Fibonacci Series and Golden Ratio in Architecture

FIBONACCI SERIES AND GOLDEN RATIO IN ARCHITECTURE

AN AESTHETICAL APPROACH AND A FUNCTIONAL ANALYSIS

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1. INTRODUCTION

1.1 INTRODUCTION

ibonacci Series is not strictly an image of "beauty" but actually an

image of "HUMANNESS". That is, it is the way we identify our

own species and individuals within our species. Like all other

animals we need a way to identify our own species for mating, bonding,

self- protection and other survival purposes. Also we need to be able to

distinguish healthy and disease free individuals within our species for

similar purposes. Other animals recognize their own species through one

or a combination of their senses.[15]

Moths and butterflies, for example, recognize each other through smell -

the olfactory sense. They are able to recognize or identify other moths by

their scents (or "pheromones") from up to 3 miles away. This is how they

identify "mothness". Dogs recognize each other by a combination of

vision and smell. They initially visually identify another animal as a "dog"

and immediately approach it to smell it and ascertain its degree of

"dogness", as well as other information about that dog. Dolphins

recognize their species and individuals within their species through the

sense of sound. The degree of one dolphin's acceptance of another is its

dependent on its perception of that other dolphin's sounds like a dolphin.

These sounds create its species identity, or "dolphinness". Elephants

appear to use a combination of sound, vision and smell. Few people

have any argument that these animal behaviours are all instinctual

behaviours that are genetically encoded and subconsciously driven. We

are animals too. Humans, however, have historically had a hard time

seeing and regarding themselves as "animals". We are, in fact, much like

other animals. And like other animals, to a tremendous degree, we are a

product of our genetic makeup. There is a school of thought among

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scientists who believe that genetics, much more than environment,

determines who we are and who we become. [15]

Humans are animals, but more specifically we are a visual animal. We

essentially recognize each other by sight. We cannot smell each other

more than a few feet away, and if someone yells our name we

immediately turn to see, or "visually" identify, who they are. Part of our

genetic code is a subconscious image of what "human" is supposed to

be. The primary image of "humanness" is the genetically coded visual

image of an "ideal" human face. The more a face resembles this "Ideal

Human Face Image" - the more we perceive it to be human. When a face

is perceived to be human, this perception sets off in us a conscious

response of "attraction" and "positive emotion". This subconscious visual

perception of "humanness", if strong enough (that is if the face one sees

looks enough like his subconscious image of "humanness"), then the

conscious response will be elevated to a combination of a sense of

"strong attraction" and a sense of "strong positive emotion".[15]

"Beauty" is defined as "the quality or combination of qualities in an entity

which evokes in the perceiver a combination of a sense of "strong

attraction" and a sense of "strong positive emotion". Thus we can

postulate that the perception or "recognition" of beauty is actually nothing

more than a strong correlation of what we subconsciously expect

"humanness" to appear to be. [15]

This leads us to believe that the image of the "ideal" human face is

indeed a subconscious image which we are born with and carry

throughout our lives. This archetype has evolved in order to help us

identify members of our own species and further sort members of our

species according to their relative health and ability to successfully

reproduce and to provide other resources to us and those who are close

to us. [15]



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Beauty, in essence, is an instinctual idea, pattern of thought, image, etc.,

inherited from the ancestors of the race and universally present in

individual psyches. [15] So, if beauty is instinctive and embedded in

psyche, then an object imitating similar proportions as that of the perfect

face would be more appealing than one without it.

1.2 AIM

The aim of the thesis is to study Fibonacci series in architectural context

along with analysing it in aesthetical as well as functional light, and

produce a design process exhibiting Le Modulor as a design tool.

1.3 OBJECTIVES

The following objectives are laid to accomplish the aim:

Study the mathematical aspect of Fibonacci Series

Study the geometrical aspect of Fibonacci Series

Find its contribution to the aesthetics of an object

Study its functionality

1.4 SCOPE

To learn about the Fibonacci Series in architectural context

To study about Le Corbusiers work and research

It would ultimately add to the academic knowledge of proportions

and design principles in architectural context

1.5 LIMITATIONS

While dealing with factors affecting aesthetical appeal, only

proportions will be dealt with and not the material used.



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Aesthetical appeal of objects is intended to be assessed only for

instinctive attraction and not influenced ones.

Not everyone will have the exact same taste for proportions and

so the maximum appreciation is aimed at.

In situations where quantitative analysis is not possible directly,

surveys will be sorted after.

1.6 RELEVANCE OF THE TOPIC

Beautiful objects are appreciated by all alike for its aesthetics,

which implies it is perceivable by all. There must be a reason for

similar perception by one individually. Analysing the principles of

aesthetics would result in guidelines for attaining the desirable

influence on the majority of audience.

At this time when the world is becoming more and more aware to

aesthetics of almost everything, there is a need to lay certain basic

guidelines to achieve the desirable.



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2. FIBONACCI SERIES

2.1 INTRODUCTION

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 ... ...

his series is called as Fibonacci Series and can be expressed as

a function of n as follows:

f (n) = f (n-1) + f (n-2)

The question arises; from where does this series come? Who discovered

or invented it? And what was the need for this discovery? This series was

discovered a number of times over the ages by different scholars and in

different ways. Acharya Hemachandra and Leonardo Fibonacci find

special mention in the history of the Fibonacci series.

2.2 ACHARYA HEMACHANDRA

Acharya Hemachandra

(1089 AD -1172 AD)

Acharya Hemchandra was born in 1089

A .D. into the Modha Vanik (merchant)

caste, in the town of Dhandhuka, sixty

miles from the city Ahmedabad in

Gujarat State. His parents were

Chachadev and Pahini. After his birth he

was name Changdeva. Acharya

Devasuri took him with himself and

initiated Changdeva into monkshood

and named him Somachandra. Acharaya Devasuri made Somachandra

an acharya when he was only twenty-one years old. At that time, he was

T

Figure 1 : Acharya Hemachandra

(Source- GAP System, School of

Mathematics and Statistics,

University of St. Andrews,

Scotland)



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given the name Hemchandra Acharya. He discovered fibonacci series

half a decade prior to fibonacci, while working on poems and syllables.[16]

In a text written about 1150 he looked at the following problem. Suppose

we assume that lines are composed of syllables which are either short or

long. Suppose also that each long syllable takes twice as long to

articulate as a short syllable. A line of length n contains n units where

each short syllable is one unit and each long syllable is two units. Clearly

a line of length n units takes the same time to articulate regardless of

how it is composed. Hemchandra asks: How many different combinations

of short and long syllables are possible in a line of length n? [16]

Hemchandra then finds the answer explicitly. Suppose that there are f (n)

possibilities for a line of length n. The line of length n either ends in a

short syllable or in a long syllable. If it is the former than there remains a

line of length n-1 which can be composed in f (n-1) ways and if the line of

length n ends in a long syllable then there is a line of length n-2

remaining which can be composed in f (n-2) ways. Hence, argues

Hemchandra, [16]

f (n) = f (n-1) + f (n-2).

