What is the Golden Ratio?
The golden ratio, also known as the golden mean or golden section, is a number often encountered when taking the measurements of objects in nature. It is approximately equal to 1.61803 (sixteen decimal places), and its numerical value appears many times in mathematics, art, architecture and other fields. In simple terms, the golden ratio can be thought of as a way to divide a line so that the longer part divided by the shorter part is equal to the sum of the two parts divided by the longer part. This relationship is referred to as an extreme and mean ratio because it establishes a division of something into two parts where one part is an extreme (the long side) and another part is a mean (the short side). The term “golden” refers to this unique property which has led to its widespread use throughout history.
The concept of dividing things into proportions based on this particular number has been around for centuries, dating back to at least 300 BC with Euclid’s Elements. In Book VI he discusses how to divide a line segment so that “the whole shall be to the greater segment as the greater segment is to the less.” He goes on further to say that this proportion “is called dividing a line in extreme and mean ratio.” However, it wasn’t until much later that this specific number was first given a name. It wasn’t until 1509 that Luca Pacioli published his book Divina Proportione which included illustrations by Leonardo da Vinci. It was here that Pacioli gave this number its now famous name: “divine proportion,” referring to God’s role in creating harmony through such mathematical relationships.
While Euclid was content with simply observing and describing these proportions, others were more interested in finding uses for them. One early example comes from Ancient Greece where architects used what we now call the golden rectangle when designing buildings such as temples. A golden rectangle can be created by starting with any rectangle and then removing a square from one corner; what remains will also be a rectangle but with different proportions than before (specifically, it will have sides whose lengths are in accordance with the golden ratio). These newly created rectangles can then have squares removed from their corners, resulting in yet smaller rectangles whose sides maintain those same special proportions…and so on ad infinitum! Not only did Greek architects find such divisions pleasing aesthetically, but they also believed there were practical benefits as well since these ratios create shapes which are easy for humans perceive and comprehend intuitively.
Another well-known occurrence of Fibonacci numbers appears when dealing with populations of animals who reproduce at rates governed by Fibonacci numbers… For example, suppose each female rabbit produces one male and one female rabbit every month starting from her second month of life onward (thus beginning reproduction immediately after reaching maturity at age 1 month). If we ignore all deaths due mainly environmental factors like predators or starvation), then after exactly 1 year there will be 768 pairs of rabbits total (counting both parents and offspring). This sequence generalizes quite easily: if each new-born pair consists of one male and one female regardless gender of either parent rabbit , then after n months there will Fn+2 pairs altogether . So for our original problem involving just females reproducing monthly according strictly to Fibonacci numbers , we would expect Fn+2=Fn+1+Fn=768 pairs rabbits total after 12 months .
