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1.1.3. GOLDEN RATIO IN ARTS AND ARCHITECTURE
What is golden ratio? The golden ratio is a structural device based on the patterns of nature, that has been used by artists and designers for centuries. Hokusai, The Great Wave off Kanagawa, 1831 The golden ratio, also known as the divine proportion, is a mathematical ratio of 1:1.618, or Phi, with a decimal that stretches to infinity, closely linked to the Fibonacci sequence. Sometimes it is also referred to as the golden section, the golden mean, the golden number, the divine proportion, or the golden proportion. Many artists and designers throughout history have adopted this mathematical equation as a means of creating balance, order and symmetry. The example below shows how the golden ratio looks – it appears as a large rectangle, divided into a series of squares. In order to find the exact proportions of the golden ratio, you can divide a rectangle into two, making sure the longer part divided by the smaller part is equal to the whole length divided by the long part. As you can see in the example below, the sequence of divisions can carry on ad infinitum in a curve. The golden ratio is a mathematical concept that has fascinated architects and designers for centuries. It is a proportion that is found in nature, arts, and architecture, and it’s said to creatye a sense of harmony and balance in design. For painters, architects, sculptors or poets, the golden ratio in art exerts a power of fascination. It rules the multiple and appear in several formulas, like the fibonacci sequence, the spiral and the golden rectangle. Architects have beenusing the golden ratio in their designs for centuries, believing that it creates a sense of balance and harmony in a building.
The one of purposes of this project is to overview the golden ratio briefly. The other is to introduce the occurrences of the golden ratio in art and architecture. The content includes the following : I.A discovery of the Golden Ratio A. A brief history of the Golden Ratio B. Definitions of the Golden Ratio related to Fibonacci sequence number II. Some Golden Geometry III. The Golden Ratio in Art and Architecture IV. Resources I. A discove ry of Golden Ratio A. A brief history of Golden Ratio There are many different names for the golden ratio; The Golden Mean, Phi, the Divine Section, The Golden Cut, The Golden Proportion, The Divine Proportion, and tau(t). The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the use of the golden ratio. The length of each side of the base is 756 feet, and the height is 481 feet. So, we can find that the ratio of the vase to height is 756/481=1.5717.. The Rhind Papyrus of about 1650 BC includes the solution to some problems about pyramids, but it does not mention anything about the golden ratio Phi. Euclid (365BC - 300BC) in his "Elements" calls dividing a line at the 0.6180399.. point dividing a line in the extreme and mean ratio. This later gave rise to the name Golden Mean. He used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (GB/AG) is the same as the ratio of the larger part, AG, to the whole line AB (AG/AB).Then the definition means that GB/AG = AG/AB. proposition 30 in book VI Plato, a Greek philosopher theorised about the Golden Ratio. He believed that if a line was divided into two unequal segments so that the smaller segment was related to the larger in the same way that the larger segment was related to the whole, what would result would be a special proportional relationship. Luca Pacioli wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. Leonardo Da Vinci first called it the sectio aurea (Latin for the golden section). Today, mathematicians also use the initial letter of the Greek Phidias who used the golden ratio in his sculptures. B. Definitions of Golden Ratio 1 ) Numeric definition Here is a 'Fibonacci series'. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .. If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers. 1/1 = 1 2/1 = 2 3/2 = 1. 5/3 = 1.6666... 8/5 = 1.
3) Algebraic and Geometric definition We can realize that Phi + 1 = Phi * Phi. Start with a golden rectangle with a short side one unit long. Since the long side of a golden rectangle equals the short side multiplied by Phi, the long side of the new rectangle is 1*Phi = Phi. If we swing the long side to make a new golden rectangle, the short side of the new rectangle is Phi and the long side is Phi * Phi. We also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or Phi + 1. (figure in page4) Since these two expressions describe the same thing, they are equivalent, and so **Phi + 1 = Phi * Phi. II. Some Golden Geometry
- The Golden Rectangle** A Golden Rectangle is a rectangle with proportions that are two consecutive numbers from the Fibonacci sequence. The Golden Rectangle has been said to be one of the most visually satisfying of all geometric forms. We can find many examples in art masterpieces such as in edifices of ancient Greece.
2) The Golden Triangle If we rotate the shorter side through the base angle until it touches one of the legs, and then, from the endpoint, we draw a segment down to the opposite base vertex, the original isosceles triangle is split into two golden triangles. Aslo, we can find that the ratio of the area of the taller triangle to that of the smaller triangle is also 1.618. (=Phi) If the golden rectangle is split into two triangles, they are called golden triangles suing the Pythagorean theorem, we can find the hypotenuse of the triangle. 3) The Golden Spiral The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle. If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi. This shape is found in many shells, particularly the nautilus.