2.3 LEONARDO PISANO FIBONACCI

eonardo Pisano Fibonacci

(1170 ad -1250 ad)

Leonardo Pisano is better known by

his nickname Fibonacci. He was the son of

Guilielmo and a member of the Bonacci

family. He was a mathematician, who was a

traveller and a scholar in former half of his

L

Figure 2:Leonardo Fibonacci,

(Source- GAP System,

School of Mathematics and

Statistics, University of St.

Andrews, Scotland)



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life and worked for Republic of Pisa in his later half. [16]

Fibonacci formulated a hypothetical problem, in the Process of finding

whose Solution he discovered a series that was later named as

Fibonacci Series. This problem later became famous by the name

Rabbit Problem.

2.4 THE RABBIT PROBLEM

The problem was a hypothetical one. It was initially a mathematics fun

problem for him. It is being extrapolated to predict population growth in

modern times.

There is a pair of baby rabbits right from the beginning. A couple of baby

rabbits take a week to become adults and then they produce another pair

of baby rabbits every week. No rabbit ever dies or loses its fertility. The

number of pairs of rabbit after any number of months is given by the

function and thus it defines the series.

The series can be expressed as a function of n:

f (n) = f (n-1) + f (n-2)

The function depicts that, to get the next number on the series, the last

and the second last numbers should be added. When each number of

the Fibonacci series is divided by its predecessor, starting with the 3rd

number i.e., 1, and this process is continued, the fraction thus achieved

approximates 1.61803... This is a non-terminating decimal number. The

Fibonacci series is also non-terminating and ever extending. The first few

numbers of the series are as follows:

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



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Month

Rabbits From Pair A:

From Pair B:

From Pair C:

D:

B1:

Total

0 A

1

1 A

1

2 A B

2

3 A B C

3

4 A B C D

B1

5

5 A B C D E

B1 B2

C1

8

6 A B C D E F B1 B2 B3 C1 C2 D1 B11 13

etc. 1 2 3 4 5 6 7 8 9 10 11 12 13 etc.

Table 1: The Rabbit Problem (Source-Author)

Figure 3: The Rabbit Problem (Source-http://www.maths.surrey.ac.uk)



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= 1

= 2

= 1.5

= 1.667

= 1.6

= 1.625

= 1.615

= 1.619

[3]

Applications of the Fibonacci Series include computer algorithms such as

the Fibonacci search technique, the Fibonacci heap data structure, and

graphs called Fibonacci cubes used for interconnecting parallel and

distributed systems. They also appear in biological settings, such as

branching in trees, arrangement of leaves on a stem, the fruit spouts of a

pineapple, the flowering of artichoke, an uncurling fern and the

arrangement of a pine cone.



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3. PROPORTION

3.1 INTRODUCTION

roportion is an ordered relationship between two comparable

entities, visible or invisible. It can be manipulated and

experienced by geometry, arithmetic ratio and visual perception.

The fact that, certain proportions have been found so generally

satisfactory, have naturally raised the question of the method by which

they have been arrived at, for instance Golden Ratio. It has been found

in natural elements such as sun flower, nautilus shell, etc. The Parthenon

also possesses it. Corbusiers Modulor is based on golden proportion

found in human body.

Proportions which are generated out of functions without considering the

subjective needs of the designer can be termed as functional

proportion.[7] A good proportion is subjective in the absence of functional

relationship. Aesthetic proportion grows form subjective tradition. They

may manipulate visually.

After all the eve must give the final judgement for even

though the object be most carefully measured, is the

even remain offended, it will not cease on that account

to censure it. The eve must decide where to take away

and where to add, as it sees defect till the due

proportions are attained.

- Georgio Vasari

The validity of proportions is attained and concept loses deeply in all

human beings but subjectively. Sense of proportion is generally affected

by surroundings. The beautiful proportions are sometimes thoroughly

immeasurable and can be achieved by inner instinct and experience.

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Architectural proportions may not be absolute but relative. Functional

proportions are precise and accurate while aesthetic proportions grow

out of subjective tradition.

Magnitudes of having equal ratio are called proportional. Ratio is a

comparison of any two quantities. Proportion is a comparison of two or

more equal ratios. The progression of ordered sequence, in which

quantities may be related to each other, is called Progression in

mathematical terms.

3.2 TYPES OF PROPORTIONS:

Continuous proportion: The comparison of equal ratios, each

has relation with the previous one is called continuous proportion.

Discontinuous proportions: The comparison of independent but

equal ratios is called discontinuous proportion.

Figure 4 : Continuous Proportion (Source-Author)



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Figure 5 : Discontinuous Proportion (Source-Author)

3.3 TYPES OF PROGRESSIONS:

Geometrical progression: The progression in which the ratio of any

two consecutive terms is the same.

Arithmetic progression: The Progression in which the difference of

any two consecutive lines remains same not the ratios or the line

increases with a constant measure.



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Figure 6 : Geometric Progression (Source-Author)

Figure 7 : Arithmetic Progression (Source-Author)



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4. BEAUTY AND AESTHETICS

raditional Indian speculative thought affirm that beauty is

capable of being known to man intrinsically and positively in the

innermost essence of his being.[1]

While human visual perception is relatively constant from one person to

another, the idea of good proportions and beauty necessarily varies,

based on experience and knowledge. There are some constants

however, such as a sense of visual balance derived from body-related

proportions, but even these vary according to cultural and other

experiential circumstances.

