The Golden Ratio is extremely relative to Fibonacci and his numbers. The Greeks
popularized the ratio and it was first acknowledged in one of Euclid's book of
Elements
. In order
to find the Golden Ratio you want to take a line segment with the points at each end labeled
A
and
B
. Between A and B you will have another point that isn't quite in the middle but closer to B,
which is labeled
C
. (
Image G)
The ratio of the segment can be found using the integers
AB
AC
=
AC
BC
. For example, if the entire segment in length is equal to 13 inches and A to C
equals 8 inches, line B to C will be equal to 5 inches. Plugging these numbers into the formula
gives you amounts that are equal to the Golden Ratio (
13
8
=
8
5
).
Image G
The coincidental aspect of the Golden Ratio is that you often find the numbers of the
Fibonacci Sequence will equal the amount of the ratio. In decimal form the ratio is 1.618
....

Another way you can spot the Golden Ratio written is
(
1
+
√
5
)
/
2.
If you take a Fibonacci
number and divide it by the previous Fibonacci number in the sequence you will always end up
with the ratio or a decimal that is extremely close to it. Take the seventh Fibonacci number in the
sequence (
13
) and divide it by the sixth number (
8
) and you get 1.625. The ninth number divided
the eighth (
34/21
) and you get 1.619. It's not until you use the
eleventh Fibonacci number in the sequence divided by the tenth
that you will see the exact ratio of 1.618 (
89/54
). I've created a
table that shows the Fibonacci numbers divided by their previous
number in order to express how their answer falls so closely to
the Golden Ratio.
The Golden Rectangle is also a very important piece of the Fibonacci puzzle. Pairing
with the knowledge of the Golden Ratio, you can find the Golden Rectangle all around you in the
world. In fact, most people don't realize how much they find this rectangle to be most pleasing
because it is the perfect shape. How do you know when a rectangle is in fact golden? When the
length or the longer side of a the rectangle (
a)
is
divided by the height or the shorter side of the
rectangle (
b)
you should find the Golden Ratio or
phi
(
(
1
+
√
5
)/
2
or
1.618...
)
.
All rectangles
that are aesthetically pleasing to the eye happen to have sides lengths equal to the numbers in the
Fibonacci Sequence.
When looking at a perfect rectangle, you wouldn't realize it but you can find the
Fibonacci numbers hidden inside. Using the sequence (
1, 1, 2, 3, 5, 8, 13,...
) you can continue to
3/2
1.5
5/3
1.6666666667
8/5
1.6
13/8
1.625
21/13
1.6153846154
34/21
1.619047619
55/34
1.6176470588

build your shape. You start with a square that is 1x1. Next to it, you add another 1x1 square.
Above those you add a 2x2, then a 3x3, a 5x5, and so forth. (
Image H)
As you add to the shape,
you'll find that you continuously create larger golden rectangles. The even more brilliant side to
this pattern is that you can also find the Golden Ratio inside the Golden Rectangles as well.

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- Winter '15
- Andrew Martino
- Math, Fibonacci number, Golden ratio