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