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# The Fibonacci Sequence and the Golden Ratio

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The Fibonacci Series was firstly introduced by Leonardo of Pisa, known as Fibonacci, in the year 1202. He researched on the populace of rabbits. Firstly he assumed a newly-born couple of rabbits, one guy, one female, are put in a field; one month later, rabbits become adult and are able to partner so that at the end of its second month a female rabbit can produce another couple of rabbits; he also assumed that rabbits never perish and a mating couple always produces one guy and one female rabbit on a monthly basis from the next month on. The question that Fibonacci posed was: just how many pairs maybe there is in one yr?

At the finish of the first month, the match mate, however they don't create a set, therefore there continues to be one only one 1 pair. At the end of the next month the few produces a new pair, so now there are 2 pairs of rabbits in the field. One of these is adolescent and the other is leverets. By the end of the third month, the initial pair produces a second couple, the leverets become adolescents hence a complete of 3 pairs in every in the field. For another month, two adolescent pairs produce two new pairs and the newly-born pair become adult. Therefore, our field consits five pairs of rabbits. The terms of the sequence receive as,

The Golden Proportion is a particular type of proportion that may be seen on many composition of living organisms and many things. It isn't only seen in the part of a complete content, but also in arts and structures for years and years. The Golden ratio gives the most appropriate sizes of geometric characters. In dynamics, The Golden Proportion is seen on the physiques of humans, shells and branches of trees and shrubs. For Platon, the secrets of the cosmical physics is this ratio. Also, this proportion is widely presumed that it is the most cosmetic ratio for a rectangle. The Golden Percentage, can be an irrational number as pi or e and its approximate value is 1, 618033988 To explain the Golden Proportion, or PHI can be used.

The Golden Percentage has been used for quite some time for different purposes. Some studies of the Acropolis, the approximate value of gold ratio can be seen on a lot of its proportions. Parthenon is a typical example of this. The Parthenon's facade including components of its facade and anywhere else are said to be circumscribed by golden rectangles. For many classical complexes, either the building itself or the elements of the buildings have a proportion which is add up to the golden ratio. This information gives a end result that their architects most probably knew the golden percentage and consciously used it in their structures. On the other hand, the architects could use their senses and found a good proportion because of their desgins, and their proportions carefully approximate the golden percentage. Beside this, some analyses can continually be questioned on the floor that the investigator chooses the points that measurements are created or where you can superimpose gold rectangles, and the proportions that are observed are afflicted by the choices of the tips.

Some scholars disagree with the idea that Greeks acquired an aesthetic association with golden ratio. For instance, Midhat J. Gazal says, "It had been not until Euclid, however, that the fantastic ratio's mathematical properties were examined. In the Elements (308 BC) the Greek mathematician basically regarded that quantity as a fascinating irrational number, regarding the the center and extreme ratios. Its event in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a normal polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who adopted, would soberly treat that quantity for what it is, without attaching to it other than its factual properties. "[1] In Keith Devlin's judgment, the declare that measurements of Parthenon is not backed by real measurements even though the golden raito is witnessed. In fact, the whole storyline about the Greeks and fantastic ratio seems to be without foundation. The thing we surely know that Euclid showed how to determine its value, in his famous textbook Elements, that was written around 300 BC. Near-contemporary resources like Vitruvius specifically discuss proportions that may be expressed in whole statistics, i. e. commensurate as opposed to irrational proportions.

A geometrical examination of the truly great Mosque of Kairouan discloses a consistent application of the gold ratio throughout the look, regarding to Boussora and Mazouz. [22] It is found in the entire proportion of the program and in the dimensioning of the prayer space, the judge, and the minaret. Boussora and Mazouz also examined earlier archaeological ideas about the mosque, and show the geometric constructions predicated on the golden percentage through the use of these constructions to the program of the mosque to test their hypothesis.

The Swiss architect Le Corbusier, famous for his efforts to the present day international style, focused his design school of thought on systems of tranquility and proportion. Le Corbusier's faith in the mathematical order of the world was closely bound to the gold proportion and the Fibonacci series, which he referred to as "rhythms noticeable to the attention and clear in their relationships with each other. And these rhythms are in the very reason behind individual activities. They resound in man by an organic inevitability, the same fine inevitability which in turn causes the tracing from the Golden Section by children, old men, savages and the discovered. "[23]

