Rectangle is a unique and important shape in mathematics. The Golden Rectangle appears in nature, music, and is often used in art and ar chitecture. Something special about the golden rectangle is that the length to the width equals approximately 1.618… . The golden rectangle has been discovered and used since ancient times. Our human eye perceives the golden rectangle as a beautiful geometric form. The symbol for the Golden Ratio is the Greek letter Phi. The Fibonacci Series was discovered around 1200 A.D. Leonardo Fibonacci discovered the unusual properties of the numeric series, that’s how it was named. It is not proven that...
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...The golden ratio is a mathematical ratio of 1 to about 1.618. The golden ratio is found naturally in the world and in nature, but we can create if we try. Most artists that use the golden ratio either use it for the purpose of showing their sheer ability in mathematics to use it correctly and efficiently in their painting, sculptures, and other works of art, or for the mere fact that mankind as a whole has an, as of yet, unexplainable attractions to the golden ratio. Or in other words, the golden ratio makes things better looking or more appealing to the human eye. And because art is seen through the eye of the beholder, having the golden ratio throughout one’s work could be the difference between a piece of art and masterpiece. The first piece of artwork I want to talk about is the Last Supper by renowned painter Leonardo da Vinci. There are golden ration rectangles everywhere. You can see the rectangles on the entire length of the table where the disciples and Jesus are eating. You can also see the rectangles about the faces of said people, split on the disciples on the left, Jesus in the middle, and the disciples on the right. The feet of all the people eating are in their own rectangles as well, once again on the left, middle, and right respectively. Additionally, if you look at the background you can see the golden ratio on the back wall and the windows with an outside view. The next piece of artwork I would like to talk about is the most famous painting in the world the...
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...The golden section proportion is seen in many different pieces of art whether it be in sculpting, painting, drawing, etc. It is a very common practice among many artists because it makes things more appealing to the eye. It’s also known as the rule of thirds because in using the golden section proportion artists tend to try to get the ratio of an object they are drawing to be in the proportion of 618:382. Not quite an even one third, but close enough to where it is referred to as such. It is also referred to as divine proportioning because that ratio is found many times throughout the body. An example of that would be the wrist to the fingertips in proportion to the entire elbow to the fingertips. The tapestry on the wall in this painting is an example of divine proportioning. The middle of the painting also follows the Golden Section, with two thirds visible and one third not. Another example is the man pouring wine from the jug is positioned in a way that allows the viewer to see 2/3rds of the people in the background. The doorway full of people would be another example, it seems to fill up a certain portion of the picture without detracting from the main subject of the painting. In this painting alone there are many different ways that we can see how the rule of thirds is implemented into artwork, and we can see it in almost all artwork...
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...The Da Vinci Code, which was released in the May of 2006, was about a murder that took place in the Louvre in France. There were clues in all of Da Vinci paintings throughout the museum. These clues lead to a discovery of a religious mystery that has been protected and kept hidden for two thousand years. This throws Robert Langdon and the victim’s granddaughter into a bizarre murder and crazy mystery, Not only is this movie a good movie and an interesting mystery to watch. But I can also connect what I’ve learned in sacred geometry to this movie in many aspects. One of the first examples of something I connected to our class, sacred geometry, was the shape of the museum that the dead body was found. The building that the body was found in was the Louvre museum. The Louvre museum is located in Paris, France and was established in 1793. In front of the actual museum there is something that is known as the Louvre pyramid. The Louvre pyramid is a large glass and metal pyramid surrounded by three smaller pyramids. The large pyramid serves as the entrance to the Louvre museum and was opened in 1989. The reason that I can connect this to our course in sacred geometry is the actually structure of the pyramid. There are many small triangles on it, which are also known as the triad. The triad is a three-sided shape and also is known as the first and oldest number. Also this is a pyramid and has many pyramids throughout it. The pyramid is something that we talked about in class to as...
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...have known this ratio for years. It's derived from something known as the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.). But this sequence is not all that important; rather, it is the ratio of the adjacent terms that possess great qualities, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. Almost everything that has dimensional properties adheres to the ratio of 1.618. 2. Prove It! Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different things throughout nature. Still don't believe it? Try measuring from your shoulder to your fingertips, and then divide this number by the length from your elbow to your fingertips. Or try measuring from your head to your feet, and divide that by the length from your belly button to your feet. The golden ratio is seemingly unavoidable. The markets have the very same mathematical base as these natural phenomena. 3. The Fibonacci Studies and Finance When used in technical analysis, the golden ratio is typically...
