Fn+2

––

2n

Fields medals 193

This pineapple has five diagonal rows of hexagonal scales in one

direction and eight rows in the other direction. (Note that the

sixth unmarked row of scales in the left picture is not a new row;

it is a continuation of the bottommost row of scales.)

sequence. For example, there are Fnways to climb n – 1

steps, one or two steps at a time (consider beginning the

climb with either a single step or a double step). There

are also Fnways to tile a 1 ×(n– 1) row of squares with

1 ×1 tiles and 1 ×2 dominoes, and there are Fn

sequences of 0s and 1s n-digits long beginning and end-

ing with 1 and containing no two consecutive 0s.

Regarding a 1 as “tails” and 0 and “heads,” and

ignoring the initial and final 1s, this latter example can

be used to show that the

PROBABILITY

of not getting

two heads in a row when tossing a coin ntimes is

. One can also use it to show that there are Fn+2

subsets of {1,2,…,n} lacking two consecutive numbers

as members.

Perhaps the most surprising appearances of

Fibonacci numbers occur in nature. The seeds in a sun-

flower’s head, for example, appear to form two systems

of spirals—often with 55 spirals arcing in a clockwise

tilt, and 34 spirals with a counterclockwise tilt. (Large

species of sunflowers have 89 and 144 spirals, again

consecutive Fibonacci numbers.) This appears to be

typical of all natural objects containing spiral floret,

petal, or seed patterns: pineapples, pinecones, and even

the spacing of branches around the trunk of a tree. The

botanical name for leaf arrangement is phyllotaxis.

It is useful to ask whether it is possible to find a

value xso that the sequence 1,x,x2,x3,… satisfies the

same recursive relationship as the Fibonacci numbers,

namely that every term after the second equals the

sum of the two preceding terms. This condition there-

fore requires xto be a number satisfying the equation

1 + x= x2. By the

QUADRATIC

formula there are two

solutions:

It follows that any combination of the form aϕn+ bτn

satisfies the same recursive relation as the Fibonacci

sequence. Choosing the constants aand bappropri-

ately, so that the first two terms of the sequence pro-

duced are both 1, yields the following formula, called

Binet’s formula, for the nth Fibonacci number:

(It is surprising that this formula yields an integer for

every value of n). One can use this result to show that

the ratio of Fibonacci numbers approaches the

value ϕas nbecomes large. This happens to be the

GOLDEN RATIO

.

The properties of the Fibonacci numbers are so

numerous that there is a mathematical periodical, The

Fibonacci Quarterly, devoted entirely to their contin-

ued study.

See also P

ASCAL

’

S TRIANGLE

;

POLYOMINO

.

field See

RING

.

Fields medals These are prizes awarded to young

researchers for outstanding achievement in mathemat-

ics. The awards are regarded as equivalent in stature to

Nobel Prizes (which do not exist for mathematics).

“International medals for outstanding discoveries in

mathematics” were first proposed at the 1924 Interna-

tional Congress of Mathematicians meeting in Toronto.

Fn

––

Fn–1