Le Corbusier explicitly used the gold ratio in his Modulor system for the range of architectural percentage. He saw this system as a continuation of the long traditions of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the task of Leon Battista Alberti, and others who used the proportions of the human body to increase the appearance and function of structures. In addition to the golden ratio, Le Corbusier based mostly the machine on real human measurements, Fibonacci amounts, and the two times unit. He needed Leonardo's advice of the golden ratio in human being proportions with an extreme: he sectioned his model individuals body's level at the navel with both sections in gold percentage, then subdivided those parts in golden proportion at the legs and throat; he used these gold proportion proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular earth plan, elevation, and internal structure closely approximate fantastic rectangles. [24]

Another Swiss architect, Mario Botta, bases many of his designs on geometric statistics. Several private homes he designed in Switzerland are comprised of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the medial side sections of the home. [25]

In a recent book, creator Jason Elliot speculated that the fantastic ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque. [26]

## Painting

Illustration from Luca Pacioli's De Divina Proportione can be applied geometric proportions to the human face.

Leonardo da Vinci's illustrations of polyhedra in De Divina Proportione (On the Divine Percentage) and his views that some physical proportions display the golden proportion have led some scholars to speculate that he included the golden percentage in his paintings. [27] But the suggestion that hisMona Lisa, for example, employs golden proportion proportions, is not recognized by anything in Leonardo's own writings. [28]

Salvador Dal explicitly used the gold ratio in his masterpiece, The Sacrament of the Last Supper. The measurements of the canvas are a gold rectangle. A huge dodecahedron, with sides in golden percentage to one another, is suspended above and behind Jesus and dominates the structure. [2]HYPERLINK "http://en. wikipedia. org/wiki/Golden_ratio#cite_note-28#cite_note-28"[29]

Mondrian used the golden section extensively in his geometrical paintings. [30]

A statistical review on 565 artwork of different great painters, performed in 1999, found that these artists had not used the gold ratio in the size of their canvases. The analysis concluded that the common ratio of both factors of the paintings analyzed is 1. 34, with averages for person artists ranging from 1. 04 (Goya) to at least one 1. 46 (Bellini). [31] Alternatively, Pablo Tosto stated over 350 works by well-known performers, including more than 100 that have canvasses with fantastic rectangle and main-5 proportions, among others with proportions like root-2, 3, 4, and 6. [32]

## 3. Mathematics in Arts:

Carla Farsi straddles two areas that many people consider are diametrically compared: as well as being a professor of mathematics at the School of Colorado at Boulder, she actually is an operating, exhibiting designer. After many years of pursuing both pursuits separately she declared 2005 her Special Yr for Artwork and Maths, and within an impressive effort placed on various exhibitions, classes, videos, lectures, concerts, works and an international convention - all to deepen the understanding of the partnership between maths and art. Plus interviewed her to find out precisely what this relationship is approximately, and what it feels as though to truly have a foot in both worlds.

## Painting by figures?

When you look at some of Carla's artwork, you may be forgiven not to recognise any maths in it. A few of her installations in particular appear impulsive, even disordered, and - created from recycled things - belong very much to the realm of reality. You will find no meticulously exercised geometrical patterns, complex fractals or properly recreated perspectives, as you might expect from an artist-mathematician. Just what exactly makes the bond between maths and art? Will there be more to it than the actual fact that maths underlies habits and point of view? "Visualisation is one of the key details, " Carla says, "especially in geometry you can show things visually, and the pictures can say up to the real theorem. But you can even go beyond geometry. A thing that is logical, which makes a numerical theorem, also makes some kind of a visual assertion about structure and structure. It's almost just like a artwork, it has its structure, logic, interpretation. In the mathematician's mind, the mathematical ideas, even if they are very abstract, can appear to be almost aesthetic, intuitive. "

Carla feels that with the move forward of computers, the visual and artistic areas of maths will become increasingly more visible: "Pcs are growing so fast and we don't really know yet what they could do for all of us in the foreseeable future. Maybe one day it will be sufficient to take into account the images involved with a numerical idea or confirmation, and a computer will compute the primary equations for us. At this time, just drawing a picture is often not enough - a proper proof has to be more rigorous than that. But pcs are already getting used to demonstrate theorems [seePlus article Welcome to the maths laboratory], and perhaps 1 day a mathematician could simply present the computer with a picture, and the computer will be able to 'read off' the maths in it. In this way, mathematicians could spend more time on the creative areas of maths - getting the ideas - and personal computers could do the boring, computerized parts. At that point maths may be nearer to art than it seems now. "

So, what does it seem like, being an designer and a mathematician at the same time? Does showing a theorem feel completely different from creating a bit of art? "No, both don't feel very different. Of course, when you're doing maths, you're destined by rules a lot more than if you are doing art work. In fine art you can transform the rules - what you in the beginning organized to do - half-way through, and I really do that a lot. In fact, even if I've made up some rules at the start, I often find that I'm struggling to adhere to them, the practicalities involved force me to seek other routes. "

Do Carla's inspiration for doing maths and her inspiration to do fine art result from the same place? "Yes, I certainly think so, I'm absolutely positive about that. There is the same kind of fascination for me in both maths and fine art. It's about expressing ideas, and sometimes maths increases results and other times it's art.