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...MTH 151, Section 042M Professor Brofft 23 May 2012 Fibonacci's Sequence The Fibonacci Sequence is one of the most famous, if not the most famous numerical sequence in history. "In the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on? The total number of pairs, month by month, forms the sequence 1,1,2,3,5,8,13,21,34,55,89, and so on. Each new term is the sum of the previous two terms." (1) As mentioned previously, "the solution of this problem leads to a sequence of numbers known as the Fibonacci sequence. Here are the first fifteen terms of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 (equivalent to 1.618). After the first two terms, in the sequence, each term is obtained by adding the two previous terms. For example, the third term is obtained by adding 1 + 1 to get 2, the fourth term is obtained by adding 1 + 2 to get 3, and so on." (2) Leonardo of Pisa, born in 1170, grew up and traveled throughout Africa and the Mediterranean during much of his childhood and early adult life, learning different mathematical styles and formulas, which were of great interest to him. During these travels, Fibonacci (nicknamed many years after his death; 'Bonacci' means "son of good fortune". While living, Fibonacci...
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...Sparrow Ostas Densmore October 31, 2013 2nd Block Section Summaries: Pages 14 – 32 Do A Number: This phrase relates to the sport of boxing. Coaches inform their boxer to hit the opponent x amount of times, which can be any number Three Sheets to the Wind: This phrase means “extremely drunk.” This phrase comes from ropes, which all have a different function. The math involved is “sheet” ropes, which control the horizontal movement of the sails. If three sails are loose, then the sailors are extremely drunk. The Third Degree: This phrase means that when people got interrogated for their past crimes, they got highly searched (third degree). This phrase includes the math were the members of an old ritual of Freemasonry, which were graded by degrees. The Fourth Estate: This phrase means the social ranks of the 1789 Estates-General. The first where the clergy, than the nobles, and finally the bourgeoisie (the wealthiest). But the Fourth Estate was the most influential on ordinary French people – newspapers and reporters. Fourth Wall: This phrase is a “wall” that separates a theatrical performance from the audience. But now this term is applies when a character “breaks” the fourth wall and addresses the audience directly. Five by Five: This phrase is a term for a NATO radio speak system. Signals are rated by one – five (five being the clearest and most understood signal). It is usually used to indicate that something is understood. Fifth Columnist: This phrase originally comes...
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...Parthenon Peristasis | | Ates Gulcugil Abstract Two golden ratio models will be constructed for the peristasis of the Parthenon and their dimensions compared with the actual one. Definitions Φ: The golden ratio, 1.618... Golden Numbers: Integer powers of Φ. Interval: Distance between two, neighboring, parallel line segments. Golden Interval: An interval which is a golden number. Normalizing: Dividing each dimension of a structure by a selected one of its dimensions. Aspect Ratio: Ratio of longer-to-shorter side of a rectangle. The Parthenon Peristasis The dimensions (in feet) of the Parthenon peristasis as measured by Francis Penrose are shown in the following diagram. There are two different kinds of intervals in the peristasis: The corner spaces (total 8), and the intercolumnia (total 38). These are shown below. Substituting the (scaled up by 101.361/101.341) intervals of the left hand side for the unmeasured right hand side, the mean values are figured out as: Mean corner space: 15.448 ft Mean intercolumnium: 14.090 ft The peristasis with the mean values is shown below. Golden Ratio Relations If the Parthenon was designed around the golden ratio there must be a golden ratio relationship between the corner space and the intercolumnium. The following diagram shows the analysis of these two intervals. The intercolumnium, because it is uninterrupted, will be considered as a golden interval, (interval a). When normalized with respect to the intercolumnium...
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...A Study of Phi and its Importance in Human Choices Concerning Beauty By: Anthony McCabe Abstract This paper aims to answer what Phi is, where it is found in nature, and how it affects humans concerning our search for beauty. This is done through graphically and mathematically finding Phi, and identifying its unique properties. The history of Phi is explored, and its usage in the past is covered. Phi is then applied to nature, through its presence in the Golden Angle, nature, and architecture. Phi is then explored in human nature, when it comes to our physique and psychological choices. This leads to a conducted survey showing human wants in facial appearance, relevant to Phi. The results show a significant amount of people prefer the face closest related to Phi, supporting the hypothesis that Phi plays an important role in human beauty. Phi is found to be a mathematical phenomenon that predates even math itself, and has always been useful to societies and to nature itself. Phi is found everywhere in our world and makes objects and patterns seem more elegant because of its presence. This is relevant to humans as well, as, concerning beauty, Phi is a powerful measurement that psychologically attracts us at our most basic and primitive levels. A Study of Phi and its Importance in Human Choices Concerning Beauty One object, one thing, can be viewed in many different ways by many different types of people. For example, a piece of wood is a tool, or a building block...
