The golden ratio has fascinated intellectuals of diverse interests for over two thousand years. Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry. The Greeks usually attributed discovery of the ratio to Pythagoras and his followers. Later, Euclid’s Elements provides the first known written definition of the golden ratio: “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.” Euclid explains a construction for dividing a segment in an extreme and mean, the golden ratio. Throughout Euclid’s masterpiece, several facts and their proofs used the golden ratio. Some of these factual statements show that the golden ratio is an irrational number. Luca Pacioli, a contemporary of Leonardo da Vinci, wrote of the golden ratio in his Divina Proportione of 1509. The golden ratio captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise.
Mark Barr was an American mathematician who, in about 1909, gave the golden ratio the name of phi φ , the first Greek letter in the name of Phidias, the Greek sculptor who lived around 450 BC. (Phidias, 480 BC – 430 BC, was an ancient Greek sculptor, painter and architect, universally regarded as the greatest of all Classical sculptors.)
Johannes Kepler once stated, “Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of Gold; the second we may name a Precious jewel.” Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer, and key figure in the 17th century astronomical revolution. Kepler incorporated religious reasoning into his work, motivated by the religious conviction that God had created the world according to an intelligible plan that is accessible through the natural light of reason.
The number itself:
In Euclid’s words: 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.
Consider the segment BC .
A is a point between B and C and let’s say A is closer to C .
If the ratio of the lengths BC to AB is equal to the ratio of the lengths AB to AC, then the segment has been cut in the extreme and mean ratio, or in a golden ratio.
Let’s assume the length of segment BC is 1.
We can call AB “x” and AC “(1 – x).”
We can solve for x.
Cross multiply the extremes and means in our proportion.
We can solve this using our quadratic formula. a = 1, b = 1, and c = -1.
We will ignore the negative solution since we are solving for a length and we know by definition that a length is always positive. So by solving the above equation we know that AB is approximately 0.6180339887, which would give AC being approximately 1 – 0.6180339887 or 0.3819660113. Remember that these are approximate values because to solve our equation we were working with the irrational number, square root of 5.
Now Euclid’s extreme to mean ratio, or golden ratio:
If we divide the “whole line” by the “greater segment”, 1 / 0.6180339887, we obtain 1.618033989.
If we divide the “greater” by the “lesser”, 0.6180339887 / 0.3819660113, we obtain 1.618033988. The two values are different at the ninth decimal place, but that’s because my TI-84 is rounding numbers that I’m working with that I rounded earlier. Remember, when you work with rounded decimals, you are working with error, and if you continue to operate with decimals that you’ve rounded, you will magnify that error. Old school — no decimals unless there’s no choice.
φ = 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204
189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661
319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454…
A scratch at the history:
According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart. In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers. Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.
Phidias applied the golden ratio to the design of sculptures for the Parthenon.
Plato, in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.
The Egyptians thought that the golden ratio was sacred. Therefore, it was very important in their religion. They used the golden ratio when building temples and places for the dead. If the proportions of their buildings weren’t according to the golden ratio, the deceased might not make it to the afterlife or the temple would not be pleasing to the gods. As well, the Egyptians found the golden ratio to be pleasing to the eye. They used it in their system of writing and in the arrangement of their temples. The Egyptians were aware that they were using the golden ratio, but they called it the “sacred ratio.” The Egyptians used both pi and phi in the design of the Great Pyramids.
The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty.
Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background sculptures to achieve balance and beauty.
Leonardo da Vinci’s drawings of the human body emphasized its proportion. The ratio of the following distances is the Golden Ratio: (foot to navel) : (navel to head).
The proportions of Michelangelo’s David conform to the golden ratio from the location of the navel with respect to the height to the placement of the joints in the fingers.
The Ark of the Covenant is a Golden Rectangle. In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten Commandments, saying, “Have them make a chest of acacia wood – two and a half cubits long, a cubit and a half wide, and a cubit and a half high.” The ratio of 2.5 to 1.5 is 1.666…, which is as close to phi as you can come with such simple numbers and is certainly not visibly different to the eye. The Ark of the Covenant is thus constructed using the Golden Section, or Divine Proportion.