We can get an approximate pentagon and pentagram using the Fibonacci numbers as lengths of lines. In above figure, there are the Fibonacci numbers; 2, 3, 5, 8. The ratio of these three pairs of consecutive Fibonacci numbers is roughly equal to the golden ratio. III. Golden Ratio in Art and Architecture A. Golden Ratio in Art
- An Old man by Leonardo Da Vinci Leonardo Da Vinci explored the human body involving in the ratios of the lengths of various body parts. He called this ratio the "divine proportion" and featured it in many of his paintings. Leonardo da Vinci's drawing of an old man can be overlaid with a square subdivided into rectangles, some of which approximate Golden Rectangles. 2) The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci We can draw many lines of the rectangles into this figure. Then, there are three distinct sets of Golden Rectangles: Each one set for the head area, the torso, and the legs.
3) Mona-Risa by Leonardo Da Vinci This picture includes lots of Golden Rectangles. In above figure, we can draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head. Then we will have a golden rectangle. Also, if we draw squares inside this Golden Rectangle, we will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. It is believed that Leonardo, as a mathematician tried to incorporate of mathematics into art. This painting seems to be made purposefully line up with golden rectangle. 4) Holy Family by Micahelangelo We can notice that this picture is positioned to the principal figures in alignment with a Pentagram or Golden star.
**3) Porch of Maidens, Acropolis, Athens
- Chartres Cathedral** The Medieval builders of churches and cathedrals approached the design of their buildings in much the same way as the Greeks. They tried to connect geometry and art. Inside and out, their building were intricate construction based on the golden section. 5) Le Corbussier In 1950, the architect Le Corbussier published a book entitled "Le modulator. Essai sur une mesure harmonique a l'echelle humaine applicable universalement a l'architecture et a la mecanique ". He invented the word "modulator" by combining "modul" (ratio) and "or" (gold); another expression for the well-known golden ratio.
III. Resoureces Internet Michael's Crazy Enterprises, Inc., The Golden Mean (http://www.vashti.net/mceinc/) The Golden Ratio (http://www.math.csusb.edu/course/m128/golden/) Ron Knott, The Golden section ratio : Phi (http://www.ee.surrey.ac.uk/Personal/R.Knott/) The Golden Ratio (http://library.thinkquest.org/C005c449/) Ron Knott, Fibonacci Numbers and Nature-part 2, Why is the Golden section the "best" arrangement? (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/) Ron Knott, The Golden Section in Art, Architecture and Music (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/) Steve Blacker, Jeantte Polanski, and Marc Schwach, The Golden Ratio (http://www.geom.umn.edu/~demo5337/s97b/) Ethan, The relations of the Golden ratio and the Fibonacci Series (http://mathforum.org/dr.math/problems) Golden Section in Art and Architecture (http://www.camosun.bc.ca/~jbritton/goldslide/) Sheri Davis and Danny Rhee, Mathematical Aspects of Arichitecture (http://www.ma.uyexas.edu/~lefcourt/SP97/M302/projects/lefc023/) Mathematics and Art (http://www.q-net.au/~lolita/) Leonardo da Vinci (http://libray.thinkquest.org/27890/) Math & Art : The golden Rectangle (http://educ.queensu.ca/~fmc/october2001/) Sue Meredith, Some Explorations with the Golden Ratio ( http://jwilson.coe.uga.edu/EMT668/) What is a Fractal? (http://ecsd2.re50j.k12.co.us/ECSD/) Ron Knott, Phi's Fascinating Figures (http://www.euler.slu.edu/teachmaterial/) Cynthia Lanius, Golden ratio Algebra (http://math.rice.edu/~lanius/) Newsletter, Mathematical Beauty (http://www.exploremath.com/news/
mathematics areas, such as Algebra, the most common expressions are numbers, sets, and functions. A mathematical sentence is the mathematical analogue of an English sentence. That is, it is a correct arrangement of mathematical symbols that state a complete thought. Hence, it makes sense to ask if a sentence is true, false, sometimes true, or sometimes false. Take a look at the table below to visualize the difference between expression and sentence in mathematics. ENGLISH MATHEMATICS NOUN (person, place, things, events, etc.) Examples: Mark, Ben, Owen
EXPRESSION
Examples: 21, 9+8, 9x9, 4x, f(x)=6x- SENTENCE Examples: The province of Albay is a province in Bicol Region, Philippines. Manila City is in National Capital Region of the Philippines.
SENTENCE
Examples: 8+7= 7+2= 6x6=
1.2.1- Mathematical expression and sentences 1.2.2- Mathematical phrases to symbols
