First of all, if you don’t know what the hell am I talking about, jeez, where have you been buddy? If you do but have not read the book, then please do yourself a favour and do so. You can buy it from here. Once you do finish reading the book, then come back here and we can talk about it.
The crux of the book relies on the golden ratio also known as the divine proportion which has a value of 1.61803399 (per the book). The book, claims an eerie prevalence of divine proportion in life and nature, which if you think is just fiction might be surprised to find out it really is not!! Whaaaaaat? Dammit, and all this time I thought I was reading just a novel!! Err.. I guess I should stop trying to be a “writer“ and get on with it.
I think some of the truth is a bit stretched in the novel, but nonetheless based on truth (of course IMHO).The Divine Proportion (represented by the Greek letter f – phi). In reality the value of phi is:
f = (1 + v5) / 2
If you actually try and solve the above, you will figure out that it is an irrational number i.e. the decimal expansion of an neither terminates or repeats.
The whole thing goes back to the Greeks, who believed the most pleasing and aesthetic and purest form is the triangle, specifically the golden triangle, which the Greeks incorporated in their architecture. The Italian mathematician Leonardo Pisano’s book called Liber Abaci brings to the west a method of writing numbers and arithmetic (that we use today) which was completed in India 500 years earlier. These new numbers eventually provided the basis for modern science and engineering. One of the many exercises in Liber Abaci leads to the Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21, …). I won’t go into the examples here but you can find the link in the bottom of this post. The Fibonacci series essentially is every one number after 1 is the sum of the previous two (so, 1+1 = 2, 1+2 = 3, 2+3 = 5, etc.). As you solve more problems (the rabbit problem, bees, etc. – again see the link at the end) with the Fibonacci series you will notice that this series occurs pretty frequently in nature. Some examples (also mentioned in the book) are:
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If you count the number of petals in most flowers you will find the total to be a Fibonacci series. E.g. an iris has 3 petals, a buttercup has 5, a delphinium 8, daisy 13, 21 or 34, etc.
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A sunflower has two beautiful spiral patterns – one clockwise and the other anti-clockwise. If you count those spirals, you will find there are 21 or 34 running clockwise and 34 or 55 running anti-clockwise.
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Similarly, other flowers exhibit the same characteristics including Pine-cones and Cauliflowers
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Leaves on trees (and stems) are arranged in a spiral pattern that is wound around the stem. If p is the number of turns you make (of the spiral) till you reach the next leaf directly above the first, and q is the number of leaves (excluding the first one), then the ratio p/q gives us the divergence of the plant. If you calculate the divergence for various plant species you will find that both the numerator and denominators represent Fibonacci series – 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, etc.
Here is where things get more interesting. If you take the Fibonacci series and divide each number into the one that follows it you would get: 1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615; 34/21 = 1.619; 55/34 = 1.6176; 89/55 = 1.6181…. Hmmmm, notice anything? The 1, 1.6, 1.61, 1.618, etc. looks like the golden ratio.
Mathematic ans have proved that the Fibonacci series gets slowly closer to f and finally equate at infinity.
What does this mean? Well I cannot say for all the stuff laid out in the book regarding the golden ratio versus the human body (e.g. measuring the distance from the tip of your head to the floor and then dividing the distance from your belly button to the floor – did you get a f or something around that number?), but the facts for plants is there in front of us and such frequent appearances cannot be accidental.
So what do you make of all this? Well we found out quite recently that it is a matter of efficiency. To achieve maximum efficiency, flower heads and plant leaves grow in a spiral fashion governed by the golden ratio and rounded off to the nearest whole number (since f is a irrational number) – because of this it will be a Fibonacci series.
Why spirals and the golden ratio? Well as new leaves are added, they need to be the least obstructive to the old leaves and the new ones (that would come later) above it. To do so, the leaves come in a spiral pattern. In the case of seeds, a spiral “packs“ the most possible – and the most efficient way to do is via spirals. We now understand that f is the ratio that gives the optimal solution to growth equations. The mathematical explanation is that of all the irrational numbers, f is in a very precise (and technical sense) farthest from being representable as a fraction.
So, as a good novel should, this one brings the plot to an end satisfactorily and also leaves one wondering about life. Also makes one wonder how much of todays religions teachings are fact and why are people so willing to accept things at so much face value? Why are we so eager to believe in everything we are told?
To conclude as it is in the book, the “Phi (f) is one H cooler than Pi (p)”.
Check out the links here and here which formed the basis of this post.
Update: I am not sure but the symbol for Phi shows up as a “f” and the Pi shows up as a “p”. I will see if I can fix that, but till then, whenever you see just a “f” or a “p”, just read it as Phi and Pi respectively.
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