The Fibonacci numbers are also exemplified by the botanical phenomenon known as phyllotaxis. Thus, the arrangement of the whorls on a pinecone or pineapple, of petals on a sunflower, and of branches from some stems follows a sequence of Fibonacci numbers or the series of fractions.

People have always taken delight in devising "problems" for the purpose of posing a challenge or providing intellectual pleasure. Thus, many mathematical recreations of early origin that have reappeared from time to time in new dress seem to have survived chiefly because they appeal to man's sense of curiosity or mystery.

A few survived from the ancient Greeks and Romans. Little was known about them during the Dark Ages, but a strong interest in such problems arose during the Middle Ages, stimulated partly by the invention of printing, partly by enthusiastic writers of arithmetic texts. Such activities were most prominent on the Continent, particularly in Italy and Germany.

The mathematician Fibonacci, also called Leonardo of Pisa, published in 1202 an influential treatise, "Liber abaci". It contained the following recreational problem: "How many pairs of rabbits can be produced from a single pair in one year if it is assumed that every month each pair begets a new pair which from the second month becomes productive?"

Straightforward calculation generates this sequence. The second row represents the first 12 terms of the sequence now known by Fibonacci' name, in which each term, except the first two, is found by adding the two terms immediately preceding.
Later, especially in the middle decades of the 20th century, the properties of the Fibonacci numbers have been studied, resulting in a considerable literature.

In short, dividing a segment into two parts in mean and extreme proportion, so that the smaller part is to the larger part as the larger is to the entire segment, yields the so-called Golden Section, an important concept in both ancient and modern artistic and architectural design.

The successive coefficients of the radical 5 are Fibonacci's sequence 1, 1, 2, 3, 5, 8, while the successive second terms within the parentheses are the so-called Lucas sequence: 1, 3, 4, 7, 11, 18.

If a golden rectangle ABCD is drawn and a square ABEF is removed, the remaining rectangle ECDF is also a golden rectangle. If this process is continued and circular arcs are drawn, the curve formed approximates the logarithmic spiral, a form found in nature.

One class of electronic devices representative of contemporary cryptomachine technology is the Fibonacci generator, named for the Fibonacci sequences of number theory. In the classical Fibonacci sequence of integers . . . 21, 13, 8, 5, 3, 2, 1, each succeeding leftmost term is the sum of the two terms to its right. By loose analogy, any sequence in which each term is the sum of a collection of earlier terms in fixed (relative) locations is called a Fibonacci sequence.