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The Golden Ratio
Transcript of The Golden Ratio
(A+B)/A= A/(B )= P
One method is starting with the left fraction. by simplifying it and substituting in B/A = 1/P
(A+B)/A = 1 + B/A = 1+ 1/P
This shows that
Then Multiply it by P
and rearrange it to
P^2 - P -1 = 0
Plug in the values to the quadratic formula
and take the positive solution, which proves that
P = (1+√5)/2 = 1.6180339887…… The Golden ratio has Invoked western thought for over 2400 years. It was researched by Leonardo da Vinci, Pythagoras, Euclid, Johannes Kepler, and many other notable figures in mathematics, art, biology, music, psychology, architecture, history, and mysticism. The Ancient Greeks first realized the high frequency that the ratio appeared in geometry, especially in pentagrams and pentagons. Pythagoras was credited with its discovery. Euclid provided the first definition of the Golden Ratio in his Elements writing, "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 less." he describes sectioning a line into the golden the ratio, as well as theorems that employ the golden ratio. Luca Pacioli inspired a new wave of interest in the golden ratio with his book De Divina Proportione, Illustrated by Leonard da Vinci. the first part of the book describes the mathematics of the inverse golden ratio and its applications in art
Micheal Maestlin gave the first decimal approximation of the Golden ratio to Johannes Kepler. He estimated that it was 0.6180340.
Kepler proved that it was limit of the ratio of consecutive Fibonacci numbers. De Divina Proportione explored the mathematics and artistic applications of the golden ratio. The author, Luca Pacioli, was a Franciscan friar who saw significance of the ratio in Catholic Religion. Though he believed in the Vitruvian system of rational proportions, many saw the book as advocating that the golden ratio had pleasing and harmonious proportions. The Parthenon's facade is circumscribed by golden rectangles (see the second picture in the presentation). The Great Mosque of Kairouan has a consistent application of the golden ration throughout its design. They are found in the overall plan, and dimensioning of the prayer space, court and minaret. Swiss architect Le Corbuisier, a pioneer of the modern international style, the architectural equivalent of modernist art, centered his design philosophy on systems of harmony and proportion. He strongly held faith in the mathematical universe, bound to the Fibonacci sequence and Golden Ratio. He said, "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned." He directly used the golden ratio in his modular system of architectural proportion. Le Corbusier also based the system on human body proportions, the Fibonacci sequence, and the double unit. He sectioned of the body in the golden ration to formulate this. Le Corbusier's system can be found in his building, Villa Stein. BY Brian Van Dyke In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. (a+b)/a= a/(b )= P is the above statement expressed algebraically P = (1+√5)/2 = 1.6180339887…… Is the golden ratio Line segments in the golden ratio A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship (a+b)/a= a/(b )= P The golden ratio in the human head and rhombicuboctahedron as illustrated by da Vinci in De divina proportione Euclid Pictures of the golden ratio in architecture Application in Painting Heinrich Agrippa first implied that the human body was proportioned by the golden ratio by drawing a man inside a pentagram. Da Vinci's Illustrations in De Divina Proportione started speculation that the golden ratio was used in his paintings, such as the Mona Lisa. Salvador Dali, influenced by Matila Ghyka, used the golden ration in his masterpiece, The Sacrament of the Last Supper. Mondrian's geometric paintings also show use of the golden section.Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6. Golden ratio in paintings. Application in book design Many books from 1550 to 1770 adhered to the golden ratio proportions of 2:3 and 1:√3 within one millimeter Depiction of the proportions in a medieval manuscript. Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section. Application in Finance The golden ration is used in stock trading algorithms, applications and strategies such as
the Fibonacci fan, arc, retraction, and time extension Industrial design applications of the golden ratio The golden ratio is shown to exist in postcards, playing cards, posters,Wide screen televisions, photographs, and light switch plates Golden Ratio in music Bela Bartok's music has been analzyzed to be comprised of two opposing forces, the golden ratio and the acoustic scale In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. Erik Satie and Debussey also use the golden ratio in their works. Pearl Drums place their air vents on the master premium drums in the golden ratio If you're dizzy from the prezi, take a break and listen to some music based off the golden ratio/Fibonacci sequence The golden ratio in nature Adolf Zeising discovered that the golden ratio was expressed in the arrangement of tree branches, leafs on plants, and veins in leaves. He also found the ratio expressed in skeletons, nerves, veins, chemical compound proportions, and crystal geometry. combined with golden ratio based human body proportions, he wrote a universal law of the golden ratio in nature "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form." golden ratio has been found to affect human perception of attractiveness, the human genome and atomic science Golden ratio in nature examples Golden Ratio conjugate Conjugate root- the negative root of an quadratic equation.
