The beauty of nature is all around. We tried planting allium bulbs in the garden for the first time this year. Late May they were looking promising (see featured image.)

But now look how they’ve now developed – long stems with huge heads, several inches across, comprising gradually emerging flowers with striking geometrical patterns.

Double click for more detail.

I’m intrigued that there are 6 petals on the individual flowers, which is not one of nature’s preferred Fibonacci numbers – but perhaps they are 3 pairs, and 3 is a Fibonacci number.

And if you look at the picture on the left, it’s impossible to count accurately, but there are over a hundred individual stems in the head, each of which will develop a flower. Since the situation is so dynamic you could not expect this to be an exact Fibonacci number, but it’s somewhere on the way between 89 and 144!

Fibonacci Grape Pips?


I was idly counting the pips in each grape off a bunch from E Leclerc (cf Tesco, Kroger). (It seems that France has not really caught on to the fashion for seedless grapes; most on sale had pips. Yes, they were more tasty.) My idle counting had spotted a potential ‘pattern’ – so far these are all Fibonacci numbers, and it is well known that Fibonacci numbers appear frequently in nature. Could it be…?? Then came the next sequence:


Now FOUR is not a Fibonacci number, so appears to be anomalous. Well, science does allow for anomalous results that don’t fit the current theory. Then comes the SIX. But here I notice two tiny black dots in the grape – putative pips that did not develop – which makes 8, another Fibonacci number. Maybe I’d missed a black dot with the 4?

So I can hang on to my theory for a while, until more anomalous data emerges. A rather trivial example of the scientific method in action? Of course, there are far too few results to draw conclusions…

Featured image by Thamizhpparithi Maari, via Wikimedia Commons

The Fibonacci Series

In my youth I was always fascinated by numbers. The Fibonacci series is one of the most interesting sequences of numbers, first mentioned by Leonardo Fibonacci (c1170–c1250), the leading mathematician of his era, who popularized the Hindu–Arabic numeral system in the Western World.

The series of integers comes from 1, the symbol of unity, followed by 2, an expression of duality. (Some people prefer to begin with two 1’s, sometimes preceded by a 0; the resulting sequence is essentially the same.)

Each subsequent number is the sum of the previous two, so we have:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…. (and so on)

I was always intrigued by the fact that 144 is the square of 12. Is it the only Fibonacci square, apart from the trivial case of 1 and maybe 0? This was long the subject of conjecture, but it was eventually proved that 144=12×12 is the only square of the sequence. Now 12 is a multiple of 2’s and 3’s, and occurs in many human systems – measurements, money, astrology, and so on. This seems somehow significant.

A similar conjecture applies to number 8 being the only cube of the series, which I believe is also the case but is far too complex a subject for this blog.

The Golden Ratio

The ratio of consecutive numbers of the series converges onto a number called the Golden Ratio, usually symbolised by the Greek letter φ (phi).

φ = 1.61803398874989484820… (and so on)

It turns out that φ is found all over the place when we measure nature and its patterns. For an excellent, but rather mathematical, overview see this fascinating post on the golden ratio on the blog ‘Rationalising the Universe’.

The Golden Ratio was seen as very important in the art of the Renaissance, and of course turns up in the work of Leonardo da Vinci mentioned in the preceding post.

In one of my particular spheres of interest, the golden ratio turns up as a key ratio in the psychological perspective on the house system in the astrological psychology of Bruno and Louise Huber. So it would appear that maybe the ratio pervades not only the outer ‘objective’ physical world, but also the inner ‘subjective’ world where it relates to space and time.

Another intriguing fact is that Golden Ratio is very similar to the ratio of kilometers to miles, e.g. 8 kilometers is approximately 5 miles. This is entirely coincidental. [Or is it?]