In the earliest cultures known, before written history, like in China, Egypt,

Islamic world and sub-Saharan Africa, beauty was a term of great

esteem linking human beings and nature with artistic practices and

works. Men and women, their bodies, characters, behaviours and virtues

are described as beautiful, together with artefacts, performances and

skills and with natural creatures and things: animals, trees and rock

formations. [1]

The early Christian (400-1400 AD) philosophy strongly emphasized,

proportion, harmony, congruence and consonance; especially in relation

to music, which was understood in Pythagorean terms to be regulated by

numbers and in terms of unity in multiplicity; light, colour, radiance,

brilliance, and clarity all were beautiful, testaments to the unity of Gods. [1]

In the Renaissance period (1400 AD) beauty was perceived in terms of

order, measure, and form; the beauty of the universe in terms of order

and perfection. The science approach of seeing things as Greeks did

found; was found again in this period and hence was called rebirth.[1]

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Beauty (also called prettiness, loveliness or comeliness) is a

characteristic of a person, animal, place, object, or idea that provides a

perceptual experience of pleasure, meaning, or satisfaction. Beauty is

studied as part of aesthetics, sociology, social psychology, and culture.

An "ideal beauty" is an entity which is admired, or possesses features

widely attributed to beauty in a particular culture, for perfection. Beauty

begins as an organic entity which can be thus altered by new means.

The qualities that give pleasure to the senses are said to be beautiful,

whereas the branch of philosophy dealing with beauty and taste is

aesthetics. Aesthetics can more elaborately be stated as a branch of

philosophy dealing with the nature of beauty, art, and taste, and with the

creation and appreciation of beauty.[17] It is more scientifically defined as

the study of sensory or sensory-emotional values, sometimes called

judgments of sentiment and taste.[18] More broadly, scholars in the field

define aesthetics as "critical reflection on art, culture and nature."[2]



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5. GOLDEN RATIO

ome of the greatest mathematical minds of all ages, from

Pythagoras and Euclid in Ancient Greece, through the medieval

Italian mathematician Leonardo of Pisa and the Renaissance

astronomer Johannes Kepler, to present-day scientific figures such as

Oxford physicist Roger Penrose, have spent endless hours over this

simple ratio and its Properties. But the fascination with the Golden Ratio

is not confined just to Mathematicians; Biologists, artists, Musicians,

historians, architects, Psychologists, and even mystics have pondered

and debated the basis of its ubiquity and appeal.[3]

The Golden Section can be defined as the ratio between two sections of

a line, or two dimensions of a plane figure, in which the lesser of the two

is to the greater as the greater is to the sum of both. It can be expressed

algebraically by the equation of two

ratios. [8]

Ancient Greek mathematicians first

studied the golden ratio because of its

frequent Appearance in geometry. The

division of a line into the golden ratio is

important in the Geometry of regular

pentagrams and Pentagons. The Greeks

usually attributed Discovery of this

concept to Pythagoras. The regular

pentagram, which has a regular

Pentagon inscribed within it, was the

Pythagoreans' symbol. But the regular pentagram, from being

Pythagoreans symbol has now become a very regular and day to day

symbolised by everyone ranging from primary teachers to scholars. [3]

S

a

b

a + b

Figure 8 : Golden Ratio (Source-Author)



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When a line is divided into two segments a and b, where a is longer and

b is shorter, in such way that [3]

Equation 1 : Golden Ratio

Equation 2 : Solution of quadratic equation



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Then, the two segments are called to be in golden ratio/proportion to

each other. It is represented by Greek letter phi or . The value of phi

is non-terminating. [3]

Hence is equated with the already formed equation from the

preconditions. As a result the linear equation in a & b is replaced by a quadratic equation in .

This value thus achieved approximates the value found by dividing each

number of the Fibonacci series by its predecessor. [3]



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6. GOLDEN SECTION/RECTANGLE

athematical systems of proportion originate from the

Pythagorean concept of all is number and the belief that

certain numerical relationships manifest the harmonic

structure of the universe. One of these relationships that has been in use

ever since the days of antiquity is the proportion known as the Golden

Section. The Greeks recognized the dominating role the Golden Section

played in the proportions of the human body. Believing that both

humanity and the shrines housing their deities should belong to a higher

universal order, they utilized these same proportions in their temple

structures. Renaissance architects also explore the Golden Section in

their work. In more recent times, Le Corbusier based his Modulor system

on the Golden System. Its use in architecture endures even today. [8]

The Golden Section has some remarkable algebraic and geometric

properties that account for its existence in architecture as well as in the

structures of many living organisms. Any progression based on the

Golden Section is at once additive and geometrical. [8]

A Golden Rectangle is a rectangle with proportions that are two

consecutive numbers from the Fibonacci sequence. The Golden

Rectangle has been said to be one of the most visually satisfying of all

geometric forms. We can find many examples in art masterpieces such

as in edifices of ancient Greece.[19]

Alternatively, a Golden Rectangle is one whose side lengths are in the

golden ratio, or approximately 1:1.618. A distinctive feature of

this shape is that when a square section is removed, the remainder is

another golden rectangle; that is, with the same proportions as the first.

Square removal can be repeated infinitely, in which case corresponding

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corners of the squares form an infinite sequence of points on the golden

spiral, the unique logarithmic spiral with this property. [19]

Steps of construction :

a) Construct a square of 1 unit by 1 unit sides.

b) Draw a line from the midpoint of one side of the square to an opposite

corner.

c) Use that line as the radius to draw an arc that defines the height of the

rectangle.

d) Complete the golden rectangle.



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Figure 9 : Steps 1 and 2 of construction of Golden Section (Source-Author)



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Figure 10 : Steps 3 and 4 of construction of Golden Section (Source-Author)



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7. GOLDEN SPIRAL

n geometry, a golden spiral is a logarithmic spiral whose growth

factor is , the golden ratio. That is, a golden spiral gets wider (or

further from its origin) by a factor of for every quarter turn it makes.

This is quite similar to a spiral constructed out of golden rectangles and

another spiral constructed out of Fibonacci rectangles in the following

manners

The green spiral is made from quarter-circles tangent to the interior of each square, while

the red spiral is a Golden Spiral. Overlapping portions appear yellow.

Figure 11 : Golden Spiral (Source-www.wikipedia.org)

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8. GOLDEN TRIANGLE

Golden triangle, also known as the sublime triangle, is an

isosceles triangle in which the smaller side is in golden ratio with

its adjacent side:

Golden triangles are found in the nets of several stellations of

dodecahedrons and icosahedrons. Also, it is the shape of the triangles

found in the points of pentagrams. The vertex angle is equal to

Equation 3 : Angle for Golden Triangle

Since the angles of a triangle sum to 180, base angles are therefore 72

each. The golden triangle can also be found in a decagon, or a ten-sided

polygon, by connecting any two adjacent vertices to the center. This will

form a golden triangle. This is because: 180(10-2)/2=144 degrees is the

interior angle and bisecting it through the vertex to the center, 144/2=72.

