If you simply draw what you believe to be the most beautiful rectangle, then measure the lengths of each side, and finally divide the longest length by the shortest, you’ll probably find that the ratio is somewhere around 1.6-which is the golden ratio, phi, rounded to the nearest tenth. And since Phidias’ time, numerous painters and musicians have incorporated the golden ratio into their work too-Leonardo da Vinci, Salvador Dalí, and Claude Debussy, among many others.īut back to the problem of figuring out the shape of the most pleasing rectangle. Of course, the Greeks knew this long before modern psychologists tested it, which is why they used golden rectangles, as well as other golden shapes and proportions adhering to the golden ratio, in their architecture and art.įor example, almost 2500 years ago, a Greek sculptor and architect named Phidias is thought to have used the golden ratio to design the statues he sculpted for the Parthenon (note the word “phi” in Phidias’ name-that isn’t a coincidence and actually inspired the naming of the number in the 20th century). We won’t go into the details right now, but there is evidence that people tend to perceive one particular shape of rectangle as being most pleasing to the eye. What’s the most beautiful rectangle? More specifically: What’s the ratio of this “most beautiful” rectangle’s height to its width? This question seems strange, but it isn’t crazy. But how did this number come to be of such importance? Oddly, it started as a question of aesthetics. If you’re interested in seeing how the actual value of phi is obtained, check out this week’s Math Dude “Video Extra!” episode on YouTube. Phi isn’t equal to precisely 1.618 since, like its famous cousin pi, phi is an irrational number-which means that its decimal digits carry on forever without repeating a pattern. This number is now often known as “phi” and is expressed in writing using the symbol for the letter phi from the Greek alphabet. So, what is this golden ratio? Well, it’s a number that’s equal to approximately 1.618. One such place is particularly fascinating: the golden ratio. But the numbers in Fibonacci’s sequence have a life far beyond rabbits, and show up in the most unexpected places. And, save a few complicating details like the fact that rabbits eventually grow old and die, this sequence does an admirable job at modeling how populations grow. Gokul Rajiv and Yong Zheng Yew are two former high-school level students in Singapore who happened to explore the idea of metallic means in a project and found it interesting enough to share.Each successive number in this sequence is obtained by adding the two previous numbers together. We will explore these in the second part of this article. Metallic ratios share many common properties: they are linked to infinite sequences reminiscent of the famous Fibonacci sequence, to very special rectangles and to logarithmic spirals. For we get the golden ratio and for the silver one. The numbers, one for each value of are called metallic ratios, or metallic means.

The positive solutions to this equation is So in analogy to our calculations above we have If we require the ratio between and to be the same as the ratio between the whole line and one of the segments of length we have Suppose we divide our line into segments of equal length, which we call, and one smaller segment of length. This suggests the possibility of further generalisation. To recap, the golden ratio involves dividing a line into two segments and the silver ratio involves cutting it into three segments, two being of equal length. Once again, we can deduce this algebraically:

Golden ratio numbers