Mathematician George Markowsky wrote a paper in 1992 called Misconceptions about the Golden Ratio, labeling these claims as simply misconceptions. The Great Pyramid of Giza has also been attributed to golden ratio-incorporating design by the ancient Egyptians, with a slope angle apparently close to one that would make it a golden pyramid. The Parthenon in Athens is often featured in introductions to the golden rectangle, with a picture of the façade (in a non-ruined state) enclosed in the rectangle. Some masterpieces of ancient architecture have been associated with the golden ratio, albeit with dubious evidence. The spiral arrangements of leaves, seeds and flowers, often given as examples of the Fibonacci sequence in nature, allow the parts to be packed closer together, and so more can fit in a given space. The golden ratio and Fibonacci sequence are important in optimization. Psychologist Adolf Zeising brought a resurgence in the popularity of the golden ratio in the 19 th century, writing of a universal law for “beauty and completeness in the realms of both nature and art”, after noting the golden ratio’s appearance in plants. Pacioli gives reasons for the ratio to be referred to as divine, arguing its simplicity, irrationality and self-similarity is representative of aspects of the Christian God. In the late 15 th century, the golden ratio first gained the pompous name of “the divine proportion” in a book on geometry and architecture, Divina Proportione, written by Italian mathematician and friar Luca Pacioli and illustrated by Leonardo da Vinci. There is a major link between the two, in that the ratios between consecutive Fibonacci numbers (of the sequence 0, 1, 1, 2, 3, 5, 13…) converge to the golden ratio:Ī golden rectangle (red) placed next to a square (blue) forms a larger golden rectangle. Fibonacci later used the golden ratio to solve geometry problems in the 11 th century, although he never related it to the Fibonacci sequence which is named after him. The golden ratio was first discovered by the ancient Greeks, in connection with its frequent appearances in geometry. The ratio can be represented by the irrational number phi, which is a solution to the quadratic equation x 2 – x – 1 = 0.įrom the golden ratio comes the golden rectangle, whose sides fit the ratio, as well as the golden angle, the golden triangle, pentagram, and pyramid. In words: two quantities a and b, with a being the greater one, are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Starting with the basic facts, the golden ratio is defined algebraically with quantities a and b, where a is greater than b, as These assertions are so widespread that they seem common knowledge, but many of the supposed instances of the golden ratio may be nothing more than myth. It has featured in architecture and paintings throughout history, and in human body proportions. Many textbooks, articles and books claim that the golden ratio, and shapes based upon it such as the golden rectangle, are aesthetically pleasing. For a mathematical ratio, it is remarkably well-known, but its appearance in works for the layperson seem to always mention something about its inherent beauty. The golden ratio is a simple relation between two quantities commonly occurring in mathematics and in nature.
Number enthusiasts are likely familiar with the idea of the golden ratio, and the artistically minded may have heard of its applications in design. Plant exhibiting a so-called golden spiral.
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