Have you ever wondered why most of the things in Nature are so visually pleasing? Why spirals, though imperfect, are so attractive? Why four petal flowers are so rare? The answers can be found in mathematical principles. The most intriguing of them is the concept of the “golden mean”, which was used by many Renaissance artists and architects. In mathematics the “golden mean”, also called the Golden Ratio, is represented by number **Phi** which approximately equals to 1.618.

Leonardo Fibonacci, the Italian mathematician, developed a series of numbers that relate to the Golden Mean. They are called Fibonacci series:

`0, 1, 1, 2, 3, 5, 8, 13, 21, 34...`

Can you guess what number comes after 34?

If you answered 55 then you either know all about Fibonacci numbers or you’re a genius and you could figure out that after two starting numbers, each number is equal to the sum of the two preceding numbers. As you can see above, 34 = 13+21, so the next number of course 21+34 = 55.

Fibonacci numbers are found in: leaf or petal arrangements in 90% of plants, petals on flowers, pine cones, pineapple, sunflower seed heads, palms… and Sea Shells.

And if you divide two adjacent numbers in the series to form a ratio, you end up with a number close to 1.618 – yes, that magic Golden Mean number (e.g. 13 divided by 8 = 1.62, and 21 divided by 13 = 1.615).

The Renaissance artists and architects claimed that the rectangles created with these proportions were the most visually pleasing.

And that’s what brings us to shells. Check out this animation to see how Fibonacci numbers and Golden Mean form a foundation of a spiral.

If you want to learn more on this subject, do visit this ultimate source for everything on Phi/Golden number.

To see what has inspired me to talk about Mathematics in this post and see where the photos in this post come from, check out this amazing Spiral Photo Gallery.

Of course, almost everything you have written about sea shells and the golden ratio is nonesense; especially with regard to the Nautilus.

Every other so-called “devine” property of the golden ratio and of Fibonacci numbers are simple results of a quadratic equation and there exists an uncountable infinity of such “devine proportions” and “extreme means.” (Look up Uncountable infinity). There also happens to be countably infinitely many odd polygons that exhibit the “devine” property that a – b= 1/a, where a is a longest diagonal and b is a second longest diagonal. Belief in the divinity of the golden ratio requires one to be a polytheist.

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hmm… thank you for a very interesting comment.

As you can see I’ve based this post on a couple of other related articles/sites on the Web. Does it mean that those sites talk nonsense as well? I think, I did define the Fibonacci series correctly as well as the Golden mean number.

In your opinion, what would’ve been the right way to explain the nature of shells?

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Well, the spirals found in sea shells are often approximated by a logarithmic spiral. In genral, however, these spirals do not fit into a rectangle with the golden ratio proportions. (1.618.. to 1). They usually fit into rectangles whose proportions range from about 1.08 to 1 for the “olive” shells to about 1.33 to 1 for the nautilus. The Fibonacci Sequence is a special case of a quadratic recursion equation. x(n+2)= p x(n+1)+q x(n). p and q can be any positive numbers. In Fibonacci, we use p=1,q=1. The golden ratio is the limit of x(n+1)/x(n), as n increases indefinitely. In the genral series, that limit is phi(p,q) = (p+Sqrt(p^2+4q))/2. These numbers phi(p,q) provide general versions of every “magical” and “divine” poperty you can ascribe to phi.

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