The golden triangle is also uniquely identified as the only triangle to have

its three angles in 2:2:1 proportion. The golden triangle is used to form a

logarithmic spiral. By bisecting the base angles, a new point is created

that in turn, makes another golden triangle. The bisection process can be

continued infinitely, creating an infinite number of golden triangles. A

logarithmic spiral can be drawn through the vertices. This spiral is also

known as an equiangular spiral, a term coined by Rene Descartes. "If a

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straight line is drawn from the pole to any point on the curve, it cuts the

curve at precisely the same angle," hence equiangular.

Closely related to the golden triangle is the golden gnomon, which is the

obtuse isosceles triangle in which the ratio of the length of the equal

(shorter) sides to the length of the third side is the reciprocal of the

golden ratio. The golden gnomon is also uniquely identified as a triangle

having its three angles in 1:1:3 proportions. The acute angle is 36

degrees, which is the same as the apex of the golden triangle.

Figure 12 : Golden triangles inscribed in a logarithmic spiral



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9. GOLDEN RATIO AND PENTAGRAM

pentagram (sometimes known as a pentalpha or pentangle or,

more formally, as a star pentagon) is the shape of a five-pointed

star drawn with five straight strokes. The word pentagram

comes from the Greek word pentagrammon, a noun form of

pentagrammos or pentegrammos, a word meaning roughly "five-lined" or

"five lines", from pente, "five" + gramm, "line".

Pentagram is closely associated with the golden ratio. It contains the

golden ratio in its lines and intersections. Pentagrams were used

symbolically in ancient Greece and Babylonia, and are used today as a

symbol of faith by many Wiccans, akin to the use of the cross by

Christians and the Star of David by Jews. The pentagram has magical

associations, and many people who practice Neo-pagan faiths wear

jewellery incorporating the symbol. Christians once more commonly used

the pentagram to represent the five wounds of Jesus. The pentagram

has associations with Freemasonry and is also utilized by other belief

systems.

The word "pentacle" is sometimes used synonymously with "pentagram",

and this usage is borne out by the Oxford English Dictionary, although

that work specifies that a circumscription makes the shape more

particularly a pentacle. Wiccans and Neo-pagans often make use of this

more specific definition for a pentagram enclosed in a circle.

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9.1 SUMER

The first known uses of the pentagram are found in Mesopotamian

writings dating to about 3000 BC. The Sumerian pentagrams served as

pictograms for the word "UB" meaning "corner, angle, nook; a small

room, cavity, hole; pitfall," suggesting something very similar to the

pentemychos (see below on the Pythagorean use for what pentemychos

means). In Ren Labat's index system of Sumerian

hieroglyphs/pictograms it is shown with two points up. In the Babylonian

context, the edges of the pentagram were probably orientations: forward,

backward, left, right, and "above". These directions also had an

astrological meaning, representing the five planets Jupiter, Mercury,

Mars and Saturn, and Venus as the "Queen of Heaven" (Ishtar) above.

9.2 PYTHAGOREANS

The Pythagoreans called the pentagram Hugieia ("health"; also the

Greek goddess of health, Hygieia), and saw in the pentagram a

mathematical perfection.

The ancient Pythagorean pentagram, with two legs up, represented the

Pentemychos (of five sanctuaries), a cosmogony written by Pythagoras'

teacher and friend Pherecydes of Syros. It was the "island" or "cave"

where the first pre-cosmic-offspring had to be put in order for the cosmos

to appear: "the divine products of Chronos" seed, when disposed in five

recesses, were called Pentemuxos".



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9.3 WU XING

Wu Xing, the five phases, or five elements, is an ancient Chinese

mnemonic and symbolic figure widely known in East Asia and used

traditionally in applications such as medicine, acupuncture, feng shui,

and Taoism. They are similar to the ancient Greek elements, with more

emphasis on their cyclic transformation than on their material aspects.

The five phases are: Fire, Earth, Metal, Water, and Wood.

9.4 GEOMETRY

The pentagram is the simplest regular star polygon. The pentagram

contains ten points (the five points of the star, and the five vertices of the

inner pentagon) and fifteen line segments. Like a regular pentagon, and

a regular pentagon with a pentagram constructed inside it, the regular

pentagram has as its symmetry group the dihedral group of order 10.

The golden ratio plays an important role in regular pentagons and

pentagrams. Each intersection of edges sections the edges in golden

ratio: the ratio of the length of the edge to the longer segment is , as is

the length of the longer segment to the shorter. Also, the ratio of the

length of the shorter segment to the segment bounded by the 2

intersecting edges (a side of the pentagon in the pentagram's centre) is

.

The pentagram includes ten isosceles triangles: five acute and five

obtuse isosceles triangles. In all of them, the ratio of the longer side to

the shorter side is . The acute triangles are golden triangles. The

obtuse isosceles triangle highlighted via the coloured lines is a golden

gnomon.



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Figure 13 : Pentagram (Source-www.wikipedia.org)



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10. GOLDEN RATIO IN GEOMETRY

aving spent 99.9% of our planetary tenure woven deep into the

wild, we humans naturally admire the weaverbirds nest, the

conchs shell, and the scales of a shimmering trout. In fact,

there are few things more beautiful to the human soul than natures

design.

Figure 14 : Division of a straight line into Golden Ratio (Source-Thesis: An Objective Search And A Subject Analysis, Drawn-Author)

[6]

H



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Figure 15 : Division of a square into Golden Section (Source-Thesis: An Objective Search And A Subject Analysis, Drawn-Author)

[6]



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Figure 16 : The lines resulting in golden ratio (Contribution- Jo Niemeyer, drawn-Author)

When the lines are laid in such a way that each has a length of one unit

and they are laid on the midpoint of the previous line, as expressed in the

figure, the project Golden Ratio on the horizontal axis.



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Figure 17 : An equilateral triangle inscribed in a circle (Drawn-Author)

An equilateral triangle inscribed in a circle expresses Golden Ratio when

the midpoints of two lines are joined and extended to meet the circle at

some point.



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Figure 18 : A square inscribed in a semi-circle (Drawn-Author)

A square inscribed in a semi-circle, results in dividing the diameter in a

way that Golden Ratio can be traced.



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Figure 19 : A pentagon inscribed in a circle (Drawn-Author)

The diagonals of the pentagon inscribed in a circle intersect each other in

such a way so as to produce golden ratio in them. The sides of the

regular pentagon are also in golden proportion with the diagonals.



