Use of golden ratio in architecture PDF

Preview

Summary

Critical Assessment of Golden Ratio in Architecture by Fibonacci Series and Le Modulor System Kunal Sahu (12BCL1034), Nataraja Sai Charan (12BCL1053), Harshit Kumar (12BCL1010) School of Mechanical and Building Science, Vellore Institute of Technology, Chennai, 600048 _______________________________...

Description

Critical Assessment of Golden Ratio in Architecture by Fibonacci Series and Le Modulor System Kunal Sahu (12BCL1034), Nataraja Sai Charan (12BCL1053), Harshit Kumar (12BCL1010) School of Mechanical and Building Science, Vellore Institute of Technology, Chennai, 600048 _______________________________________________________________________________________________

Abstract: Golden Proportion or Golden Ratio is usually denoted by the Greek letter Phi (φ), in lower case, which represents an irrational number, 1.6180339887 approximately. Because of its unique and mystifying properties, many researchers and mathematicians have been studied about the Golden Ratio which is also known as Golden Section. Renaissance architects, artists and designers also studied on this interesting topic, documented and employed the Golden section proportions in eminent works of artifacts, sculptures, paintings and architectures. The Golden Proportion is considered as the most pleasing to human visual sensation and not limited to aesthetic beauty but also be found its existence in natural world through the body proportions of living beings, the growth patterns of many plants, insects and also in the model of enigmatic universe. This project seeks to represent a panoptic view of the miraculous Golden Proportion and its relation with the nature and architecture. This also 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. Geometrical substantiation of the equation of Phi, based on the classical geometric relations, is also explicated in this study.

__________________________________________________________________________ Introduction: The interrelation between proportion and good looks has made a lot of discussion in science because of the accidental occurrence of the shapes in various designs of objects like books, paintings, edifices and so on. The designs are approximated by a rectangle shaped such that the ratio of its length and height is equal to the ‘Golden Ratio,’ φ = (1 + 51/2)/2 = 1. 6180339887 (approx.). The ‘φ’ also called as the divine proportion, golden section, golden cut, golden ratio, golden mean etc. which is the result of dividing a segment into two segments (A + B) such that A/B = (A+B)/A = 1.6180339887 (approx.) where A > B. 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, . . . The ratio of each successive pair of numbers in the sequence approximates phi (1.618. . .), as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60. After the 40th number in the sequence, the ratio is accurate to 15 decimal places as shown in the given graph. 1.618033988749895 . . .

The natural proportioning system provides the foundation of the work of many artists and designers. The proportion known as the Golden Mean has always existed in mathematics and in the physical universe and it has been of interest to mathematicians, physicists, philosophers, architects, artists and even musicians since antiquity. In the early days of the 19th century it was suggested that the Greek letter ‘φ’ (Phi), the initial letters of Phidias’s name, should be adopted to designate the golden ratio. In the medieval age and during the Renaissance, the ubiquity of ‘φ’ in mathematics aroused the involvement of many mathematicians. It is unknown exactly when the idea was first discovered and applied by mankind. It appears that the primitive Egyptian engineers may have used both Pi (π) and Phi (φ) in the structural design of the Great Pyramids. The Greeks based the design of the Parthenon (example of Doric architecture, the main temple of the goddess Athena built more than 400 years BC) on this proportion.

Phi and its Relation with Geometry For a line segment, golden section can be considered as a point where the line is divided into two sections containing a unique property such as the ratio between the bigger segment and the shorter segment should be equal to the ratio between the line and its bigger segment [13, 14, 15, 37]. The ratio is approximately 1.6180339887 and is indicated by the Greek letter φ. If a line AC is considered according to Figure below, the point B is its golden section where the line is divided into two sections p and q (AB = p and BC = q). According to the theory, the ratio of p and q is equal to the ratio between (p + q) and p where p > q. This relation can be represented by the following equation,

From this relation it is clear that if any of the segments of the line AC is considered as 1, the other segment can easily be found with the value of φ. For example, if q is considered as 1 (q = 1) then p will be φ (p = φ) and if p is considered as 1 (p = 1) then the value of q will be the inverse of φ that

is considered as φ/ (1/φ = φ/). From the most ancient time it is often been claimed that the golden section is the most aesthetically pleasing point at which the line is sectioned. Consequently, the idea has been incorporated into many art works, architectural design and mathematical analysis.

Fig. 1: A line segment AC is divided in the golden section point B.

Golden Section and the Beauty of Nature Fibonacci numbers are said as one of the Nature's numbering systems because of its existence not only in the population growth of rabbits, but also everywhere in Nature, from the leaf arrangements in plants to the structures in outer space. The special proportional properties of the golden section have a close relationship with the Fibonacci sequence. Any number of the series divided by the contiguous previous number approximates 1.618, near to the value of φ. Golden section preferences are considered as an important part of human beauty and aesthetics as well as a part of the remarkable proportions of growth patterns in living things such as plants and animals. Many flowers have the arrangement in petals that are to the Fibonacci numbers. Some display single or double petals. Three petals are more common like Lilies and Iris. Some have 5 petals such as Buttercups, Wild Rose, Larkspur and Columbines. Some have 8, 13, 21, 34, 55 and even 89 petals. All these numbers are consecutive Fibonacci numbers. The petal counts of Field Daisies are usually thirteen, twenty-one or thirty-four. The seed heads are also follow the Fibonacci spiral arrangement. Other flowers having four or six petals also have a deep relation with Fibonacci numbers where they can be grouped into two and three respectively having two members each. Passion flower also known as Passiflora Incarnata is a perfect example having the existence of the Fibonacci Numbers.

