One of the most common places to see Fibonacci numbers is in the growth patterns of plants. Growth spirals are characterized by both a circular motion, and elongation. As a branch grows, it produces leaves at regular intervals, but not after each complete circle of its spiral. The reason the leaves are not directly above each other is because all of the leaves would not be able to get the necessary elements. It appears that leaves are generated on the stem in phyllotactic ratios where the numerator and denominator are both Fibonacci numbers.
The numerator is the number of turns, and the denominator is the number of leaves past until there is a leaf directly above the original. The number of leaves past, ad both directions of turns produce 3 consecutive Fibonacci numbers. For example, in the top plant on this transparency, there are 3 clockwise rotations before there is a leaf directly above the first leaf, passing 5 leaves along the way. Notice that 2, 3, and 5 are all consecutive Fibonacci numbers. The same is true for the bottom plant, except that it rakes 5 rotations for 8 leaves.
We would write this as 3/5 clockwise rotations per leaf on the top one and 5/8 for the bottom. Although, these are just computer-generated plants, the same is true in real life. A few real life examples of these phyllotactic ratios are 2/3 elm, 1/3 black berry, 2/5 apple, 3/8 weeping willow, and 5/13 *censored* willow. Daisies display Fibonacci numbers in their own unique way. If we look at this enlarged seed head of a daisy, and took the time to count the number of seeds spiraling in clockwise and counter clockwise rotations we would arrive at 34 and 55.
Note that these are consecutive Fibonacci numbers. Many other flowers exhibit Fibonacci numbers not only in their buds and seeds, but also in the count of their petals. Most daisies have 13, 21, or 34 petals. Other examples would be as follows 3 – lilies, 5 –wild roses, 8- delphiniums, 13- ragworts and the list goes on forever. Please note though, that some flowers have an exact number of petals e. g. buttercups, others have an average that is a Fibonacci numbers. Fibonacci numbers are also associated with animals as well as plants.
The most fascinating example of the sequence in the animal kingdom is the remarkable spiral that characterizes some animal growth. If you draw a square 1 unit long, then continue to add squares around that one you will get this following diagram. Then quarter circle arcs can be drawn connecting opposite corners of the squares on such a way that the arcs connect sequentially. What develops is a close approximation of the spiral shell of here of the chambered nautilus. Spirals such as this also occur in the horns of wild sheep, parrot beaks, and elephant tusks. Fibonacci numbers in Art and Architecture:
Fibonacci numbers and ratios have has a curios influence on art and architecture for many centuries. There seems to be a visually pleasing quality to these numbers and their relationship to each other that has appealed to humanity’s sense of beauty since recorded history. Today 3×5 and 5×8 index cards and booklets are extremely popular. This obviously shows the appeal of Fibonacci numbers that characterize such things as playing cards, writing pads, windows, mirrors, calculators, and credit cards, to mention only a few items from the endless list of Fibonaccian shapes.
In fact these shapes are close approximations of a rectangle best known as the golden rectangle. It strikes people as quite perfect, being neither too fat and stubby nor to long and skinny. It lies somewhere between a square and a double square, but not exactly in the middle. About 100 years ago, Gustav Fechrer conducted studies of the crosses in graveyards and discovered an interesting relationship between the crosses and the golden rectangle. The main stem of the crosses tended to be cut by the cross bar in the same proportion as that found in the sides of the golden rectangle.
This happens also to be the same proportion that exists between a side and a diagonal of a regular pentagon. This proportion has intrigued mathematicians, artists, and architects for over 4000 years. Thousands of years ago the Greeks named this the golden ratio. This proportion has many unique qualities. If the small part is called S and the large part is called L the proportion can be mathematically stated as follows S/L = L/S+L. More simply, the ratio of the small to the large is the same as the ratio of the large to the total.
In the case of the cross, the relationship between the smaller top and the large bottom if the same art the relationship between the large bottom and the entire main stem. This proportion best known as Phi. Other evidence of the use of the golden rectangle in architecture can be found throughout history and all over the world an example would be the exterior dimensions of the Partehenon in Athens, built in about 440 BC. The dimensions form a perfect golden rectangle and the proportion can be found elsewhere in the structure. The Greeks appear to have bee strangely influenced by Golden Proportions.
