Download the PDF version of this lesson plan. Introduction Fibonacci numbers are an interesting mathematical idea.
We can now draw a new square — touching both one of the unit squares and the latest square of side 2 — so having sides 3 units long; and then another touching both the 2-square and the 3-square which has sides of 5 units.
This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles. If we now draw a quarter of a circle in each square, we can build up a sort of spiral.
The spiral is not a true mathematical spiral since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller but it is a good approximation to a kind of spiral that does appear often in nature.
Such spirals are seen in the shape of shells of snails and sea shells. The image below of a cross-section of a nautilus shell shows the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide buoyancy in the water.
Fibonacci numbers also appear in plants and flowers. Some plants branch in such a way that they always have a Fibonacci number of growing points.
Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! A particularly beautiful appearance of fibonacci numbers is in the spirals of seeds in a seed head. The next time you see a sunflower, look at the arrangements of the seeds at its centre.
They appear to be spiralling outwards both to the left and the right. At the edge of this picture of a sunflower, if you count those curves of seeds spiralling to the left as you go outwards, there are 55 spirals.
At the same point there are 34 spirals of seeds spiralling to the right. A little further towards the centre and you can count 34 spirals to the left and 21 spirals to the right. The pair of numbers counting spirals curving left and curving right are almost always neighbours 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.
Nature seems to use the same pattern to arrange petals around the edge of a flower and to place leaves round a stem.
So just how do plants grow to maintain this optimality of design? Golden growth Botanists have shown that plants grow from a single tiny group of cells right at the tip of any growing plant, called the meristem. There is a separate meristem at the end of each branch or twig where new cells are formed.
Once formed, they grow in size, but new cells are only formed at such growing points. Cells earlier down the stem expand and so the growing point rises. Also, these cells grow in a spiral fashion: These cells may then become a seed, a new leaf, a new branch, or perhaps on a flower become petals and stamens.
The leaves here are numbered in turn — each is exactly 0. The amazing thing is that a single fixed angle of rotation can produce the optimal design no matter how big the plant grows. But where does this magic number 0. Ratio of successive Fibonacci terms. It has a value of approximately 1. The closely related value which we write asa lowercase phi, is just the decimal part of Phi, namely 0.
But why do we see phi in so many plants? The number Phi 1. After two turns through half of a circle we would be back to where the first seed was produced.
Over time, turning by half a turn between seeds would produce a seed head with two arms radiating from a central point, leaving lots of wasted space. A seed head produced by 0.Fibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence.
He discovered the sequence - the first recursive number sequence known in Europe - while considering a practical problem in the “Liber Abaci” involving the growth of.
Fibonacci asked how many rabbits a single pair can produce after a year with this highly unbelievable breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month). D E Smith's History of Mathematics Volume 1, (Dover, - a reprint of the orignal version from ) gives a complete list of other books that he wrote and is a fuller reference on Fibonacci's life and works.
Playful introduction for kids to the life of Fibonacci. Not exactly historically accurate, but no matter; the point of the book is how Fibonacci saw fascinating patterns in numbers all around him, and they didn't look like a math srmvision.coms: A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which " that served to introduce Indian-Arabic numerals and methods and contributed to the mastering of the problems of daily life.
Here Fibonacci became the teacher of the masters of computation and of the. Fibonacci (c. – c. ) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
The name he is commonly called, "Fibonacci" (Italian.