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Figure 20 : Harmonic subdivision of line in Golden Ratio (Source-Thesis: An Objective Search And A Subject Analysis, Drawn-Author)

[6]

Figure 21 : Harmonic subdivision of line in horizontal and vertical direction in Golden

Ratio (Source-Thesis: An Objective Search And A Subject Analysis, Drawn-Author) [6]



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Certain beautiful geometry can be achieved by arranging circles and

squares and scaling them repeatedly by phi = 1.618...

Figure 22 : Ten squares seen as the projection of a three-dimensional pyramid-like

structure (Source- Janusz Kapusta, Redrawn-Author)[4]



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Figure 23 : Two lines touching the corners of the ten squares (Source- Janusz Kapusta, Redrawn-Author)

[4]



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Figure 24 : The appearance of a new virtual square with an inscribed upward pointed

triangle (Source- Janusz Kapusta, Redrawn-Author) [4]



40 | P a g e

Figure 25 : Circles are inscribed within the squares (Source- Janusz Kapusta, Redrawn-Author)

[4]



41 | P a g e

Figure 26 : Two lines tangent to the circles define a pair of circles within the virtual

square with diameters in the golden proportion (Source- Janusz Kapusta,

Redrawn-Author) [4]



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Figure 27 : The virtual square is seen in exploded view (Source- Janusz Kapusta, Redrawn-Author)

[4]



43 | P a g e

Figure 28 : A sequence of tangential circles is created with the inverse powers of the

golden mean as their diameters (Source- Janusz Kapusta, Redrawn-Author)

[4]



44 | P a g e

Figure 29 : Odd, inverse powers of the golden mean sum to unity (Source- Janusz

Kapusta, Redrawn-Author) [4]



45 | P a g e

Figure 30 : All the inverse powers of the golden mean with the exception of 1/ sum to

unity (Source- Janusz Kapusta, Redrawn-Author) [4]



46 | P a g e

Figure 31 : Another surprising relation of odd, inverse powers. Notice that the squares

that circumscribe the sequence of the golden circles touch the side of the

upward pointed triangle (Source- Janusz Kapusta, Redrawn-Author) [4]



47 | P a g e

Figure 32 : An infinite sequence of a half golden circles tangent to their diameters and to

the side of a upward pointed triangle (Source- Janusz Kapusta, Redrawn-

Author) [4]



48 | P a g e

Figure 33 : Another way to view the odd, inverse powers of the golden mean as a

sequence of circles (Source- Janusz Kapusta, Redrawn-Author) [4]



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Figure 34 : They can also be seen as a sequence of squares (Source- Janusz Kapusta,

Redrawn-Author) [4]



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Figure 35 : An infinite sequence is seen to be a geometric sequence of squares of

decreasing size (Source- Janusz Kapusta, Redrawn-Author) [4]



51 | P a g e

Figure 36 : The Pythagorean Theorem is expressed by this sequence of squares. The

sequence of vertices of the squares upon the hypotenuse lies against the right edge of

the framing square (Source- Janusz Kapusta, Redrawn-Author) [4]



52 | P a g e

Figure 37 : The Golden rectangles (Source- http://www.maths.surrey.ac.uk)

Three Golden rectangles are placed perpendicular to each other passing

through each others centre. The 12 vertices thus gained, when treated

as the mid points of 12 pentagons, a dodecahedron is formed; and when

those 12 vertices are treated as a solid geometry made of triangles with

their vertices on them, icosahedrons is constructed.



53 | P a g e

Figure 38 : Icosahedrons circumscribing the three golden rectangles

Figure 39 : Dodecahedron circumscribing the three golden rectangles (Source- http://www.maths.surrey.ac.uk)



54 | P a g e

11. GOLDEN RATIO IN ART

he 16th-century philosopher Heinrich Agrippa drew a man over a

pentagram inside a circle, implying a relationship to the golden

ratio.

Leonardo da Vinci illustrated polyhedral in De divina proportione (On the

Divine Proportion) and expressed his views that some bodily proportions

exhibit the golden ratio. Salvador Dal, influenced by the works of Matila

Ghyka, explicitly used the golden ratio in his masterpiece, The

Sacrament of the Last Supper. The dimensions of the canvas are a

golden rectangle. A huge dodecahedron, in perspective so that edges

appear in golden ratio to one another, is suspended above and behind

Jesus and dominates the composition.

Figure 40 : The sacrament of the Last Supper by Salvador Dali (1904-1989) (Source-www.wikipedia.org, editing-author)

T



55 | P a g e

Figure 41 : Mona Lisa by Leonardo Da Vinci exhibiting relationship with the Golden rectangles arranged in the pattern as in the construction of Golden Spiral (Source of image-www.wikipedia.org, editing-author)



56 | P a g e

Figure 23 includes lots of Golden Rectangles. A rectangle whose base

extends from the woman's right wrist to her left elbow can be drawn and

extended vertically until it reaches the very top of her head. Now the

rectangle thus drawn is a golden one.

Also, if squares are drawn inside this Golden Rectangle, the edges of

these new squares come to all the important focal points of the woman:

her chin, her eye, her nose, and the upturned corner of her mysterious

mouth.

It is believed widely that Leonardo, as a mathematician tried to

incorporate mathematics into art. This painting seems to be made

purposefully line up with golden rectangle.

Mondrian has been said to have used the golden section extensively in

his geometrical paintings. [3]

Figure 42 : Self-portrait by Rembrandt (Source-jwilson.coe.uga.edu)



57 | P a g e

Three straight lines can be drawn into this figure. Then, the image of the artist

can be included into a triangle. Moreover, if a perpendicular line would be

dropped from the apex of the triangle to the base, the triangle would cut the

base in Golden Section.

Figure 43 : Afrodita's sculpture created by Agesander is considered to be the masterpiece

of woman's beauty (Source-http://milan.milanovic.org)[20]

The sculpture of the Greek Goddess of fertility Afrodita created by

Agesander illustrates golden proportion in the womans body.

There was a time when deviations from the truly beautiful page

proportions 2:3, 1:3, and the Golden Section were rare. Many books

produced between 1550 and 1770 show these proportions exactly, to

within half a millimetre.

The figure illustrates The Vitruvuan Man, a man inscribed in a square

and a circle in two different postures. Three different sets of Golden

0.382

0.618



58 | P a g e

Rectangles can be traced in the figure; each one for the head area, the

torso, and the legs.

