No one knows for sure when or where the first appearance of the Golden Ratio in civilization occurred. The oldest example that has been found is located near Cairo, Egypt at the Great Pyramid of Giza. Built around 2560 BCE, this pyramid is the largest and oldest of the three pyramids in the Giza Necropolis. Hemiunu, the architect of the pyramid, may have used dimensions that yield the golden ratio for the aesthetic beauty of the structure; however, it could have also been a product of chance. No one can be absolutely sure. Over the years archaeologists have studied everything that one can possibly study about this pyramid, including the measurements of the outside dimensions. When considering the dimensions, these scientists have determined that the ratio of the slant height of the pyramid to half the base is the golden ratio. (Posamentier & Lehmann, 2012).
The next sighting of this marvelous ratio in history is in the works of Phidias, the Greek sculptor responsible for construction the Parthenon in Athens, built in the 5th century BCE. It is said that his design for the building itself and the sculptures that are found with it reflect the Golden Ratio. Posamentier & Lehmann (2012) demonstrate that the Parthenon “fits nicely into a golden rectangle—that is, a rectangle where the quotient of the sides is the golden ratio” (p. 45). The famous statue of Zeus located there is also reflective of the aesthetically pleasing Golden Ratio. Additionally, the ratio’s numerical representation, 1.618…, is commonly represented by the Greek letter F (Phi) because it is the first letter of Feidiaz, which is the name “Phidias” in Greek (Posamentier & Lehmann, 2012).
Although these structures have dimensions that conform to the Golden Ratio, the foundational thinking that has contributed to the language and understanding of it today was laid in the work of Plato and his students in the 4th century BCE. Plato was interested in explaining how the universe works and came into being. He posited that matter was structured by five regular solids: the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron (Livio, 2004). Plato and his students thought that earth, water, air, and fire were the four basic elements of matter. They argued that each of these elements corresponded with one of the solids. Earth was associated with the cube, fire with the tetrahedron, air with the octahedron, and water with the icosahedron. Since there was not an element to be assigned the dodecahedron, some of Plato’s followers thought that there might be a cosmic fifth element that pervades all matter. For this reason, these mathematicians were fascinated with the five Platonic solids. Each of these solids has dimensions that are related to the Golden Ratio. Therefore, scholars have come to the conclusion that the Greeks became interested in studying this ratio as a result of attempts to construct the five solids (Livio, 2002).
The interest in the Golden Ratio fostered by Plato and his associates led to its first definition. The first written reference to it is found in the Elements, a thirteen volume compilation of everything that the world knew about math at the time. This compilation of knowledge was written around 300 BCE by Greek mathematician Euclid (Posamentier & Lehmann, 2012). In one of these volumes Euclid provides the first written explanation of how to find what we now call Golden Ratio. He describes a line segment that is sectioned in such a way that the ratio of the whole segment to the larger portion is equal to the ratio of the larger portion to the smaller portion. Euclid called this the segment’s “extreme and mean ratio.” Therefore, if one can prove this type of proportional division exists on any segment, then the Golden Ratio has been found (Livio, 2002). Euclid used this description to show how the ratio could be found inside many geometric shapes, such as pentagons and the five Platonic solids (Bentley, 2008).
Next in the line of people who made history with the Golden Ratio was Fibonacci. This mathematician played an important role in the Arabic decimal system replacing the use of Roman numerals in Europe. He discovered an unusual set of numbers that is now named after him, the Fibonacci sequence; he may not have realized that his sequence of numbers had a connection with Phi. Phi is an irrational number, meaning that no ratio of two whole numbers will yield its value. The sequence of whole numbers that Fibonacci is responsible for gets the closest to achieving this. As the sequence continues, the ratio of a term and the one before it gets increasingly closer to Phi (Hemenway, 2005). Bentley states that Fibonacci’s sequence of whole numbers “acts like a light that steadily illuminates more and more of phi. The larger the numbers in the sequence, the more we see of the true value of phi” (2008, p. 76).
In about 1487, Leonardo da Vinci created the famous Vitruvian Man, in which he drew a picture of a man’s body that has measurements that approximate the Golden Ratio. The male figure is drawn with his arms and legs apart in two superimposed positions. The ratio is shown in that if one takes the distance from the top of his head to the navel and divides it by the distance from the navel to the sole of the man’s feet, the result is .0656, a number that is close to f (.618). Interestingly, the Vitruvian Man is drawn inscribed within a circle and a square that are tangent at one point at the bottom of the portrait. If the upper vertices of the square would have been closer to the circle or tangent to it, Da Vinci would have attained the Golden Ratio. In any event, the drawing is important as it relates to Phi because it was generally considered a breakthrough for demonstrating the proportions for the ideal human form. (Posamentier & Lehmann, 2012). In 1509 Da Vinci contributed to a three volume work, written by Franciscan friar and mathematician Fra Luca Pacioli. In this work, the two mathematicians examined and drew constructions of the five Platonic solids using the Golden Ratio (Posamentier & Lehmann, 2012). As an ordained Franciscan friar, and one who had studied theology, Pacioli entitled the work The Divine Proportion, because he believed that the properties of this special number were related to and expressions of certain characteristics of God; therefore, the ratio should be known as the Divine Proportion (Livio, 2002).
Kepler was the next person in history that made waves by using the Golden Ratio. He was one of the earliest proponents of the Copernican solar system. While most people at that time believed that the Earth was the center of the solar system, Kepler held the view that the Earth and other planets revolved around the sun. In calculating the orbits of the planets around the sun, Kepler theorized that the aforementioned Platonic solids acted as spacers between the orbits of the planets. For him, this was evidence that God used math to create the universe (Bentley, 2008).
Another name for the Divine Proportion developed in the 1830’s. German mathematician Martin Ohm wrote a book called Die Reine Elementar-Mathematik (The pure elementary mathematics) in which he leaves a footnote saying that a line that is divided in two parts in the manner that Euclid explained in the Elements is called the Golden Section. Although the language used implies that Ohm was not the first person to use the term in math circles, he is credited with being the first to officially use it in a written work. One thing that is certain is that the term began to be used frequently in German mathematical and art history literature after Ohm published his work. The term was first used in English in an article on aesthetics in the Encyclopaedia Britannica in 1875. The earliest mathematical use of the Golden Section in the English language appeared in an article of that name written by E. Ackerman in 1895. It was published in the American Mathematical Monthly (Livio, 2002).
Kepler was the next person in history that made waves by using the Golden Ratio. He was one of the earliest proponents of the Copernican solar system. While most people at that time believed that the Earth was the center of the solar system, Kepler held the view that the Earth and other planets revolved around the sun. In calculating the orbits of the planets around the sun, Kepler theorized that the aforementioned Platonic solids acted as spacers between the orbits of the planets. For him, this was evidence that God used math to create the universe (Bentley, 2008).
Another name for the Divine Proportion developed in the 1830’s. German mathematician Martin Ohm wrote a book called Die Reine Elementar-Mathematik (The pure elementary mathematics) in which he leaves a footnote saying that a line that is divided in two parts in the manner that Euclid explained in the Elements is called the Golden Section. Although the language used implies that Ohm was not the first person to use the term in math circles, he is credited with being the first to officially use it in a written work. One thing that is certain is that the term began to be used frequently in German mathematical and art history literature after Ohm published his work. The term was first used in English in an article on aesthetics in the Encyclopaedia Britannica in 1875. The earliest mathematical use of the Golden Section in the English language appeared in an article of that name written by E. Ackerman in 1895. It was published in the American Mathematical Monthly (Livio, 2002).