PROJECT 03
GOLDEN RECTANGLE
AND GOLDEN RATIO
‘Rectangles’ and ‘ratios’ are the two concepts which have great importance in our day-to-day life. Due to this, they are studied in one form or the other at every stage of school mathematics. It is also a fact that whenever there is some discussion on rectangles and ratios, people start recalling something about ‘Golden rectangle’ and ‘Golden ratio’. Keeping in view the above, it was felt to know something about these two phrases ‘Golden rectangle’ and ‘Golden ratio’.
OBJECTIVE
To explore the meanings of ‘Golden rectangle’ and ‘Golden ratio’ and their relationship with some other mathematical concepts.
DESCRIPTION
‘Golden rectangle’ and ‘Golden ratio’ are very closely related concepts. To understand this, let us first understand the meaning of a golden rectangle.
(A) Golden Rectangle A rectangle is said to be a golden rectangle, if it can be divided into two parts such that one part is a square and other part is a rectangle similar to the original rectangle.
In the following figure, rectangle ABCD has been divided into a square APQD and a rectangle QPBC.
If the rectangle
QPBC is similar to rectangle
ABCD, then we can say that ABCD is
a golden rectangle. Let AB = l and BC = b. Therefore, QP = b.
Now, as ABCD ~ QPBC, we have
AB = QP BC PB
l b
or b = l – b
or l2 – lb = b2
i.e., l2 – lb – b2 = 0
æ l ö 2
or ôè b ôø
l
– b –1= 0
(1)
Let
l
b = x
So, from (1), we have
x2– x – 1 = 0
or x =
1± (–1)2 – 4 ´ (1) (–1)
2 ´1
= 1± 5 (Solving the quadratic equation).
2
Now, as x cannot be negative, therefore x =
5 + 1 .
2
l
Thus, b =
5 + 1
2
i.e., for a rectangle to become a golden rectangle, the ratio of its length and
breadth
æ l ö
ôè bôø
must be
5 + 1
Thus, it can be seen that the golden ratio is the ratio of the sides of a golden rectangle.
(B) Golden ratio and a continued fraction Let us consider a continued fraction 1 1 1 1 1 ... + + + We may note that it is an infinite continued fraction.
We may write it as 1 x= +1 x
So, x 2 = x + 1 or, x 2 – x – 1 = 0
It is the same quadratic equation as we obtained earlier.
So, again we have 5 1 2 + x = (Ignoring the negative root).
Thus, it can be said that the golden ratio is equal to the infinite continued fraction 1 1 1 1 1 ... + + + in the limiting form.
(C) Golden Ratio, Continued Fraction and a Sequence Having seen the relationship between golden ratio and the continued fraction 1 1 1 1 1 1 1 ...
let us examine the value of this fraction at different stages as shown below:
Considering 1, we get the value as 1;
considering 1+ 1 1 , we get the value as 2 1 ;
considering 1+ 1 1 1 1 + , we get the value as 3 2 ;
considering 1 1 1 1 1 1 1 + + + , we get the value as 5 3 ;
considering 1 1 1 1 1 1 1 1 1 + + + + , we get the value as 8 5 ; and so on
Thus, the values obtained at different stages are : 1, 2 3 5 8 13 , , , , ,... 1 2 3 5 8
The numerators of these values are 1, 2, 3, 5, 8, 13, ...
These values depict the following pattern: 3 = 1 + 2, 5 = 2 + 3, 13 = 5 + 8 and so on
Note that by including 1 in the beginning,
it will take the following form: 1, 1, 2, 3, 5, 8, 13, ...
This is a famous sequence called the Fibonacci sequence.
It can be found that the n th term of the Fibonacci sequence is 1 5 5 1 2 5 1 2 + + n n
It can also be seen that in the above expression, 5 1 2 + is the golden ratio.
(D) Golden Ratio and Trigonometric Ratios
It can be found that cos 36° = sin 54° = 5 1 4 + That is, 2 cos 36° = 2 sin 54° = 2 5 1 4 + = 5 1 2 + = Golden Ratio
Thus, it can be said that twice the value of cos 36° (or twice the value of sin 54°) is equal to the golden ratio.
(E) Golden Ratio and Regular Pentagons
We know that a pentagon having all its sides and all its angles equal is called a regular pentagon. Clearly, each interior angle of a regular pentagon will be 540 108 5 ° = ° .
Let us now draw any regular pentagon ABCDE and draw all its diagonals AC, AD, BD, BE and CE as shown in the following figure:
It can be observed that inside this regular pentagon, another pentagon PQRST is formed.
Further, this pentagon is also a regular pentagon.
It can also be seen that AP PQ , AP PT , AQ PQ , DS SR , ... are all equal to 5 1 2 + .
That is, the ratio of the length of the part of any diagonal not forming the side of the new pentagon on one side and the length of a side of the new pentagon is equal to the golden ratio.
(1) Further, it can also be seen that AE AP , AB BQ , CD DS , BC CS ,... are all equal to 5 1 2 + . That is, the ratio of the length of any side of the given regular pentagon and that of the part of the diagonal not forming the side of the new pentagon on one side is equal to the golden ratio.
(2)
Combining the above two results (1) and (2), it can be seen that in the above
two regular pentagons ABCDE and PQRST,
2
AB BC 5 1
...
PQ QR 2
+
= = =
, i.e., (Golden Ratio)2
That is, ratio of the corresponding sides of the two regular pentagons
ABCDE and PQRST is equal to (golden ratio)
2
.
We also know that the ratio of the areas of two similar triangles is equal to
the square of the ratio of their corresponding sides.
In fact, it is true for all the similar polygons.
Further, all regular polygons are always similar. So, it can also be said that ratio of the areas of above pentagons ABCDE and PQRST = 2 2 AB BC ... PQ QR = =
Therefore, ratio of the areas of the above two pentagons = 2 2 2 AB 5 1 PQ 2 + = = 4 5 1 2 + = (Golden ratio)4
Thus, areas of the above two regular pentagons is equal to (Golden ratio)4 .
[Note : The above results relating to trigonometric ratios and regular pentagons can be, in fact, proved using simple trigonometrical knowledge of Class XI].
CONCLUSION
Golden rectangle and golden ratio are very closely related concepts involving other mathematical concepts such as fractions, similarity, quadratic equations, regular pentagons, trigonometry, sequences, etc.
After getting some basic understanding of these at the secondary stage, they may be studied in a little details in higher classes at appropriate places and time.
APPLICATION
Project is useful in designing buildings, architecture and structural engineering.
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