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