In Liber Abacci,
a problem is posed that gives rise to the sequence of numbers 1, 1,
2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on to infinity, known
today as the Fibonacci sequence. The problem is this:
How many pairs of rabbits placed in
an enclosed area can be produced in a single year from one pair of
rabbits if each pair gives birth to a new pair each month starting
with the second month?
In arriving at the solution, we find
that each pair, including the first pair, needs a month's time to
mature, but once in production, begets a new pair each month. The
number of pairs is the same at the beginning of each of the first
two months, so the sequence is 1, 1. This first pair finally doubles
its number during the second month, so that there are two pairs at
the beginning of the third month. Of these, the older pair begets a
third pair the following month so that at the beginning of the
fourth month, the sequence expands 1, 1, 2, 3. Of these three, the
two older pairs reproduce, but not the youngest pair, so the number
of rabbit pairs expands to five. The next month, three pairs
reproduce so the sequence expands to 1, 1, 2, 3, 5, 8 and so forth.
Figure 3-1 shows the Rabbit Family Tree with the family growing with
logarithmic acceleration. Continue the sequence for a few years and
the numbers become astronomical. In 100 months, for instance, we
would have to contend with 354,224,848,179,261,915,075 pairs of
rabbits. The Fibonacci sequence resulting from the rabbit problem
has many interesting properties and reflects an almost constant
relationship among its components.
![](images/FIG3-1.gif)
Figure 3-1
The sum of any two adjacent numbers
in the sequence forms the next higher number in the sequence, viz.,
1 plus 1 equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5
equals 8, and so on to infinity.
The Golden Ratio
After the first several numbers in
the sequence, the ratio of any number to the next higher is
approximately .618 to 1 and to the next lower number approximately
1.618 to 1. The further along the sequence, the closer the ratio
approaches phi (denoted f) which is an irrational number,
.618034.... Between alternate numbers in the sequence, the ratio is
approximately .382, whose inverse is 2.618. Refer to Figure 3-2 for
a ratio table interlocking all Fibonacci numbers from 1 to 144.
![](images/FIG3-2.gif)
Figure 3-2
Phi is the only number that
when added to 1 yields its inverse: .618 + 1 = 1 ÷ .618. This
alliance of the additive and the multiplicative produces the
following sequence of equations:
.6182 = 1 - .618,
.6183 = .618 - .6182,
.6184 = .6182
- .6183,
.6185 = .6183
- .6184, etc.
or alternatively,
1.6182 = 1 + 1.618,
1.6183 = 1.618 + 1.6182,
1.6184 = 1.6182
+ 1.6183,
1.6185 = 1.6183
+ 1.6184, etc.
Some statements of the interrelated
properties of these four main ratios can be listed as follows:
1) 1.618 - .618 = 1,
2) 1.618 x .618 = 1,
3) 1 - .618 = .382,
4) .618 x .618 = .382,
5) 2.618 - 1.618 = 1,
6) 2.618 x .382 = 1,
7) 2.618 x .618 = 1.618,
8) 1.618 x 1.618 = 2.618.
Besides 1 and 2, any Fibonacci number
multiplied by four, when added to a selected Fibonacci number, gives
another Fibo-nacci number, so that:
3 x 4 = 12; + 1 = 13,
5 x 4 = 20; + 1 = 21,
8 x 4 = 32; + 2 = 34,
13 x 4 = 52; + 3 = 55,
21 x 4 = 84; + 5 = 89, and so on.
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