The conjugate root of the golden ratio is -1/p = 1-p = (1-√5)/2 = - 0.6180339887......
This is just the ratio taken in reverse order
P = 1/p =1/1.6180339887.... = 0.6180339887.....
P can also be expressed as
P = p-1 = 1.6180339887.... = 0.6180339887.....
This is a unique property of the golden ratio among other positive numbers because it and its inverse (1/p = p-1 and 1/P=P+1 respectively) are 0.61803......:1=1:1.61803...... The golden ratio is irrational Here is two reasons why:
Contradiction from an expression in lowest terms- In the golden ratio, the whole is the longer part plus the shorter part and the whole is to the longer part as the longer part is to the shorter part. if the longer part is X and the whole is Y the second statement is: Y is to X as X is to Y-X, or Y/X = X/(Y-X). If the golden ratio(p) is rational, then p is a fraction Y/X and X and Y are integers. You can take Y/X to lowest terms and Y and X to be positive. If Y/X is in lowest terms then Y/X = X/(Y-X) says that X/(Y-X) can be in lower terms, a contradiction that proves irrationality.
Derivation from irrationality of √5- a shorter way to prove that the golden ratio is irrational uses the closure of rational numbers under addition and multiplication. If (1+√5)/2 is rational, then 2((1+√5)/2 -1/2) = √5 is also rational.this is a contradiction because a square root of a non-square natural number is irrational. A picture to help understanding If P (the golden ratio) were rational, then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so P cannot be rational. Alternative forms of the golden ratio (p) p= 1+1/p can be expanded to find a continued fraction for the golden ratio: p=[1;1,1,1,...] = 1+1/1+1/1+1/1+...
Here is the reciprocal
p^-1 =[0;1,1,1,...] =0+1/1+1/1+1/1+...
The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers.
p^2=1+p produces the continued square root of the golden ratio:p=√(1+√(1+√(1+√(1+⋯)) ) )
This is an infinite series derived to express p:
p= 13/8+ ∑_(n=0)^∞▒((〖-1)〗^((n+1) ) (2n+1)!)/(n+2)!n!4^((2n+3))
Here are some statements that correspond to the fact that the length of a regular pentagon is p times the length of its side:
p=2sin(3pi/10)=2sin54 Geometry pt1 : overview The golden ratio is present frequently in geometry. It is present in all figures of pentagonal symmetry. the length of a regular pentagon diagonal is p's (the golden ratio) its side.The vertices of icosahedron are golden rectangles. Logarithmic expressions graphed inside golden rectangles. Geometry 2: Geometry Harder This algorithm shows a line that is divide into two segments in the golden ratio.
1. Draw a right triangle and name it triangle ABC
2. Draw a circle with a center of C and a radius of BC have the arc inside the triangle intersect the hypotenuse at D
3. Draw a circle with the center of A and a radius of AD. Where it intersects AB is called S and it divides the line into the golden ratio. Geometry pt 3: Live Free or Geometry Hard The golden triangle- the golden triangle is an isosceles triangle ABC that when bisected from C forms a new, similar triangle. The Original angles are 36 72 72. The similar daughter triangle has the same angles as the original and the dissimilar triangle formed when the originals is divided angles are 36 36 108. A triangle's scalenity must always be less then the golden ratio.
The pentagon- In a regular pentagon the ratio between a side and a diagonal is p, while intersecting diagonals section each other in the golden ratio. George Odom had a special way of constructing a pentagon. He found that if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. then you use the intersecting chords theorem (AP*AQ=AR*AS) to draw a pentagon.
The pentagram- The golden ratio is present in the geometry of pentagrams. Every intersection of the edges in a pentagram is in the golden ratio. The ratio of the length of the shorter segment to the segment composed of two intersecting edges is the golden ratio. Five of the isosceles triangles in a pentagram are golden triangles (the acute ones) and the obtuse isosceles triangles are golden gnomons (The triangles that were not similar to the original triangle when we divided the golden triangle).
