Nothing is as it seems

Things are not always as they seem.

We could describe the peculiar appearance of the number π. The mathematical constant π represents the ratio of the circumference of a circle to its diameter.

Several scientists were all posed the following question: “What is π (pi) ?”
The engineer said: “It is approximately 3 and 1/7”
The physicist said: “It is 3.14159”
The mathematician thought a bit, and replied “It is equal to pi”.
(A nutritionist could think that “Pie is a healthy and delicious dessert!”)

But, what makes π so special?

\Pi appears all over math and Nature. There is another famous mathematical constant: φ.
\Phi is defined as the positive, irrational solution to the quadratic equation x^2-x-1=0 (we’re only interested in the positive solution \frac {1+ \sqrt 5}{2}.
Another definition of φ involves what Euclid, an ancient Greek mathematician, called “division in extreme and mean ratio“.
If we have a line segment of length 1 and consider the ratios 1/x and x/(1-x), we can obtain:

foto 1

Robert Everest discovered that we can express φ as a function of π and the numbers 1, 2, 3 and 5 of the Fibonacci series:

\phi = 1 - 2 cos (\frac{3 \pi}{5})

The Fibonacci sequence obeys the recursion relation P(n) = P(n-1) + P(n-2). In such a sequence the first two values are arbitrarily chosen ( they are called the “seeds” of the sequence ). When 0 and 1 are chosen as seeds, or 1 and 1, or 1 and 2, the sequence is called the Fibonacci sequence. The sequence formed from the ratio of adjacent numbers of the Fibonacci sequence converges to a value of 1.6180339887…., called “phi”, whose symbol is φ.
\Phi is found in the realm of plant life. It is common to find φ in the realm of plant life. Botanists have shown that, if you look at closely the central stem of a plant, the plant grows upward, leaves or branches sprout off of the stem in a spiral pattern.
Pine trees are also influenced by Fibonnaci numbers and the φ proportion.
The needles that grow from pine tree branches make use of different Fibonacci numbers – most often 3, 5, or 8.
As with plants, the proportion of 1:1.618 can be found in the structure of many animal bodies.
Probably the most famous example of use of φ related proportions is that of the Nautilus shell, which adheres directly to the Golden Spiral. The Golden Spiral is the most perfect spiral when considering self replication through continual growth.

Schermata 2015-10-15 alle 18.13.51.png The Golden Rectangle is the basis for generating this logarithmic curve.
The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains. Thus a smaller square can be removed, and so on, with a spiral pattern ( i.e. the golden spiral) resulting.
The φ proportion can be found throughout the human body. An artist who is perhaps the most famous for his use of the golden ratio, is Leonardo Da Vinci. The idea of the pentagonal symmetry of the human body may have been most popularized by his artworks.
These are not all of the φ proportions that can be found in Nature, but they are certainly enough to show that φ is not only a number that can be found all around us: it is also something that is within us!

Nothing is as it seems…


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