Use of golden ratio in architecture

Describe how golden ratio is used in making of finest structures. It present three paradigmatic case studies where Golden Mean rectangles allegedly apply in architecture: (i) The Parthenon in Athens; (ii) The United Nations Secretariat Building in New York City; and (iii) The Great Pyramid of Giza.

1. Architecture and Town Planning Critical Assessment of Golden Ratio in Architecture by Fibonacci Series and Le M odulor System

2. Golden Ratio In everyday life, we use the word “proportion” either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term “proportion” is used to describe an equality of the type: nine is to three as six is to two. The Golden Ratio provides us with an intriguing mingling, it is claimed to have pleasingly harmonious qualities . The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C by the founder of geometry as a formalized deductive system , Euclid of Alexandria .

3. 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 lesser. If the ratio of the length AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio. The Golden Ratio is thus the ratio of the larger sub segment to the smaller. 

4. If the whole segment has length 1 and the larger sub segment has length x, then: Thus X is a solution of the quadratic equation             X 2 = 1–X   or   X 2 +x-1=0   This equation has two solutions   X1= (-1+ 5) / 2 ≈ 0.618 and X2 = (-1-  5) / 2 ≈ - 1.618   The length X must be positive, so   X = (1+ 5) / 2 ≈  1.618 or  Φ   (phi)

5. GOLDEN RATIO AND THE ANCIENT EGYPT 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."

6. The Egytians used both Pi (Π) and Phi (Φ) in the design of the Great Pyramids . T he  Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet).  This creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy.  If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.5367. This varies from the estimated actual dimensions of the Great Pyramid by only 0.0367 meters (1.4 inches) or 0.025%, which could be just a measurement or rounding difference.

7. Fibonacci Sequence In the 12th century, Leonardo Fibonacci wrote in Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi.  This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa . Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . Leonardo Fibonacci

8. He wanted to calculate the ideal expansion of pairs of rabbits over a year.  After the calculation he found that the number of pairs of rabbits are following a certain sequence. It turns out, though, that he was really on to something. Mathematicians and artists took this sequence of number and coated it in gold. The first step was taking each number in the series and dividing it by the previous number. At first the results don't look special. One divided by one is one. Two divided by one is two. Three divided by two is 1.5. Riveting stuff. But as the sequence increases something strange begins to happen. Five divided by three is 1.666. Eight divided by five is 1.6. Thirteen divided by eight is 1.625. Twenty-one divided by thirteen is 1.615.

9. Examples of the Golden Ratio in Nature As the series goes on, the ratio of the latest number to the last number zeroes in on 1.618. It approaches 1.618, getting increasingly accurate, but never quite reaching that ratio. This was called The Golden Mean, or The Divine Proportion, and it seems to be everywhere in art and architecture. Fibonacci spiral not only found in architecture but also widely present in nature. The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on. 

10. The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.  Even the microscopic realm is not immune to Fibonacci. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

11. Le Modulor System The  Modulor  is an anthropometric scale of proportions devised by the Swiss-born French architect Le Corbusier  (1887–1965). It was developed as a visual bridge between two incompatible scales, the Imperial system and the Metric system. It's a stylised human figure, standing proudly and square-shouldered, sometimes with one arm raised: this is Modulor Man, the mascot of Le Corbusier's system for re-ordering the universe.   This Modulor Man is segmented according the "golden section", so the ratio of the total height of the figure to the height to the figure's navel is 1.61. In devising this system, Corbusier was joining a 2000-year-old hunt for the mathematical architecture of the universe, a search that had obsessed Pythagoras, Vitruvius and Leonardo Da Vinci.

12. All these three; Fibonacci series, Golden ratio and Le Modulor System are interconnected. We can see the golden ratio in the alternative numbers of Fibonacci series. And the whole Le Modulor System is based on Golden ratio only. Keeping all these in mind a architect design a building. This golden ratio is considered to be one of the most pleasing and beautiful shapes to look at, which is why many artists have used it in their work . The two artists, who are perhaps the most famous for their use of the golden ratio, are Leonardo Da Vinci and Piet Mondrian .  It can be found in art and architecture of ancient Greece and Rome, in works of the Renaissance period, through to modern art of the 20th Century. However , various features of the Mona Lisa have Golden proportions, too. The Parthenon was perhaps the best example of a mathematical approach to art.

13. The Parthenon and Phi, the Golden Ratio The Parthenon in Athens, built by the ancient Greeks from 447 to 438 BC, is regarded by many to illustrate the application of the Golden Ratio in design.  Others, however, debate this and say that the Golden Ratio was not used in its design. It was not until about 300 BC that the Greek’s knowledge of the Golden Ratio was first documented in the written historical record by Euclid in “Elements.”

