The document discusses the use of the golden ratio in architecture and its origins. It provides examples of how the golden ratio was used in ancient Egyptian architecture and the Great Pyramids. It also discusses Fibonacci's discovery of the Fibonacci sequence and how the golden ratio appears in this sequence as the terms grow large. Examples of the golden ratio in nature are also given. Le Corbusier's Modulor system for architectural proportions is described, which was based on the golden ratio. Analysis of the Parthenon and UN Secretariat building show they incorporate golden ratio proportions in their design.

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

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. Overlay 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.

32. 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.

33. 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.

34. 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…

35.  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.

36. 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.

37. 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

38. 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

39. 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.

40. 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.