The golden rhombus - A golden rhombus is a rhombus whose diagonals are in the golden ratio. Geometry pt 4: Geomocalyse 2012 Relationship to the Fibonacci sequence golden triangle Odom's pentagon construction A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another. A golden rhombus The Fibonacci sequence is a sequence which starts out with a zero and two ones and every term is the sum of the two numbers before it except for the first two. The terms are: 0,1,1,2,3,5,8,13,21,34,55,89, 144, 233,377,610,987...
Two terms of the Fibonacci sequence divided by each other (987/610 for example) are the golden ratio. This is know as the limit of the sequence. The higher the terms are, the more accurate the sequence is to the golden ratio. the success powers of the Golden ratio obeys the Fibonacci recurrence, meaning that they are in a similar pattern to the Fibonacci sequence.
A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to 34 Reflection When I first heard of the golden ratio, I grouped it along the lines of “aliens control the pentagon” or “our lives are just giant governmental experiments run by a supercomputer.” To me, there was no way that one number would be so omnipresent in so many facets of our life. Out of interest, I did this as my math project topic, and my mind was sufficiently blown. Reading and learning about all the truth behind the golden ratio just astounded me. I never knew the Mona Lisa, the patterns on my fish, my mini palm tree, and Egyptian pyramids were all connected by one number. Now, when I look around my house I see it. Painting- Boom golden rectangles, bowl- golden spiral, my arms- in golden ratio, even the teenage mutant ninja turtle on my desk is in the golden ratio. This ratio just kept popping up in my everyday life. I feel like I know a secret about how the universe was created just by recognizing these numbers. There is not one subject area in school that does not relate to the golden ratio. As I did more research I found out about new mathematical concepts I had no idea about. There are ten different ways two write the golden ratio! I want to learn more about how you can write one ratio ten ways. Before this project I never knew what a limit is, was a recurrence is, how to prove irrationality, what an algorithm is, and how seemingly complex formulas can be rewritten into something I could understand in eighth grade. Another thing this project taught me was how to use a prezi. The incredibly high standard set by the project required me to actually impress somebody with what I was doing. In Junior High, I would go to the extreme with all of my projects. I had the longest essays, the largest piñata, the strongest, most earthquake proof house, the grandest utopian city, the most detailed vacation plan, the largest piece of clay ever to be fired in the Junior High’s kiln; I even combined elements of seven different skyscrapers into my ninth grade project building. My project policy was shock and awe and I held it with an iron fist. In high school that policy turned to appeasement, do the bare minimum to get the maximum grade and a satisfied teacher. The radical ideas, mountains of creative effort, and mind-blowing concepts were gone, replaced by a cookie-cutter, right-off-the-rubric, project accompanied by truckloads of persuasion to pass it off as a miracle. I did most of my projects the night before, or even the day of and I presented the first topic I thought about, not what I really wanted to do. Aside from the biology ecosystem project, that was how my projects were. Then I got the rubric for this project, and I found out that 100% actually was 80% and in order to reach “perfection” I had to do 120% effort. Scrambling for unconventional ideas and practices that were rusty from a year of neglect, I ran into a mass media format called a prezi. Learning how to incorporate pictures, videos, top-notch content and wild transitions in to a free-form project was hard, but I learned how to do all of it. This is the first project I invested significant effort and time into for an entire year, and I got back a new outlook on school and a renewal of interest in the unknown. Bibliography Bejan, Adrian. "Researcher Explains Mystery of Golden Ratio." Researcher Explains Mystery of Golden Ratio. Duke University, n.d. Web. 20 Dec. 2012.
"Golden Ratio." - Encyclopedia of Mathematics. N.p., n.d. Web. 20 Dec. 2012.
"Golden Section." Demonstrations RSS. N.p., n.d. Web. 20 Dec. 2012.
Green, Thomas M. "The Pentagram and The Golden Ratio." N.p., n.d. Web. 20 Dec. 2012. <http://web.archive.org/web/20071105084747/http://www.contracosta.cc.ca.us/math/pentagrm.htm>.
Knott, Ron. "The Golden Section Ratio: Phi." The Golden Section. N.p., n.d. Web. 20 Dec. 2012.
Weisstien, Eric W. "The Golden Ratio." Mathworld.com. Wolfram.com, n.d. Web. 20 Dec. 2012. <http://mathworld.wolfram.com/GoldenRatio.html>. Videos Recognize the shape ? Some pictures to help understanding