14. Challenges There are several challenges in determining whether the Golden Ratio was used is in the design and construction of the Parthenon: The Parthenon was constructed using few straight or parallel lines to make it appear more visually pleasing, a brilliant feat of engineering. It is now in ruins, making its original features and height dimension subject to some conjecture. Even if the Golden Ratio wasn’t used intentionally in its design, Golden Ratio proportions may still be present as the appearance of the Golden Ratio in nature and the human body influences what humans perceive as aesthetically pleasing. Photos of the Parthenon used for the analysis often introduce an element of distortion due to the angle from which they are taken or the optics of the camera used.

15. O verlay to the entire face This illustrates that the height and width of the Parthenon conform closely to Golden Ratio proportions.  

16. This construction requires a assumption though: The bottom of the golden rectangle should align with the bottom of the second step into the structure and that the top should align with a peak of the roof that is projected by the remaining sections.   Given that assumption, the top of the columns and base of the roof line are in a close golden ratio proportion to the height of the Parthenon.  This demonstrates that the Parthenon has golden ratio proportions, but because of the assumptions is probably not strong enough evidence to demonstrate that the ancient Greeks used it intentionally in its overall design, particularly given the exacting precision found in many aspects of its overall design.

17.   To elements of the Parthenon The grid lines appear to illustrate golden ratio proportions in these design elements.

18. Height of the columns  – The structural beam on top of the columns is in a golden ratio proportion to the height of the columns.  Note that each of the grid lines is a golden ratio proportion of the one below it, so the third golden ratio grid line from the bottom to the top at the base of the support beam represents a length that is phi cubed, 0.236, from the top of the beam to the base of the column. Dividing line of the root support beam -  The structural beam on top of the columns has a horizontal dividing line that is in golden ratio proportion to the height of the support beam. Width of the columns  – The width of the columns is in a golden ratio proportion formed by the distance from the center line of the columns to the outside of the columns .

19. The photo below illustrates the golden ratio proportions that appear in the height of the roof support beam and in the decorative rectangular sections that run horizontally across it.  The gold colored grids below are golden rectangles, with a width to height ratio of exactly 1.618 to 1.

20. The animated photo provides a closer look yet at the quite precise golden ratio rectangle that appear in the design work above the columns. The photo below illustrates how this section of the Parthenon would have been constructed if other common ratios of 2/3′s or 3/5′s had been intended to be represented by its designers rather than the golden ratio.

21. The UN Secretariat Building, Le Corbusier and the Golden Ratio The building, known as the UN Secretariat Building, was started in 1947 and completed in 1952. The architects for the building were  Oscar Niemeyer  of Brazil and the Swiss born French architect  Le Corbusier . Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion . Some claim that the design of the United Nations headquarters building in New York City exemplifies the application of the golden ratio in architecture.

22. Le Modulor system: Le Corbusier developed the Modulor in the long tradition of Vitruvius, Leonardo da Vinci’s Vitruvian Man, the work of Leone Battista Alberti , and other attempts to discover mathematical proportions in the human body and then to use that knowledge to improve both the appearance and function of architecture. The system is based on human measurements, the double unit, the Fibonacci numbers, and the golden ratio. Le Corbusier described it as a “range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things.”

23. Design of the UN Secretariat Building The  United Nations Secretariat Building is a 154 m (505  ft ) tall skyscraper and the centerpiece of the headquarters of the United Nations, located in the Turtle Bay area of Manhattan, in New York City. As much as Corbusier may have loved the golden ratio, it’s not easy to divide a 505 foot building by an irrational number like the golden ratio, 1.6180339887…, into its 39 floors and have them all come out equal in height and exactly at a golden ratio point. The building was designed with 4 noticeable non-reflective bands on its facade, with 5, 9, 11 and 10 floors between them. Interestingly enough, this configuration divides the west side entrance to the building at several golden ratio points

24. An interesting aspect of the building’s design is that these golden ratio points are more precise because The first floor of the building is slighter taller than all the other floors. the top section for mechanical equipment is also not exactly equal to the height of the other floors. The photo on the left shows lines based on Le Corbusier’s Modulor system, which are created when each rectangle is 1.618 times the height of the previous one. The photo on the right shows the golden ratios lines which are created when the dimension of the largest rectangle is divided again and again by 1.618. Both approaches corroborate the presence of golden ratio relationships in the design .

25. UN Secretariat Building West 3, Golden Ratios with PhiMatrix

26.  The building has 39 floors, but the extended portion for mechanical equipment on the top makes it about 41 floors tall. 41 divided by 1.618 creates two sections of 25.3 floors and 15.7 floors.  The golden ratio point indicated by the green lines is midway between the 15th and 16th floors, or 15.5 floors from the street. This means that the building was designed with a golden ratio as its foundation. Approximately 41 floors ÷ 1.618² ≅ 15.7 floors, and the visual dividing line is midway between the 15th and 16th floor.

