Category archives: Golden section

Fibonacci Numbers and the Golden Ratio

Fibonacci 1Leonardo Fibonacci was a mathematician who lived in the early 1200's. He discovered a sequence of numbers that relates to all of nature as well as the golden section.

In mathematics, the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence

Fibonacci Sequence 1

or (often, in modern usage):

Fibonacci Sequence 2

The pattern to get the numbers is starting at the lowest number, from the left, add the first two numbers and that will give you the third number. Add the second and the third number and that will give you the fourth number and so on ie 0+1=1; 1+1=2; 1=2+3; 2=3+5 ;

Now, if you divide any two of the numbers, you will find that the answer will get closer and closer to either 0,618a or 1,618a, depending which way you divide, which is the ratio in the golden section. (The Fibonacci sequence gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively)

The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series.

Fibonaccia's RabbitsFibonacci Rabbits

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

  1. At the end of the first month, they mate, but there is still one only 1 pair.
  2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
  3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
  4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs

Fibonacci 3

Seed Heads

Fibonacci numbers can also be seen in the arrangement of seeds on flower heads.Fibonacci 4

You can see that the orange "petals" seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places? You will see that the pair of numbers (counting spirals in curing left and curving right) are adjacent numbers in the Fibonacci series.

The same happens in many seed and flower heads in nature. The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges.

The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.

So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers!

Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks. In the seeming randomness of the natural world, we can find many instances of mathematical order involving the Fibonacci numbers themselves and the closely related "Golden" elements.

Humans and Spirals

Humans exhibit Fibonacci characteristics, too. The Golden Ratio is seen in the proportions in the sections of a finger.

Fibonacci 6

  • It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each hand,3 bones in each finger,2 bones in 1 thumb, and 1 thumb on each hand.
  • The ratio between the forearm and the hand is the Golden Ratio!

 

Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi (Golden Ratio - 0,618.). We also use these principles in things we do and make, even in musical instruments, in guitar making and violin making, etc.Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of Equiangular spirals. An Equiangular spiral itself is a special type of spiral with unique mathematical properties in which the size of the spiral increases but its shape remains the same with each successive rotation of its curve.  The curve of an equiangular spiral has a constant angle between a line from origin to any point on the curve and the tangent at that point, hence its name.  In nature, equiangular spirals occur simply because they result in the forces that create the spiral are in equilibrium, and are often seen in non-living examples such as spiral arms of galaxies and the spirals of hurricanes.  Fibonacci spirals, Golden spirals and golden ratio-based spirals generally appear in living organisms, as illustrated below:

Fibonacci 7

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The Golden Section

In most objects that are aesthetically pleasing, you will often find that the proportions that make them up are in the golden section, also known as the golden ratio, golden rectangle, golden proportion, golden cut, golden number.

The ancient Pythagoreans discovered the golden ratio which is approx. 1:1,618033988749895. This ratio is used extensively in art, painting, architecture, music, design, book design, the human body Golden sec. 9

and can even be found in nature. This use of the Golden Ratio results in various objects being aesthetically pleasing to the eye.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

 

Golde sec. 1 Golden sec. 2.

where the Greek letter phi (I) represents the golden ratio. Its value is:

Golden sec. 3

 

 

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Architecture

The Parthenon's faade as well as elements of its faade and elsewhere are said by some to be circumscribed by golden rectangles.Golden sec. 6 A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz. They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. 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.

Music

Ern Lendva analyzes Bla Bartk's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patenton this innovation.

In musical instruments this is also prevalent and in Guitar making and Violin making there are plenty of examples and if examined closely Golden sec. 4you will find these ratios everywhere. These are adapted and combined with functionality, aesthetics and tonal quality to produce the musical instrument, which over time, the shape has not varied that much, but tweaked here and there to facilitate small improvements toward making the perfect instrument.

Golden sec. 5

Fibonacci

In a future blog I will discuss the Fibonacci Series and its presence everywhere around us................