Numbers are something with built-in exactness. When we think of five in whatever context it is precise, there is no ambiguity about it. The preponderance of numbers in our lives is such that we hardly pay much attention to this feature. There is hardly any activity in which numbers do not figure. Numbers come in different sizes but all can be expressed as a collection of ten primary or natural numbers from zero to nine. There is some argument as to whether zero is a finite number but we shall not go into that here. We simply note that it does not have the exactness referred to earlier.
Numbers are also classified in various ways like prime, rational, irrational, and also transcendental. Some of these terms are anthropomorphic and are used as mere definitions. But whatever be their classification numbers are just symbols to be used in expressing some idea. By themselves they mean nothing. There is nothing seven-like in seven; it acquires a meaning only in association with something being counted. Although we deal with numbers all the time in our daily lives, we hardly ever see them as existing in their own rights. However, if we look at them closely, we find that some are special with distinct ‘personality’. For example, zero and one; multiply any number by zero and it puts its face on it; as for one it creates all whole numbers by adding itself repeatedly.
Then there are groups of numbers in special combinations – series are sequences that are unique in their own way. One of them is encountered conspicuously in nature, although there is as yet no satisfactory explanation for that. This is a sequence of numbers starting from one*, with each subsequent number obtained by adding up the previous two. This is known as Fibonacci sequence in the literature. The first few numbers are one, one, two, three, five, eight, thirteen, twenty-one, and the sequence continues on indefinitely.
Numbers in Nature
There is well-documented evidence of occurrence of these numbers in nature. Pinecones, pineapples, daisy flowers, sunflowers all exhibit these numbers. If you look closely at a pineapple or a pinecone, at any point on the surface you find the little protrusions (or bumps) are arranged in spirals (helices in three dimensions) going in mutually perpendicular directions. The same is true for the arrangement of seeds at the center of the flowers. A curious fact is that in each case the numbers of spirals in the two directions are always two consecutive numbers of Fibonacci sequence. A pinecone may have spiral count of (3,5), (5,8), or (8,13). A ripe pineapple has a spiral count of (8,13). A coneflower has (13,21) spiral count at its center. The center of a daisy flower has a spiral count of (21,34). The sun flower has a spiral count of (34,55) or (55,89) depending on its size. Similar patterns are found in many other plants and even vegetables.
The sequence is named after Fibonacci because he found it by studying the reproduction progress of a pair of rabbits in a year assuming that the pair produced a pair of babies each month and the babies at one-month age were able to produce a pair likewise. The number of pair of rabbits follows the sequence. He published the results in a book and the sequence came to be known by his name. The genealogy of honeybees also follows this sequence.
Even though the sequence became well publicized only in early thirteenth century, it had been known earlier albeit without a name. In Sanskrit poetry syllables are classified as short or long. A line of a poem contains a combination of these. In the metrical construction a problem arose as to in how many ways the short and long syllables could be arranged, assuming that every line takes the same time to recite. Studying this problem in relation to the number of time units, in 1150 AD Aachaarya Hemachandra in India found that the number of ways in which short and long syllables could be arranged for sequentially increasing time units followed this very sequence. This was more than seventy years before Fibonacci published his book Liber Abaci .
Numbers and Philosophy
Although spirals with adjacent numbers of the sequence occur in nature so abundantly the reason for this is not clear. There are two aspects to this problem. One is the presence of interlocked spirals, the other is their numbers. The current belief is that during the growth of the plants, flowers, and fruits this arrangement gives the optimum space utilization for a given surface. We can also think of the spirals as interconnecting links for all the elements as well as connecting each one to the single source – the root, symbolically representing the fact that everything in the universe is connected to everything else.
One way of looking at the numbers in the sequence is to consider the start of the life cycle itself. One seed germinates into one plant with two leaves to begin with. In some plants the next two stages are three and five leaves. Thus the building blocks for the sequence are in place. Now we carry this idea to philosophy considering, specifically Vedanta and its theory of the origin of the universe. To begin with there is only one Ultimate Reality – the Universal consciousness. The other aspect of consciousness is energy and now we have two unities. In Vedanta and Sankhyathese are called Purusha and Prakriti . Next there are three universal qualities (trigunas) and five intrinsic natures of matter (panchatattva ; there is no exact English equivalent of the word tattva unless we coin a word ‘thatness’). Consciousness and energy together with trigunas and panchatattva create everything in the universe. So the origin of the universe itself is based on the numbers one, one, two, three, and five, the building blocks of the Fibonacci sequence.
*Strictly speaking the starting numbers should be zero and one because their addition gives the second one in the sequence. Conventionally though zero is not included in the sequence. The only meaningful operation with zero is addition, which itself sets it apart from finite numbers.
Acarya Hemcandra and the (so-called) Fibonacci Numbers, Int. J. of Mathematical Education, Vol. 20 (1986), pp. 20 – 30.
Dharmbir Rai Sharma is a retired professor with electrical engineering and physics background. He obtained his M. S.degree in physics in India and Ph. D.in electrical engineering at Cornell University. He has taught in universities here and also in Brazil, where he spent sometime. He maintains a website http://www.cosmosebooks.com devoted mainly to philosophy and science.