Figure 44 : The Vetruvian Man (The Man in Action) by Leonardo Da Vinci (Source-www.wikipedia.org)

Figure 45 : Crucifixion by Raphael (Source-http://milan.milanovic.org, editing-author) [3]



59 | P a g e

Crucifixion by Raphael is a well-known example, in which we can find a

Golden Triangle and also Pentagram. A golden triangle can be traced

when a line joining the centre of the crossing of the cross and the lowest

points of the disciples position is drawn. And a pentagram can be traced

by joining the centre of the cross with the shoulders of the two kneeling

disciples and the legs of the angles.



60 | P a g e

12. GOLDEN RATIO IN MONUMENTS AND BUILDINGS

12.1 THE PYRAMID OF GIZA

he papyrus of Egypt gives an account of the building of the Great

Pyramid of Giaz in 4700 B.C. with proportions according to a

"sacred ratio."

The Greek sculptor Phidias sculpted many things including the bands of

sculpture that run above the columns of the Parthenon. Even from the

time of the Greeks, a rectangle whose sides are in the "golden

proportion" has been known since it occurs naturally in some of the

proportions of the Five Platonic. This rectangle is supposed to appear in

many of the proportions of that famous ancient Greek temple in the

Acropolis in Athens, Greece. It's also believed that the numeric value

assigned to the Golden Ratio, Phi, was named in Phidias honour.

Figure 29 : The Pyramid of Giza (Source- www.jwilson.coe.uga.edu) [3]

Figure 460 : The angle subtended at the centre of the base by the slope of the pyramis (Source- www.jwilson.coe.uga.edu)

[3]

T



61 | P a g e

12.2 NOTRE DAME

Figure 47 : Notre Dame (Source-http://www.goldennumber.net)

Notre Dame in Paris, which was built in the 1163 and

1250 exhibits the use of Golden Section through its

front facade.[22]

12.3 THE CNN TOWER

The CN Tower in Toronto, the tallest tower and

freestanding structure in the world, expresses the

golden ratio in its design. The ratio of observation deck

at 342 meters to the total height of 553.33 is 0.618 [22]

Figure 48 : The CN

Tower, Toronto



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12.4 TAJ MAHAL

The Taj Mahal at Agra, India also has pentagrams and golden sections in

its elevation (perspective view from a distance).

Figure 49 : The Front Facade of Taj Mahal showing two Golden Rectangles : the red

and the cyan (Source- www.wikipedia.org, editing-author) [22]

The different bands formed by continuous decorative panels show the

golden relation when compared with one another in width. There are four

Golden Section in the figure, namely, red, yellow, blue and cyan. [22]



63 | P a g e

Figure 50 : A zoomed in view of the entrance showing different golden rectangles

(Source- Author) [22]



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12.5 VILLA SAVOYE

Villa Savoyes Plan expresses the deliberate use of Golden Section by

Le Corbusier. The central square forms two golden sections alternatively

by combining with the two rectangles on the sides.

Figure 51 : Le Corbusiers Villa Savoye, France.[23]

The graphic analysis illustrates the use of Golden Section in the

proportioning of the facade of the Parthenon. It is interesting to note that

while both analyses begin by fitting the facade into a Golden Rectangle,

each analysis then varies from the other in its approach to providing the

existence of the Golden Section and its effect on the dimensions and the

distribution of elements across the facade.



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12.6 VILLA GARCHES

Figure 52 : Villa Garches (Source-www.wikipedia.org)

Figure 53 : Front facade-Villa Garches, Vaucressen, France, by Le Corbusier (1926-27) showing golden relationships (Drawn-Author)



66 | P a g e

12.7 PARTHENON

Figure 54 : The Front facade of the Parthenon (Source- www.jwilson.coe.uga.edu) [3]

The Parthenon exhibits Golden Section and the series of Golden

Rectangles used to construct the Golden Spiral at various places;

ranging from the front facade to the plan.

Figure 55 : Plan of the Parthenon (Source- www.jwilson.coe.uga.edu) [3]



67 | P a g e

Figure 56 : Parthenon-Front facade, tracing Golden rectangles (Source-Author)

Figure 57 : Parthenon-Front facade, Golden rectangles (Source-Author)



68 | P a g e





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13. GOLDEN RATIO AND FIBONACCI SERIES IN

NATURE

n botany, phyllotaxis or phyllotaxy is the arrangement of the leaves

on the stem of a plant. The basic patterns are alternate, opposite,

whorled or spiral. With an alternate pattern, leaves switch from side

to side. An alternate distichously phlyllotaxis means that each leaf

growing at a single node is disposed in a single rank along with their

branch (such as in grasses). In an opposite pattern, if successive leaf

pairs are perpendicular, this is called decussate. A whorled pattern

consists of three or more leaves at each node. An opposite leaf pair can

be thought of as a whorl of two leaves. A whorl can occur as a basal

structure where all the leaves are attached at the base of the shoot and

the internodes are small or non-existent. A basal whorl with a large

number of leaves spreads out in a circle is called a rosette. A mitigate

pattern is a spiral composed of whorls. The pattern has also been

observe to emerge in at least one animal cell (the red blood cell). During

process that perturb cellular fluid dynamics. [5]

The leaves on a stem are positioned

over the gaps between the lower leaves

as they spiral up the stem. What is most

remarkable about this spiral spacing is

that irrespective of species, the rotation

angle tends to have only a few values.

By far the most common of which is

137.5o. This is considered an efficient

arrangement to allow maximum sunlight

to reach each set of leaves. This angle is none other than the golden

proportion relate to the perimeter of a circle.

I

Figure 60 : 137.5o

as a part of a circle (Source- Communication of ACM, July 2003, Vol 43.)



70 | P a g e

A repenting spiral can be represented by a fraction describing the angle

of windings leaf per leaf. Alternate leaves will have an angle of of a full

rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5,

in polar and pear it is 3/8, in willow and almond the angle is 5/13,. The

numerator and denominator normally consist of Fibonacci number and ts

second successor. The number of leaves is

sometimes called rank, in the case of simple

Fibonacci ratios, because the leaves line up in

vertical rows. With larger Fibonacci pairs, the

ratio approaches phi and the pattern becomes

complex and non-repeating. This tends to occur

with a basal configuration. Examples can be

found in composite flowers and seed heads.

