A golden rectangle is one whose lengths are in the ratio 1:1.6180339875, the golden ratio. The most interesting feature of the golden rectangle is that on partitioning the rectangle into a square and another rectangle, results into a rectangle with similar properties. The flag of Togo also closely approximates the golden rectangle.
The golden ratio, also referred to as the golden mean or section, is the irrational number (1+ √5)/2 represented by the Greek letters τ or φ. Due to its appearance in many civilizations in history, it is reasonable to assume it has been discovered and rediscovered and probably that’s why it goes by several names. Nevertheless, according to the history of mathematics, it was understood and used by ancient Egypt mathematicians first. Mathematicians from Greece also studied it. An example is the Greek mathematician and sculptor Phidias, who studied and used it in many design of his sculptures. However, the Greeks attributed this concept extreme and mean ratio -another name of the ratio used between the third until about the nineteenth century- to Pythagoras and his followers. Similarly, Euclid- another Greek mathematician- in his book the Elements (1482), stated "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less," the first written evidence of what we now call the golden ratio. Johannes Kepler, a German astronomer, termed the ratio a “precious jewel” after he discovered, it was the limit of the ratio of consecutive numbers of the Fibonacci sequence. It is approximately 1.6180 (Joyce, 1997).
In algebra, if a line is divided into two segments, x and y being the larger and shorter segments respectively, then the segments are in golden ratio if the sum of x and y divided by x is equal to x divided by y, that is (x+y)/x =x/y= φ. In addition, if the shorter segment is 1 unit and the longer one x units, we get the equation (x+1)/x= x/1 which can be rewritten to the quadratic equation x2-x-1=0 whose positive root is x = (1 + √5)/2, phi.
A 2008 BBC documentary titled the story of mathematics, suggests that the concept of the golden ratio was hidden in the proportions of the pyramid. In mathematics, a pyramid where the slant height is equal to phi time the semi-base is known as the golden pyramid. The great pyramid of Giza, an ancient architecture and wonder of the world, is one where the golden ratio appears and is remarkably close to the golden pyramid. It is reported to be 148.2m in height and 116.4m in semi-base, yielding a ratio of the slant height to the base of 1.6189. The Greek Parthenon, built between 447 and 472BC, shows some of the golden ratio properties. Both the façade and elements of its façade are said to be circumscribed in golden rectangle by some historians (Sautoy & Okuefuna, 2008).
Da Vinci, a great mathematician and sculptor, used the golden ratio in some of his most famous pieces of art. The face of Mona Lisa, a famous painting, actually appears in a golden rectangle. Scientific comparison between the widths of her forehead to the length from the top of her head to her chin shows the golden ratio. Heinrich Agrippa, a philosopher of the 16th century implied a relationship to the golden ratio by drawing a man over a pentagram inside a circle. Other pieces of art that show evidence of use of this ratio include the drawing of the last supper and the statue of Athena.
In nature the golden ratio appears in both plants and animals. A honeybee colony consists of a queen, a few drones and a lot of workers. A drone has one parent since it hatches from an unfertilized egg which implies that in the family tree, it has one parent, two grandparents and three great grandparents and so on according to the Fibonacci series. Looking at the centre of sunflower, the seeds appear to form curves at the center curving left and right. Spiral patterns in pineapples, pinecones and cauliflower also reflect the appearance of the Fibonacci sequence and to some extent the golden ratio. It has also been observed by scientist that displacement of leaves around a stem, phyllotaxis, occurs in patterns defined by the Fibonacci series. Petals of the rose flower are separated an angle congruent to the divine proportion of 137.5 degrees.
Engineering also applies concepts of the golden ratio in many of its applications from the construction of ancient buildings to recent skyscrapers. For instance, California Polytechnic State University has made provisions to build a magnificent iconic engineering plaza relying on the Fibonacci numbers. Jeffrey Gordon, a former student and the project designer, explains that the guiding principle was the use of the Fibonacci series as a representation of engineering knowledge. Also, the Eden project in St. Austell, England has an education centre- The core- which was designed using plant spirals and Fibonacci numbers. It is also claimed that the United Nations headquarter building in the New York also shows this properties.
The golden ratio, one of the most important numbers in mathematics, is a universal law which contains the ground-principle of all formative striving for completeness and beauty in art and nature. All things follow a set of laws of behavior from galaxies and plants to sea shells and humans. The pattern is always the same though, hidden in plain sight and the ratio repeats itself as 1.16180 over and over again.
References
Joyce D.E. (1997). Euclid, Elements, Book 6, Definition 3. Retrieved from http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/defVI3.html
Livio, M. (2002). The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. New York: Broadway Books.
Sautoy, M. & Okuefuna, D. (2008). The Story of Mathematics [Documentary]. United Kingdom: BBC.
