Geometry Notes — History — The Golden Ratio

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.

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.

Math History — Chinese Mathematicians — Father Zu Chongzhi and Son Zu Gengzhi

Zu Gengzhi was a Chinese mathematician. He lived from about 450 to 520 in China. He was the son of another famous Chinese mathematician, Zu Chongzhi. Zu Gengzhi discovered the “Zu Gengzhi’s Principle”, stating that “The volumes of two solids of the same height are equal if the areas of the plane sections at equal heights are the same.” This is same as Cavalieri’s principle, but was discovered about 1100 years earlier.

He is credited with the invention of the astronomical sighting tube, which Shen Kuo and Wei Pu would later improve during the 11th century.

The Zu family was an extremely talented one with successive generations being astronomers with special interests in the calendar. They handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted in China at this time. Zu Gengzhi, in the family tradition, was taught a variety of skills as he grew up. Zu Gengzhi’s greatest achievement was to compute the diameter of a sphere of a given volume. He had demonstrated that a previously accepted formula was incorrect by using various constructions for comparison, but he was unable to derive the correct formula.

His father Zu Chongzhi and Zu Gengzhi wrote a mathematical text entitled Zhui Shu (Method of Interpolation). This book most likely contained astronomical calculation techniques due to the accuracy of his calendars. It most likely contained formulas for the volume of the sphere, cubic equations and the accurate value of pi. Unfortunately, this book didn’t survive to the present day, it has been lost since the Song Dynasty, which began in the middle to late tenth century.

Their mathematical achievements included:

• the Da Ming calendar introduced by the father in 465. The midwinter was the starting point in making the calendar; therefore it was of extremely important to pinpoint the position of the sun on that day. However, the astronomers before Zu all believed the position was fixed, which caused errors in the calendar-making task from the very beginning. To solve the problem, Zu introduced the concept of procession (of the sun) in making the Da Ming Calendar, greatly enhancing the accuracy of calendar computation.
• distinguishing the Sidereal Year and the Tropical Year. The sidereal year is the time taken for the Sun to return to the same position with respect to the stars of the celestial sphere. It is the orbital period of Earth. A tropical year (also known as a solar year) is the length of time that the Sun takes to return to the same position in the cycle of seasons.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today. His estimate is only 46 seconds different from the modern estimate.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today. Using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving an approximation of pi that was correct to the seventh decimal place, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355 ⁄ 113 (detailed approximation) and 22 ⁄ 7 (rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 sided polygon. This was an impressive feat for the time, especially considering that the only device he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. No one discovered more of pi for nearly 1000 years.
• finding the volume of a sphere as 4πr³/3, where r is radius.
• discovering the Cavalieri’s principle, 1000 years before Bonaventura Cavalieri in the West.

Not too shabby. Very talented mathematicians without technology. I guess talent, hard work, and perseverance count for something huh. Outstanding.