The most famous example is the sun flower

head. This phyllotactic pattern creates an optical

illusion of criss-crossing spirals. In the botanical

literature, these designs are described by the

number of counter-clockwise spirals and the

number of clock wise spirals. These turn out to

be Fibonacci numbers. In some cases, the

numbers appears to be multiples of Fibonacci

numbers because the spiral consists of the

whorls. [5]

Leonardo Da Vinci was the first to suggest that

the adaptive advantage of the Fibonacci pattern is to maximize exposure

to dew. Current thinking supports this interpretation. Phyllotactic

architecture optimizes access to the moisture, rainfall and sunlight. [5]

In further study it is seen that this spiral pattern in nature occurs from the

very early stage, at cell stage where the cells are blocked by auxin in a

particular manner and so the leaf started to grow in this pattern. It is also

Figure 61 : Leaves on a stem demonstrating the Fibonacci Series as they spiral up the stem (Source- Communication of ACM, July 2003, Vol 43.)



71 | P a g e

seen that the angle between the leaves is almost near to the golden ratio.

[5]

Figure 62 : Pine cone showing the Golden Spiral (Source-http://oregonexpat.wordpress.com)

The human body is a perfect exhibit for the relation of golden ratio with

fibonacci series and the manifestation of both of them together in nature.

In the figure above the golden ratio exists in the ratio between each

letters upper and lower case. Viz. A : a, b : b, c : c, etc.



72 | P a g e

Figure 63 : The Fibonacci Series in human anatomy



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Figure 64 : A tendril of a plant spiralling along the Golden Spiral (Source-www.flicker.com)

Figure 65 : The Broccoli exhibiting Golden Spiral (Source-www.wikipedia.org)



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14. PROPORTIONS DERIVED FROM HUMAN

ANATOMY

ccording to Vitruvius the navel is the natural centre of the body.

If a man is placed flat on his back with his hand and feet

extended, is a pair of compasses centred at the navel, the

figures and toes of his two hands and feet will touch the circumference of

a circle[9] A mans arm stretched out has been found to be the same

length as his height. The scrutinized mans figure by Vitruvius, explain

the relationship of the parts to parts and part to the whole. The face from

the chin to the top of the forehead and the lowest root of the hair is n

tenth part of the total height. The open head from the chin to the crown is

an eight. The neck and the shoulder, from the top of the breast to the

lowest root of the hair is a sixth.[12]

On the face, the bottom of the chin to the underside of the nostril is 1/3rd

of it. The nose from the underside of the nostril to the line between the

eyebrows is the same, from there to the lowest root of the hair is also one

third. [10]

Leonardo illustrates a mans body fixed in the centre of a superimposed

circle and square easily and arms raised in V, to touch the circle and

with feet together, arms stretched wide to touch the square.[11]

Various systems of proportions have been established from Vitruvius to

le Corbusier. People have used them in a different ways. The use of

system is to guide a designer.[12]

A



75 | P a g e

Figure 66 : The image establishing that the human arm is in Fibonacci Series

Figure 67 : Relevance of Fibonacci Series and Golden Ratio in human body



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15. THE MODULOR BY LE CORBUSIER

It is a scale of proportions which makes the bad

difficult and the good easy.

- Albert Einstein

e Corbusier invented the word "modulator" by combining "modul"

(ratio) and "or" (gold); another expression for the well-known

golden ratio. He developed his proportioning system, the Modulor,

to order the dimensions of that which contains and that which is

contained. He saw the measuring tools of the Greeks, Egyptians, and

other high civilizations as being infinitely rich and subtle because they

formed part of the mathematics of the human body, gracious, elegant,

and firm, the source of that harmony which moves us, beauty. He

therefore based his measuring tool, the Modulor, on both mathematics

(the aesthetics dimensions of the Golden Section and the Fibonacci

Series), and the proportions of the human body (functional

dimensions).[8]

You know, it is life that is right and the architect who is wrong.

- Le Corbusier

Le Corbusier began his study in 19942, and published The Modulor: A

Harmonious Measure to the Human Scale Universally Applicable to

Architecture and Mathematics in 1948. A second volume, Modulor II, was

published in 1954. [8]

L



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Le Corbusier saw the Modulor not merely as a series of numbers with an

inherent harmony, but as a system of measurements that could govern

lengths, surfaces, and volumes, and maintain the human scale

everywhere. It could lend itself to an infinity of combinations; it ensures

unity with diversity...the miracles of numbers. [8]

Figure 68 : The Modulor

270 + 430 = 700

430 + 700 = 1130

700 + 1130 = 1830

860 + 1400 = 2260

Thus,

270

430

700

1130

1830

And

860

1400

2260



78 | P a g e

are two Fibonacci series from the decisive points of a six feet tall mans

occupation of space. Extrapolating both of them on either sides we get

two ever extending series in which the ratio between any pair of

consecutive numbers is phi.

The first series has the 1130 mm, it is the height of the solar plexus of the

six feet man. This series was termed as red series by Le Corbusier. The

other series having 2260 mm is the height of the upraised arm of the man

and also the distance between his index fingers when standing with arms

stretched. This series was termed as blue series.

15.1 DEFINING MODULOR

Modulor consisted of two limitless series namely, the red series and the

blue series. The red series is generated by multiplying the measure 113,

the solar plexus height from ground of a man six feet tall, by golden

number, 1.618 and dividing it by the same. Similarly, the double unit,

226, the height of the arm upraised of a six feet man is divided

consecutively by 1,618 and multiplied by the same to get blue series. The

following will make the process more lucid.

Hence when 113 is multiplied by 1.618 repeatedly, we get at each step

182.9, 295.9, 478.8 and so forth and when 113 is divided by 1.318

repeatedly, we get 69.8, 43.2, 26.7 and so forth. Thus this forms the non-

terminating Red Series. Similarly we get 53.4, 86.3, 139.7, 226, 365.8,

591.8, . as the Blue Series when the same process is repeated with

226. The following table gives a part of the Red and the Blue series in

metric system. The same can be obtained in imperial system by taking

44.5 and 89 as a substitute for 113cm and 226cm and following the

same process of dividing and multiplying repeatedly.



