The Fibonacci Sequence and the Golden Ratio

The Fibonacci Sequence was firstly introduced by Leonardo of Pisa, known as Fibonacci, in the year 1202. He studied on the population of rabbits. Firstly he assumed that a newly-born pair of rabbits, one male, one female, are put in a field; one month later, rabbits become adult and are able to mate so that at the end of its second month a female rabbit can produce another pair of rabbits; he also assumed that rabbits never die and a mating pair always produces one male and one female rabbit every month from the second month on. The question that Fibonacci posed was: how many pairs will there be in one year?

At the end of the first month, the pair mate, however they don’t produce a pair, therefore there is still one only 1 pair. At the end of the second month the couple produces a new pair, so now there are 2 pairs of rabbits in the field. One of them is adolescent and the other is leverets. At the end of the third month, the original pair produces a second pair, the leverets become adolescents hence a total of 3 pairs in all in the field. For the next 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 are given as,

The Golden Ratio is a special type of ratio that can be seen on many structure of living organisms and many objects. It is not only observed in the part of a whole subjects, but also in arts and architecture for centuries. The Golden ratio gives the most compatible sizes of geometric figures. In nature, The Golden Ratio can be seen on the bodies of human beings, shells and branches of trees. For Platon, the keys of the cosmical physics is this ratio. Also, this ratio is widely believed that it is the most aesthetic ratio for a rectangle. The Golden Ratio, is an irrational number just as pi or e and its approximate value is 1,618033988… To define the Golden Ratio, Φ or PHI is used.

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The Golden Ratio has been used for many years for different purposes. Some studies of the Acropolis, the approximate value of golden ratio can be seen on many of its proportions. Parthenon is a typical example of this. The Parthenon’s facade including elements of its facade and elsewhere are said to be circumscribed by golden rectangles. For many classical buildings, either the building itself or the elements of the buildings have a proportion which is equal to the golden ratio. This information gives a result that their architects most probably knew the golden ratio and consciously employed it in their buildings. On the other hand, the architects may use their senses and found a good proportion for their desgins, and their proportions closely approximate the golden ratio. Beside this, some analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and the proportions that are observed are affected by the choices of the points.

Some scholars disagree with the idea that Greeks had an aesthetic association with golden ratio. For instance, Midhat J. Gazalé says, “It was not until Euclid, however, that the golden ratio’s mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties.”[1] In Keith Devlin’s opinion, the claim that measurements of Parthenon is not supported by actual measurements even though the golden raito is observed. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we surely know that Euclid showed how to calculate its value, in his famous textbook Elements, that was written around 300 BC. Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[22] It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis.

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as “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.”[23]

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci’s “Vitruvian Man”, the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo’s suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body’s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier’s 1927 Villa Stein in Garches exemplified the Modulor system’s application. The villa’s rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[24]

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed 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 side sections of the house.[25]

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

[edit]Painting

Illustration from Luca Pacioli’s De Divina Proportione applies geometric proportions to the human face.

Leonardo da Vinci’s illustrations of polyhedra in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.[27] But the suggestion that hisMona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo’s own writings.[28]

Salvador Dalí explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[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 study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[31] On the other hand, 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.[32]

3. Math in Arts:

Carla Farsi straddles two fields that many people believe are diametrically opposed: as well as being a professor of mathematics at the University of Colorado at Boulder, she is a working, exhibiting artist. After years of pursuing both interests separately she declared 2005 her Special Year for Art and Maths, and in an impressive effort put on various exhibitions, classes, movies, lectures, concerts, plays and an international conference – all to deepen the understanding of the relationship between maths and art. Plus interviewed her to find out just what this relationship is about, and what it feels like to have a foot in both worlds.

Painting by numbers?

When you look at some of Carla’s artwork, you might be forgiven not to recognise any maths in it. Some of her installations in particular appear impulsive, even disordered, and – made from recycled objects – belong very much to the realm of reality. There are no meticulously worked out geometrical patterns, intricate fractals or perfectly recreated perspectives, as you might expect from an artist-mathematician. So what makes the connection between maths and art? Is there more to it than the fact that maths underlies patterns and perspective? “Visualisation is one of the main points,” Carla says, “especially in geometry you can prove things visually, and the pictures can say as much as the actual theorem. But you can even go beyond geometry. Something that is logical, that makes a mathematical theorem, also makes some kind of a visual statement about structure and composition. It’s almost like a piece of art, it has its own structure, logic, meaning. In a mathematician’s head, the mathematical ideas, even if they’re very abstract, can appear to be almost visual, intuitive.”

