. The Golden Ratio
What is the Golden Ratio?
“NCTM
Student Notes: "[The Golden Ratio] is found by dividing a segment into two
parts so that the length of the smaller part is to the length of the larger part
as the length of the larger part is to the length of the entire segment" (Yunker,
1986)” It is said however that a
definition of the golden ratio is easier to understand when illustrated in
algebraic terms. “Labeling
the shorter part of the segment as x and the longer part as the unit length, 1,
(see Figure 1) we conclude that the whole segment has length 1 + x and then the
Golden Ratio may be defined in algebraic terms by means of the proportion:
.
”
What is the history of the Golden Ratio?
Who studied the Golden Ratio and believed it was important to art and
architecture?
Euclid
studied the Golden ratio, as the construction for the golden section point is
found in “Euclid’s Elements” The rectangle
that is created by the golden ratio
is very appealing to the eye. “This
rectangle is supposed to appear in many of the proportions of that famous
ancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece” So, as you can see the
golden ratio has been an important tool since ancient times!!!
Art-“Luca
Pacioli (1445-1517) in his Divina proportione (On Divine
Proportion) wrote about the golden section also called the golden
mean or the divine proportion:
A M B
| 1-x | x |
The line AB is divided at point M so that the ratio of the two
parts, the smaller to the larger (AM and MB), is the same as the ratio of the
larger part (MB) to the whole AB.
If AB is of length 1 unit, and we let MB have length x, then the
definition (in bold) above becomes
the ratio of 1-x to x is the same as the ratio of x to 1 or, in symbols:
1 - x = x which simplifies to 1-x = x2
x 1
This gives two values for x, (-1-
5)/2 and (
5-1)/2.
The first is negative, so does not apply here. The second is just phi (which has
the same value as 1/Phi and as Phi-1).”
The work of Pacioli influenced, and can
even be seen and demonstrated in many artists’ works. Some of these artists
are Leonardo da Vinci, Albrecht Durer, Georges
Seurat, and Paul Signac.
Also, many modern books about art and drawing will demonstrate that placing the
objects in a picture around one third, or two two thirds across the page will
“make
the picture design more pleasing to the eye and relies again on the idea of the
golden section being "ideal".”
How does the Fibonacci Sequence relate to the
Golden Ratio?
What are other names for the Golden Ratio?-There
are many names
for the golden ratio. Some of them are the golden section, the golden mean, and
the golden number.
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