the Greeks. In the world of
mathematics, the numeric value
is
called
"phi",
named
for the
Greek sculptor Phidias.
Both the rectangles ABCD and
PBCQ are golden rectangles.
Golden Rectangle

What is a Golden Rectangle?
A rectangle can be drawn of such a shape that if it is cut into a square
and a rectangle, the smaller rectangle will be similar in shape to the
larger rectangle.
1
x
1 + x
x
1
1
This is the golden rectangle whose sides are in the” golden ratio” of 1+x: 1,
where x is a non-ending decimal
whose value can be calculated in a
number of ways, including the construction of a simple continued fraction.

Since the two rectangles are similar, their sides are in the same ratio as follows:
Or simply x can be replaced on the right-hand side by
or
Continue replacing x by eq 1
Continue the process, we will arrive at the following equation after eight iterations

Iteration means repeating a
process over and over again.
In Mathematics, it means the
repeated application of an
operation on a given function
over and over again.
The golden ratio is also given by the ratio
of the two sides
of the golden
rectangle.
After the largest square is cut off, the leftover piece is again a
golden rectangle.
The largest square is cut again from the leftover
rectangle, and so on. Since the squares get smaller by scaling factor,
they are self-similar golden squares.

Original golden rectangle is cut up into ever-decreasing squares
+
+
+
+
+
+

Exercises Set
Answer the following problem:
1.
If you have a wooden board that is 0.75 m wide, how long
should you cut it such that the golden ratio is observed?
Use
1.618 as the value of the golden ratio.
2.
The golden ratio( shoulder to waist) is the most important
ratio for achieving the body proportions like that of a Greek
god.
Now,
measure
your shoulder circumference s and
then your waist size w.
Then divide s by w.
Is the result
roughly the golden ratio?
If not, then what must be your
waist size to get the golden ratio?

Exercises Set
3.
Cut out
the golden rectangle of different dimensions and
show that a considerable number of cutouts give out the golden
ratio.
( example:
10cm by 16.2 cm)

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- Fibonacci number, Golden ratio