Interesting facts and samples of the golden ratio found in nature, art, math, and architecture
What is the Golden Ratio? If you ask different people you might a find a few that have been exposed to it in their field of work or in their hobby. If you ask an artist, you might be told that the golden ratio is a method of geometrical use of space in a painting. A biologist will explain how the golden ratio is found in the beauty of nature. A mathematician can provide solutions ranging from simple algebraic and geometric problems to more advanced problems in number theory. A medical doctor will share with you how the golden ratio is found in human body’s anatomy. What is truly amazing is that all these explanations, and many others, for the golden ratio are all correct!
In fact, the ancient Greeks saw this ratio as being so special they gave it its own name and they called it Phi (Greek symbol:φ). The same name still used today. The mathematical value is approximately 1.618….
The Greeks used this ratio to construct proportional rectangles which they believed as being the “most beautiful rectangles” These rectangles have their sides formed using this ratio. See figure below: (With figure "b" just a 90 degree rotation of the smaller rectangle to the left in figure "a".
Figure #1 Credit Reference #1
If you insert rectangle (b) into rectangle (a), the resulting space will be a square of sides “y.”
Figure #2 Credit Reference #1
As shown in the figure above, this process can be repeated over and over again resulting in smaller and smaller rectangles. This spiraling effect is what makes these rectangles so unique and the reason the Greeks saw this ratio as being so important. Let us explore a few examples is architecture:
The Golden Ratio in Architecture
1) The Great Pyramid of Giza
The Great Pyramid of Giza in 4700 B.C. with proportions according to a "sacred ratio."
Picture #1, Credit Reference #2
In above figure, “h” is the height of the pyramid, “b” is half the length of the base, and “a” is the height of a triangular face. The ancient Egyptians constructed the Great Pyramids in such a way that the ratio (b : h : a) is approximately equal to (1 : √φ : φ).
The Greek sculptor Phidias sculpted many things including the bands of sculpture that run above the columns of the Parthenon.
Picture #2, Credit Reference #3
See how each major portion of the design fits into a red box or rectangle.
3) United Nations Building
The ratio for width to height for every 10 floors is the golden ratio.
Picture #3, Credit Reference #2
As far back as the ancient Egyptians, the use of the golden ratio has been witnessed in the construction of their pyramids.
Euclid was the first person to write about the golden ratio in his mathematical work entitled Elements. At the time, and prior, mathematicians pondered the relationship between the ratio of two numbers “x” and “y” and equating that to the ratio of “x” and “x + y.” Such length would make two triangles similar Called Golden Triangles:
Figure #3 Credit Reference #1
One of the simplest methods to solve this ratio is to fix one variable and solve for the other. If we let x = 1, we have the following ratio and after cross multiplying the following formula:
Recall from the quadratic formula equation #2 can be solved by letting a = 1, b = -1, and c = -1.
In equation #4, there are two values. One is negative and cannot reflect a real value for length of a triangle while the other is a positive value. The positive value is what is called the Golden Ratio and this ratio is surprisingly found throughout nature, art, architecture, and science.
1) Statue of Athena:
Picture #4 Credit Reference #2
When viewed from the side the golden ratio can be seen. One golden ratio is the length from the front head to the opening of the ear compared to length of the forehead to the chin. The other ratio is the length of the nostril to the earlobe compared to the length from the nostril to the chin
2) Leonardo Da Vinci’s Mona Lisa
Picture #5 Credit Reference #3
In this world famous painting, we see each major section divided in accordance to the golden ratio.
4) Modern abstract art such as Penrose Tilings
“The British physicist and mathematician, Roger Penrose, developed a periodic tiling which incorporates the golden section. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B). When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the Golden Ratio."
Figure #4 Credit Reference #3
"In addition to the unusual symmetry, Penrose Tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean.” (3)
If a quarter of a circle is placed inside each of these resulting squares and the quarter circles are connected, we have a continuous spiral called a logarithmic spiral also called the golden spiral. See the below figure.
Figure #5 Credit Reference #1
Picture #6 Credit Reference #3
Golden spiral seen in sea shells and pine cones
Picture #7 Credit Reference #3
See how the Golden Spiral appears in this Nautilus Shell
Picture #8 Credit Reference #3
Nautilus shell cut in half
The Golden Ratio and the Fibonacci series
The Fibonacci series is a unique mathematical series that first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci (fi-bo-na-chee) in 1202. The series is a solution to simple mathematical problem about the population of reproducing rabbits.
The solution is:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (Each number is the sum of the previous two).
How the does the Golden Ratio relate to something completely different like the Fibonacci Series? Amazedly the answer lies when you compare pairs of Fibonacci numbers. If you take the preceding number and divide it into the subsequent number in the series then the answer you get will be become the Golden Ratio with ever large pairs of Fibonacci numbers! See table below:
Table #1, Credit Reference #4
The Golden ratio and the Fibonacci Series in Nature:
In nature one can see numerous examples of the Golden Ratio in the number of petals in plants:
Picture #9 Credit Reference #3
The Golden Ratio in all of us!
Since the ancient Greeks, people have known that the human body has symmetry. What is interesting is how the major portions of the human body follow the Golden Ratio. The bones making up your fingers are in the golden ratio.
Picture #10, Credit Reference #3
Red/Yellow, Yellow/Green, and Green/Blue are all the reciprocal of the Golden Ratio.
Golden Ratio in the Human Body:
Picture #11 Credit Reference #3
1) Height and length between naval point and foot
2) Length between shoulder line and length of the head
3) Length between finger tip to elbow and length between wrist and elbow
4) Length between naval point to knee and length between knee and foot
5) Length of the face and width of the face
6) Length of mouth and width of nose
7) Width of nose and length between nostrils
8) Length between pupils and length between eyebrows
All the ratios are very close to the Golden Ratio.
The golden ratio is a beautiful mathematical, biological, architectural, and artistic construct that is found everywhere we look. Now that you have an idea of what to look for, try to find out if you can find other interesting places where the Golden Ratio is hiding.
PS: The number of words in this article equates to the sum of the 14th and 16th Fibonacci numbers!
#1) "Schaum's Mathematics for Liberal Arts Majors," By Christopher Thomas, PhD, McGraw Hill Companies, 2009, Pages 208-210.
#2) "The beauty of the golden ratio," Oracle Think Quest.
#3) "The Golden Ratio in Art and Architecture," by Samuel Obara published by The University of Georgia, Department of Mathematical Education, J. Wilson
#4) www.mathamazement.com Lesson 5.3.2