Architecture and Town

PlanningCritical Assessment of Golden Ratio in

Architecture by Fibonacci Series and Le Modulor

System

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.

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.

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

X2= 1–X or X2+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 orΦ (phi)

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."

The Egytians used both Pi (Π) and Phi (Φ) in the design of

the Great Pyramids. 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 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.

Fibonacci Sequence

In the 12th century, Leonardo Fibonacci wrote in Liber Abaci of asimple numerical sequence that is the foundation for anincredible mathematical relationship behind phi. This sequencewas known as early as the 6th century AD by Indianmathematicians, but it was Fibonacci who introduced it to thewest after his travels throughout the Mediterranean world andNorth Africa.

Starting with 0 and 1, each new number in the sequence issimply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

Leonardo Fibonacci

He wanted to calculate the ideal expansion of pairs of rabbits over ayear. After the calculation he found that the number of pairs ofrabbits are following a certain sequence. It turns out, though, thathe was really on to something. Mathematicians and artists took thissequence of number and coated it in gold.

The first step was taking each number in the series and dividing itby the previous number. At first the results don't look special. Onedivided by one is one. Two divided by one is two. Three divided bytwo is 1.5. Riveting stuff. But as the sequence increases somethingstrange begins to happen. Five divided by three is 1.666. Eightdivided by five is 1.6. Thirteen divided by eight is 1.625. Twenty-onedivided by thirteen is 1.615.

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.

The Fibonacci sequence can also be seen inthe way tree branches form or split. A main trunkwill grow until it produces a branch, which createstwo growth points. Then, one of the new stemsbranches into two, while the other one liesdormant. This pattern of branching is repeated foreach of the new stems.

Even the microscopic realm is not immune toFibonacci. The DNA molecule measures 34 angstromslong by 21 angstroms wide for each full cycle of itsdouble helix spiral. These numbers, 34 and 21, arenumbers in the Fibonacci series, and their ratio1.6190476 closely approximates Phi, 1.6180339.

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.

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.

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.”

Challenges

There are several challenges in determining whether the Golden Ratio was usedis in the design and construction of the Parthenon:

The Parthenon was constructed using few straight or parallel lines to make itappear more visually pleasing, a brilliant feat of engineering.

It is now in ruins, making its original features and height dimension subject tosome conjecture.

Even if the Golden Ratio wasn’t used intentionally in its design, Golden Ratioproportions may still be present as the appearance of the Golden Ratio innature and the human body influences what humans perceive as aestheticallypleasing.

Photos of the Parthenon used for the analysis often introduce an element ofdistortion due to the angle from which they are taken or the optics of thecamera used.

Overlay to the entire face

This illustrates that the height and width of the Parthenon conform closely to Golden Ratio proportions.

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.

To elements of the Parthenon

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.

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.

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.

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.

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.

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.”

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

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.

UN Secretariat Building West 3, Golden Ratios with PhiMatrix

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.

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.

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.

The design of the front entrance

This attention to detail in the consistent application of design principleswelcomes visitors as they enter the UN Building. The front entrance of theSecretariat Building reveals golden ratios in it proportions in the following ways:

The columns that surround the center area of the front entrance are placedat the golden ratio point of the distance from the midpoint of the entrance tothe side of the entrance.

The large open framed areas to the left and right of the center entrance areaare golden rectangles.

The doors on the left and right side of the center entrance are goldenrectangles.

The left and center frame sections of the center section is a goldenrectangle.

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.

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.

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…

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.

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.

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

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

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.

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.