Golden Section in Architecture 1. The Parthenon in Athens The Parthenon in Athens, built by Iktinos and Kallikrates around 440 BC, is widely perceived as an extremely attractive building. One of several reasons for its appeal is its scaling hierarchy and highly ordered complexity. And yet Golden Mean enthusiasts attribute its informational success to its supposed design using Golden Mean rectangles, and the Parthenon is promoted as being one of the paradigmatic examples of buildings designed according to the Golden Mean. But 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.” It states, “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.” There are several challenges in determining whether the Golden Ratio was used is in the design and construction of the Parthenon: 

 

Fig. 2: Some flowers having different number of petals related to Fibonacci numbers, (a) White Calla Lily having one petal, (b) Euphorbia having two petals, (c) Trillium with three petals, (d) Hibiscus having five petals, (e) Buttercups with five petals, (f) Bloodroot with eight petals, (g) Black Eyed Susan having thirteen petals, (h) Shasta Daisy having 21 petals and (i) Daisy with 34 petal.

This golden ratio has been observed in flowers as shown above also in fruits, vegetables, on the branches of trees, on animals, their eyes tails. The same properties also are found on the beautiful design of butterfly wings and shapes. The natural design of Peacock’s feather also goes to the golden ratio.



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.

To provide better insight into the answer to this question, the photos below selected for photographic analysis are of very high resolution and were taken from an angle that is almost exactly perpendicular to the face of the Parthenon. The grids overlayed on each of the photos are from PhiMatrix golden ratio software so each line of the grid is in perfect golden ratio proportion to other grid lines.

The photo below (Figure 3) shows a Golden Rectangles with a Golden Spiral overlay to the entire face of the Parthenon. This illustrates that the height and width of the Parthenon conform closely to Golden Ratio proportions. This construction requires an assumption though, that 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.



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.

Fig. 4: Application of Golden Ratio on the elements of the Parthenon.

A magnified of the above photo view reveals that each of these golden ratio proportions is very close to perfect, but perhaps not as exact as one might hope, particularly given the preciseness of the design and construction of the Parthenon.

Fig. 3: Golden Rectangles with a Golden Spiral overlay to the entire face of the Parthenon

In the next photo (Figure 4), however, applies golden ratio grid lines to elements of the Parthenon that remaining standing. The grid lines appear to illustrate golden ratio proportions in these design elements: 



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.

Fig. 5: A closer look on the roof support beam.

The photo below illustrates the golden ratio proportions that appear in the height of the roof support beam and in the decorative rectangular

Fig. 6: A closer look on the roof support beam.

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:

After checking all dimensions of the Parthenon, we find a variety of numbers and proportions. A floor plan view shows eight columns across the front view and seventeen columns from the side view. Six columns are the inside entry way, with five by ten columns enclosing the large interior temple room. Several interior rooms are found, some with proportions that are close to a golden rectangle, but clearly not exactly a golden rectangle.

Fig. 8: Plan of the Parthenon.

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. If the Greeks had intended the Parthenon to highlight the golden ratio in its design, they could have taken advantage of many more opportunities to do so, or done it with the level of exacting precision in the various places that it seems to appear that is found throughout its design and construction. If, however, the golden ratio was intended to be included among the many numbers and proportions included, then one can find some rather compelling evidence that they applied it, whether through a simple geometry construction below or with the deeper knowledge recorded by Euclid some 150 years later.

2. The United Nations Secretariat Building, Le Corbusier and the Golden Ratio Some claim that the design of the United Nations headquarters building in New York City exemplifies the application of the golden ratio in architecture. Debunkers of the golden ratio say no, that this is just another groundless myth to be dispelled. Here we will review its design, the sources of the claims and mathematics of the dimensions. 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. 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.”

Le Corbusier first devised the Modulor system in 1943, presented his concepts in the US in 1946 and published the Modulor in 1948. Construction on the UN Secretariat Building was started in 1947 and completed in 1952, so we can assu e it was i the forefro t of Le Corbusier’s thinking at the time the building was designed.

Fig. 9: The UN Secretariat Building

Richard Padovan states on page 316 of his book “Proportion: Science, Philosophy, Architecture: “Le Corbusier placed systems of harmony and proportion at the centre of his design philosophy, and his faith in the mathematical order of the universe was closely bound to the golden section and the Fibonacci series, which he described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in Man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages, and the learned.” The other lead architect, Niemeyer, was heavily influenced by Le Corbusier and used golden ratios in a previous building designed with Le Corbusier. Le Corbusier conceived a tall central building that would house all the Secretariat offices. Le Corbusier’s plan, known as project 23A, was taken as the basis for the design.

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, illustrated with PhiMatrix software in the photos below (Figure 10). 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, and 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. The photo on the left illustrates a Modulor progression in golden ratio dimensions, starting with the height of the entrance and building to the bottom of the first two dividing lines. In the photo on the right, the first golden ratio point defines the middle of the second non-reflective band. This is based on the height from the base at street level to the top of the building, as illustrated by the green lines. 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 red 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.

Fig. 10: UN Secretariat Building West 3, Golden Ratios with PhiMatrix.

Approximately 41 floors ÷ 1.618² ≅ 15.7 floors, and the visual dividing line is midway between the 15th and 16th floor. 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 nonreflective 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. The front ...