It is for Phidias, considered the greatest of Greek sculptors, that the golden ratio was named phi. The proportion can be found abundantly in his work, including the hands of sculpture that run above the columns of the Parthenon. Much Renaissance art and architecture was inspired by the Greek sense of beauty and proportion. It is not surprising then, that a lot of buildings and statues are based on the golden ratio. Artists from 12th century Le Corbusier all of the way to Leonardo Divinche have purposely used the golden rectangle.
The link between nature’s proportions and the proportions found in art and architecture has been pondered by philosophers over the years. It has been suggested that the golden ratio might be an underlying explanation for what we call beauty both in the natural object and in an artistic masterpiece. Fibonacci Numbers in Music and Poetry: Perhaps the clearest link between Fibonacci numbers and music can be found on the keyboard of a piano. An octave on a keyboard is made up of 8 white keys and 5 black keys. The black keys are positioned in-groups of 2 and 3.
There are 13 keys altogether in one octave, an analysis of which involves each of the first six Fibonacci numbers. The keyboard aside, those 13 notes belong to what is known as the chromatic scale, the most complete scale to have developed in Western Music. Its principal predecessor was the 8-note diatonic scale, better known as the octave, which was preceded by the 5-note pentatonic scale. The pentatonic scale was used in Early European music and is 6 the basis today of the American Kodaly method of music education for young children. Any 5 consecutive black keys on a keyboard constitute such a scale.
A number of well-known folk tunes can be played using just the se kids including Mary Had A Little Lamb and Amazing Grace. The musical intervals considered by many to be the most pleasing to the ear are the major sixth and minor sixth. A major sixth for example, consists of C, vibrating at about 264 vibrations per second, and A, vibrating at about 440 vibrations per second. The ratio of 264 to 440 reduces to 3/5, a Fibonacci ratio. An example of a minor sixth would be E about 330 vibrations a second, and C about 528 vibrations per second. That ratio of 330 to 528 reduces to 5/8, the next Fibonacci Ratio.
The vibrations of any sixth interval reduce to a similar ratio. It has been suggested that the Fibonacci numbers are part of a natural harmony that not only looks good to the eye but sounds good to the ear. Perhaps this is why great composers have used these systems. It is not difficult to find Fibonacci numbers in poetry. Consider, for example, the limerick, made up of 5 lines with a total of 13 beats grouped in 3’s and 2’s. For example “A fly and a flea in a flue Were imprisoned so what could they do? Said the fly, Let us flee, let us fly said the flea so they fled through a flaw in the flue.
It’s easy to see the first 2 and last line have 3 beats, and the middle two have 2 beats for a total of 13 beats in five lines. The Mathematics of Fibonacci Numbers: To the casual observer, Fibonacci numbers may appear to be nothing more than random numbers. Some are odd, and some are even, some are prime, some are composite, and the distances between them vary. But the intuitive or informed eye will note the sequence I have described in so many contexts in this speech; each Fibonacci number is the sum of the previous two numbers beginning with one. The simplest of all numbers is of course 1.
By following it with another 1, I can generate an infinite sequence of numbers. Any two adjacent numbers create the next number by addition. The mathematical properties of Fibonacci numbers are fascinating and extensive. The following are just a few of the vast number of examples. No two consecutive Fibonacci numbers have any common factors. Twice any Fibonacci number minus the next Fibonacci number equals the second number preceding the original one. The product of any two alternating Fibonacci numbers differs from the square of the middle number by 1.
If Fibonacci numbers are squared and the adjacent squares are added together, a sequence of alternate Fibonacci numbers emerges. The difference of the squares of alternate Fibonacci numbers is always a Fibonacci number. For any four consecutive Fibonacci numbers, the difference of the squares of the middle two numbers equals the product of the smallest and largest numbers. The diagonals of Pascal’s triangles add up to Fibonacci numbers. Any Fibonacci number of a prime term is prime. This concludes my studies on the Fibonacci series, Thank You for your time.