27. A second golden ratio point defines the position of the third of the four non-reflective bands. This is based on the distance from the top of the building to the middle of the first non-reflective band, as illustrated by the yellow lines. Approximately (41 – 5.5 floors) ÷ 1.618² ≅ 21.9+5.5 floors ≅ 27.4 floors, and the visual dividing line is midway between the 26th and 27th floor. A third golden ratio point defines the position of the first and second of the four non-reflective bands. This is based on the distance from the base of the building to the top of the second non-reflective band, as illustrated by the blue lines. Mathematically, the 16 floors would be divided by 1.618 to create an ideal golden ratio divisions of 9.9 floors and 6.1 floors. This second dividing line on the building is at the 6th floors. 16 floors ÷ 1.618² ≅ 6.1 floors, and the visual dividing line is at the 6th floor.

28. Design of the windows and curtain wall of the building Other golden rectangles and golden ratios dividing points have been designed into the intricate pattern of windows. This is illustrated by golden ratio grid lines shown in the photos below. 

29. The design of the front entrance This attention to detail in the consistent application of design principles welcomes visitors as they enter the UN Building. The front entrance of the Secretariat Building reveals golden ratios in it proportions in the following ways: The columns that surround the center area of the front entrance are placed at the golden ratio point of the distance from the midpoint of the entrance to the side of the entrance. The large open framed areas to the left and right of the center entrance area are golden rectangles. The doors on the left and right side of the center entrance are golden rectangles. The left and center frame sections of the center section is a golden rectangle.

30. The interior floor plans reflect golden ratios in their design The pattern of golden ratios continued in the interior. Below is one of the representative floor plans, with the hallway dividing the floor at the golden ratio of the buildings cross-section. There is also a central conference room in the shape of a golden rectangle.

31. The Great Pyramid of Giza The   Great Pyramid of Giza  (also known as the  Pyramid of Khufu  or the  Pyramid of Cheops ) is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza,  Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact . There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty.

32. There is evidence, however, that the design of the pyramid embodies these foundations of mathematics and geometry: Phi, the Golden Ratio that appears throughout nature. Pi, the circumference of a circle in relation to its diameter. The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c² . Phi is the only number which has the mathematical property of its square being one more than itself:  Φ + 1 = Φ², or 1.618… + 1 = 2.618…

33. By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this: This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272.  The ratio of the height to the base is 0.636.

34. According to Wikipedia, the Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet).  This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy.  If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.5367. This varies from the estimated actual dimensions of the Great Pyramid by only 0.0367 meters (1.4 inches) or 0.025%, which could be just a measurement or rounding difference.

35. A pyramid based on golden triangle would have other interesting properties.  The surface area of the four sides would be a golden ratio of the surface area of the base.  The area of each triangular side is the base x height / 2, or 2 x Φ/2 or Φ.  The surface area of the base is 2 x 2, or 4.  So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base

36. It may be possible that the pyramid was constructed using a completely different approach that simply produced the phi relationship. The writings of Herodotus make a vague and debated reference to a relationship between the area of the surface of the face of the pyramid to that of the area of a square formed by its height. If that’s the case, this is expressed as follows: Area of the Face = Area of the Square formed by the Height (h) (2r × s)  / 2 = h² By the Pythagorean Theorem that r² + h² = s², which is equal to s² – r² = h², so r × s  =   s² – r²

37. Let the base r equal 1 to express the other dimensions in relation to it: s = s² – 1 Solve for zero: s² – s – 1 = 0 Using the quadratic formula, the only positive solution is where s = Phi, 1.618….. If the height area to side area was the basis for the dimensions of the Great Pyramid, it would be in a perfect Phi relationship, whether or not that was intended by its designers. If so, it would demonstrate another of the many geometric constructions which embody Phi.

38. Conclusion Using Fibonacci numbers, the Golden Ratio becomes a golden spiral, that plays an enigmatic role everywhere, from the nature such as in shells, pine cones, the arrangement of seeds in a sunflower head and even galaxies to the architectural design for structure as old as the pyramid of Giza to modern building like The Farnsworth House, designed by Ludwig Mies van der Rohe designed in 1950s. Adolf Zeising , a mathematician and philosopher, while studying the natural world, saw that the Golden Ratio is operating as a universal law. On the other hand, some scholars deny that the Greeks had any aesthetic association with golden ratio. Midhat J. Gazale says that until Euclid the golden ratio's mathematical properties were not studied. In the “ Misconceptions about the Golden Ratio ”, Dr. George Markowsky also discussed about some misconceptions of the properties and existence golden ratio in various structures and design. Basically the Golden Ratio should not be considered as a convention to all circumstances like a law of nature but it needs deeper study and analysis to establish the relation with the ratio as it is a curiosity of researchers to fulfil the demand of this field of research.