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...No of flowers | Fibonacci | Non Fibonacci | 30 | 18 | 12 | Proportion | .6 | .4 | Z value | 1.09 | | P value | .1366>.05 | Null acceptance | As we can see the above table, the p value is greater than .05, (level of significance) the alternate hypothesis that the number of petals in flowers follows Fibonacci series is not conclusively proved. However even the proportion of 0.65 would have proved the Fibonacci hypothesis. We feel that sample size of 30 is insufficient. GOLDEN RATIO What is it exactly? Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887. What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can be produced simply by adding 1 and who’s reciprocal by subtracting 1. The golden ratio is also difficult to pin down: it's the most difficult to express as any kind of fraction and its digits - 10 million of...
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...arithmetic and geometric sequence. An arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms. 1, 3, 5, 7, 9,…. The numbers in the sequence are called terms. Thus, 1 is the first term, 3 is the second term, 5 is the third term, and so forth. The symbol Un denotes the first term of a sequence. Since the first term is 1, we can express it as Un= 1. Each consecutive terms has a difference (denoted as d). Meanwhile, a geometric sequence is a sequence in which the same number is multiplied or divided by each term to get the next term in the sequence. 3, 9, 27, 81, 243,…. This is an example of geometric sequence. The quotient of a term with its previous term is called ratio. In geometric sequence, the ratio between each successive terms is constant. What is a Fibonacci sequence? Fibonacci sequence is a special sequence. It is founded by an Italian mathematician named Leonardo Pisano Bigollo, known commonly as Fibonacci. According to WhatIs.com, it is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... (1) 1, 1, 2, 3, 5, 8, 13, 21, 34 ... Those are the two common forms of Fibonacci sequence. The sequence is denoted F ( n ), where n is the first term in the sequence. Thus, in...
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...phenomena in the brand’s history. It communicates the brand in a timeless manner and with an expression of clarity. Pepsi BREATHTAKING builds on this knowledge. True innovation always begins by investigating the historic path. Going back-to-the-roots moves the brand forward as it changes the trajectory of the future. 1910 1970 2009 BREATHTAKING Universal Design Principles BREATHTAKING is a strategy based on the evolution of 5000+ years of shared ideas in design philosophy creating an authentic Constitution of Design. This chart documents the origin and evolution of intellectual property. 3000 BC 600 BC 300 BC 278 BC 70 BC 1455 1637 1858 1948 2009 Vāstu Śāstra: Musica Mundana: Golden Ratio: Feng Shui: Vitruvian Principle: The Art of Building: Vitruvian Renaissance: La Géométrie: Möbius Strip: The Modulor:...
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...Mathematical symbols are all around us yet most people to not seem to realize this usage of what many have dreaded as they made their way through various levels of the education system. Some uses of symbols are right in plain sight every time we drive our vehicles to various locations. For example, think about what greets you at your steering wheel as you enter the highway and decide to engage your cruise control. There right in front of your face is the “+” and “-“ symbols that we all equate with increase or decrease. Two of the most basic of mathematical symbols from algebra and they stare at us virtually every day. Another area that struck me as I was recently searching for a chemical for my work was how often mathematical symbols make their way into the logos for companies. I first noticed it while looking for some chemicals and came across Sigma-Aldrich which uses the uppercase Greek letter “” prominently as part of their logo. Of course as anyone with even a minor exposure to calculus and statistics knows this symbol is used extensively to indicate a summation of values. Another work related logo that just about poked me in the eye is a SPC software package I use on a regular basis from Infinity QS. This company uses the familiar symbol “∞” as part of its’ logo on Proficient QS software. I decided to do some further research into companies that incorporate mathematical symbols in their corporate logos. What I quickly found was the spectrum of companies...
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...Man and Nature Have you ever stopped to think about the relationship between yourself and Mother Nature? For most people chances are slim to none, in fact many may not even consider the fact that there might even be any kind of relationship between nature and themselves. As far as anyone might be concerned in today’s society, nature could just mean their backyard, or neighborhood park. In reality there is much more to you and I and this wilderness we refer to as nature. In this paper I argue that there exists a higher connection between man and nature that serves to unify all living things. Today, man and nature are commonly referred to in opposition of one another. Man destroys nature in order to expand and urbanize while nature destroys all man creates over time. People tend to see nature as some uncontrollable wild factor full of danger and chaos. Many think like Thomas Hobbes who would say that the very state of nature is chaotic; that if man were without society he would be inherently evil selfish with only self interest in mind and life would be lonely, difficult and short. However, if taken from a Rousseauian stand point, nature and man share an interest for self-preservation giving them a natural sense of compassion and the state of nature is calm and peaceful. I would have to say that the Rousseauian perspective makes more sense and ties into reality better than Hobbes’s state of nature. The main reason being that all nature moves towards a state of homeostasis...
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