Carla thinks that with the advance of computers, the visual and artistic aspects of maths will become more and more prominent: “Computers are developing so fast and we don’t really know yet what they could do for us in the future. Maybe one day it will be sufficient to think about the images involved in a mathematical idea or proof, and a computer will compute the underlying equations for us. Right now, just drawing a picture is often not enough – a proper proof has to be more rigorous than that. But computers are already being used to prove theorems [seePlus article Welcome to the maths lab], and maybe one 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 aspects of maths – having the ideas – and computers could do the boring, automatic parts. At that point maths may be closer to art than it appears now.”

So, what does it feel like, being an artist and a mathematician at the same time? Does proving a theorem feel very different from creating a piece of art? “No, the two don’t feel very different. Of course, when you’re doing maths, you’re bound by rules much more than when you’re doing art. In art you can change the rules – what you initially planned to do – half-way through, and I do that a lot. In fact, even if I’ve made up some rules at the beginning, I often find that I’m unable to stick to them, the practicalities involved force me to seek other routes.”

Do Carla’s motivation for doing maths and her inspiration to do art come 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 art. It’s all about expressing ideas, and sometimes maths works better and other times it’s art.

“Maths and art are just two different languages that can be used to express the same ideas.”

In some periods of my life I’m more attracted by the rigour and formality of maths, and at other times I prefer art. I think maths and art are just different languages that can be used to express the same ideas.”

What are these ideas? “That’s a very difficult question! I think it’s how I relate to the world, how I see and understand the world. I feel a relationship with certain objects, or objects of the mind, and I want to express that. For example, I may be touched by the idea of an explosion [Carla indeed painted a series of pictures on the subject of Hiroshima], and to express it, I may prefer to use art, bright colours. If I want to express or understand something more formal, maths may be better-suited.”

Numbers by painting

But Carla didn’t put on her Special Year just in order to contemplate those deep connections. First and foremost, she wants to open up the world of maths to those who are scared of it, or feel that it has nothing to do with real life. “Emphasising the visual and creative aspects of maths might make people like it more. I created a course at my university, aimed at non-maths students, which teaches maths using the visual arts. I think this could also be of great benefit to maths students, and here we could teach the more formal mathematical ideas.”

Carla uses paintings and sculptures both to give an overall feel for the subject and to illustrate concrete maths objects and problems. An area that benefits most from the visual approach is topology. This branch of maths studies the nature of geometric objects by allowing them to distort and change. Think of a knot in an elastic band: its defining feature, the way the band winds around itself, remains the same even when you stretch the band. In this spirit, topologists regard any two objects that can be deformed into each other without tearing to be one and the same thing – have a look at Plusarticle In space, do all roads lead home? to see how a coffee cup can be turned into a doughnut.

Carla teaches topological ideas and methods using the sculptures of North American artistHelaman Ferguson, and also the works of Catalan architect Antoni Gaudí.

“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 usually present first the ‘puzzle’ and then give them some hints to see if we can work out the solution together. Usually I also ask students to bring playdough to this class, so that we can work ‘hands on’. After we have worked out the maths I show them a piece by Ferguson that beautifully illustrate the result.”

“With Gaudí I am a bit more loose. I introduce him when I talk about topological transformations of surfaces and also when I talk about spirals. Some of his work illustrates well the concept of topological deformation and I use it for that, as a general example. This is also useful when students ask (as they often do) how mathematics relates to things they see in the real world.”

Of course, no class on maths and the visual arts would be complete without fractals. Their often astonishing beauty comes from their infinite intricacy: no matter how closely you zoom in on a fractal, what you see is still extremely complicated and crinkly. What’s more, it often looks similar to the whole fractal, a phenomenon called “self-similarity” (see the box on the Von Koch Snowflakebelow). There are various mathematical techniques to measure the crinkliness of a fractal, and Carla teaches them in her classes with the aid of fractals that occur in nature and art: “I teach my students how to compute the fractal dimension of a fractal. First I show them some examples from art and other fields, especially nature. Then we study the technique formally, and then apply it to images of fractal art. We also work out the fractal dimension of some of the original examples I presented them with.”

As Carla points out, there are paintings containing fractals that were never consciously intended by the artist: mathematicians have shown that the drip paintings by abstract expressionist Jackson Pollock can be identified by their own particular fractal structures (see Plus article Fractal expressionism).

Symmetry is another concept that is as visual as it is mathematical. We can perceive it almost subconsciously – and it has been argued that it plays a vital role in our perception of beauty – yet it opens the door to a wealth of mathematical structure. A square, for example, has 8 symmetries: you can reflect it in the vertical, horizontal or diagonal axes, you can rotate it through 90, 180 or 270 degrees, or you can simply do nothing and leave it as it is. Each of these transformations is called a symmetry, because after you’ve done it, the square appears to be exactly as it was before. If you put all these 8 symmetries together, you get a self-contained system: whenever you combine two of them, by first doing one and then the other, you get one of the other symmetries in your set – try it! Such a self-contained system of symmetries is called a group, and symmetry groups are the gateway to abstract algebra. A simple visual consideration lands you in the thick of some quite advanced mathematics!