## "Maths and art are just two different dialects you can use to express the same ideas. "

In some periods of my life I'm more seduced by the rigour and formality of maths, and at other times I favor art. I think maths and fine art are just different dialects you can use expressing the same ideas. "

What are these ideas? "That's a very hard question! I believe it's by domain flipping relate to the entire world, can certainly make money see and understand the world. Personally i think a marriage with certain items, or objects of the mind, and I wish to express that. For example, I might be touched by the idea of an explosion [Carla indeed colored a series of pictures on the subject of Hiroshima], and express it, I might would rather use art, smart colours. If I want expressing or understand something more formal, maths may be better-suited. "

## Numbers by painting

But Carla didn't put on her Special Calendar year just in order to contemplate those deep connections. To start with, she needs to start the world of maths to those who are terrified of it, or feel that it has nothing in connection with true to life. "Emphasising the aesthetic and creative aspects of maths will make people enjoy it more. I created a course at my university, aimed at non-maths students, which demonstrates to maths using the aesthetic arts. I think this may also be of great profit to maths students, and here we could teach a lot more formal mathematical ideas. "

Carla uses paintings and sculptures both to give a standard feel for the topic and to demonstrate concrete maths items and problems. An area that benefits most from the aesthetic way is topology. This branch of maths studies the nature of geometric items by allowing them to distort and change. Think of a knot in an elastic band: its defining feature, what sort of band winds around itself, remains the same even when you expand the band. On this spirit, topologists respect any two items that may be deformed into one another without tearing to be one and the same thing - take a look at Plusarticle In space, do all roads lead home? to observe how a coffee glass can be converted into a doughnut.

Carla demonstrates to topological ideas and methods using the sculptures of UNITED STATES artistHelaman Ferguson, and also the works of Catalan architect Antoni Gaud.

## "I usually ask students to bring playdough to the maths class. "

Ferguson's work in particular is good for illustrating the solutions to concrete mathematical problems, such as how to unknot a knot: "I present first the 'puzzle' and then give them some clues to see if we can work out the perfect solution is jointly. Usually I also ask students to bring playdough to the class, so that people could work 'hands on'. Directly after we have worked out the maths I suggest to them a piece by Ferguson that beautifully illustrate the effect. "

"With Gaud I am somewhat more loose. I add him once i discuss topological transformations of areas and also once i speak about spirals. A few of his work illustrates well the concept of topological deformation and I make use of it to the, as a general example. That is also useful when students ask (as they often do) how mathematics pertains to things they see in the real world. "

Of course, no category on maths and the visible arts would be complete without fractals. Their often astonishing beauty originates from their infinite intricacy: no matter how tightly you focus in on the fractal, what the thing is that continues to be extremely complicated and crinkly. Also, it often looks like the whole fractal, a occurrence called "self-similarity" (start to see the pack on the Von Koch Snowflakebelow). There are many mathematical ways to measure the crinkliness of any fractal, and Carla teaches them in her classes using fractals that happen in nature and fine art: "I train my students how to compute the fractal dimensions of an fractal. First I show them some examples from skill and other fields, especially aspect. Then we review the technique formally, and then apply it to images of fractal fine art. We also work out the fractal dimensions of some of the original cases I provided them with. "

As Carla highlights, there are paintings filled with fractals that were never consciously designed by the designer: mathematicians show that the drip paintings by abstract expressionist Jackson Pollock can be identified by their own particular fractal constructions (see Plus article Fractal expressionism).

Symmetry is another theory that is as aesthetic as it is numerical. We can perceive it almost subconsciously - and it's been argued which it plays a vital role inside our perception of beauty - yet it starts the door to a wealth of mathematical framework. A square, for example, has 8 symmetries: you can echo it in the vertical, horizontal or diagonal axes, you can turn it through 90, 180 or 270 levels, or you can merely do nothing at all and leave it as it is. Each one of these transformations is named a symmetry, because after you have done it, the square appears to be just as it was before. In the event that you put all these 8 symmetries alongside one another, you get a self-contained system: once you combine two of them, by first doing one and then your other, you get one of the other symmetries in your set - check it out! Such a self-contained system of symmetries is named an organization, and symmetry communities will be the gateway to abstract algebra. A straightforward visual awareness lands you in the solid of some quite advanced mathematics!

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