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295.9

182.9

MULTIPLIED BY 1.618

591.8

365.8

226 MULTIPLIED BY 1.618

113 DIVIDED BY 1.618

69.8

43.2

DIVIDED BY 1.618

86.3

53.4

Figure 69 : Producing Modulor Series



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RED SERIES BLUE SERIES

cm m cm m

95,280.0

58,886.7

36,394.0

22,492.7

13901.3

8,591.4

5,309.8

3,281.6

2,028.2

1,253.5

774.7

478.8

295.9

182.9

113.0

952.80

588.86

363.94

224.92

139.01

85.91

53.10

32.81

20.28

12.53

7.74

4.79

2.96

1.83

1.13

117,773.5

72,788.0

44,985.5

27,802.5

17,182.9

10,619.6

6,563.3

4,056.3

2,506.9

1,549.4

957.6

591.8

365.8

226.0

1177.73

727.88

449.85

278.02

171.83

106.19

65.63

40.56

25.07

15.49

9.57

5.92

3.66

2.26

69.8

43.2

26.7

16.5

10.2

6.3

3.9

2.4

1.5

0.9

0.6

0.70

0.43

0.26

0.16

0.10

0.06

0.04

0.02

0.01

139.7

86.3

53.4

33.0

20.4

12.6

7.8

4.8

3.0

1.8

1.1

1.40

0.86

0.53

0.33

0.20

0.12

0.08

0.04

0.03

0.03

0.01



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15.2 UNITE DHABITATION, AT MARSEILLES

The principle work of Le Corbusier that exemplified the use of the

Modulor to bring human scale to a building that is 140 meters long, 24

meters wide, and 70 meters high.

The Unit d'Habitation (French, means Housing Unit) is the name of a

modernist residential housing design principle developed by Le

Corbusier, with the collaboration of painter-architect Nadir Afonso. The

concept formed the basis of several housing developments designed by

him throughout Europe with this name.

The first and most famous of these buildings, also known as Cit

Radieuse (radiant city) and, informally, as La Maison du Fada (French -

Provenal, "The House of the Mad"), is located in Marseille, France, built

1947-1952. One of Le Corbusiers's most famous works, it proved

enormously influential and is often cited as the initial inspiration of the

Brutalist architectural style and philosophy.

The Marseille building, developed with Corbusier's designers Shadrach

Woods and George Candilis, comprises 337 apartments arranged over

twelve stories, all suspended on large piloti. The building also

incorporates shops with architectural bookshop, sporting, medical and

educational facilities, a hotel which is open to the public, and a

gastronomic restaurant, Le Ventre de l'Architecte ("The Architect's

Belly"). [24]The flat roof is designed as a communal terrace with sculptural

ventilation stacks, a running track, and a shallow paddling pool for

children. The roof, where a number of theatrical performances have

taken place, underwent renovation in 2010. It has unobstructed views of

the Mediterranean and Marseille.



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Inside, corridors run through the centre of the long axis of every third

floor of the building, with each apartment lying on two levels, and

stretching from one side of the building to the other, with a balcony.

Unlike many of the inferior system-built blocks it inspired, which lack the

original's generous proportions, communal facilities and parkland setting,

the Unit is popular with its residents and is now mainly occupied by

upper middle-class professionals.

The building is constructed in bton brut (rough-cast concrete), as the

hoped-for steel frame proved too expensive in light of post-War

shortages. The Unit in Marseille is pending designation as a World

Heritage site by UNESCO. It is designated a historic monument by the

French Ministry of Culture.

Figure 70 : The front facade of Unite DHabitation



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16. REGULATING LINES

16.1 INTRODUCTION

To leave a mass intact in the splendour of its form in

light, but, on the other hand, to appropriate its surface

for needs which are often utilitarian, is to force oneself

to discover in this unavoidable dividing up of the surface

the accusing and generating lines of the form.

- Le Corbusier

(Towards a New Architecture)

The idea of regulating lines was not original to Le Corbusier. The principle

of an ordering geometry had been in use since before antiquity.

Renaissance architects onwards used ratios as a means of constructing

an ordered geometry in a building, most visibly in elevation, but also in

plan and section.

Le Corbusier argues from historical evidence that great architecture of

the past has been guided by the use of what came to be known in

English as "regulating lines." These lines, starting at significant areas of

the main volumes, could be used to rationalize the placement of features

in buildings. Le Corbusier lists off several structures he claims used this,

including a speculative ancient temple form, Notre-Dame de Paris, the

Capitol in Rome, the Petit Trianon, and lastly, his prewar neoclassical

work in Paris and some more contemporary modern buildings. In each

case, he attempts to show how the lines augment the fine proportions

and add a rational sense of coherence to the buildings. In this way, the

order, the function, and the volume of the space are drawn into one

architectural moment. Le Corbusier argues that this method aids in



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formalizing the intuitive sense of aesthetics and integrating human

proportions as well.

Le Corbusier claims in the text that no architects trained in the Beaux-

arts technique use regulating lines, because of contradictory training, but

most of the Grand Prix architects did use them, even if they were

supplementing the basic techniques.

If the diagonals of two rectangles are either parallel or perpendicular to

each other, they indicate that two rectangles have similar proportions.

These diagonals, as well as lines that indicate the common alignment of

elements, are called regulating lines. They can also be used to control

the proportion and placement of elements in other proportioning systems

as well.

A regulating line is an assurance against

capriciousness; it is a means of verification which can

ratify all work created in fervour... It confers on the work

the quality of rhythm. The regulating line brings in this

tangible form of mathematics which gives the reassuring

perception of order. The choice of regulating line fixes

the fundamental geometry of the work... It is a means to

an end; it is not a recipe.

- Le Corbusier

(Towards a New Architecture)



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16.2 VILLA GARCHES, BY LE CORBUSIER

The fundamental of Regulating lines have been used ectensively in the

front and rare facades of the Villa Garches. The figure demonstrates the

various regulating lines being used in the two facades.

Figure 71 : Front facade-Villa Garches, Vaucressen, France, by Le Corbusier (1926-27) showing regulating lines (Drawn-Author)



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Figure 72 : Rear facade-Villa Garches, Vaucressen, France, by Le Corbusier

(1926-27) showing regulating lines (Drawn-Author)



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16.3 CANDORO MARBLE WORKS SHOWROOM

This little gem can be found in the small community of Vestal just across

the river from downtown Knoxville. It was designed in 1923 by Charles

Barber.

We are inspired by its simplistic beauty. The building has total command

of its site. Look at the view down the entrance road through the alley of

cedar trees, what an amazing, formal entry! The garage to the side is

more of a Mediterranean style, while the front facade is Classical, yet

they blend well. The intricate details in the stonework show that this

came from the shop of some very skilled craftsmen.

The regulating lines below show that a great amount of thought went into

the facade. Two overlapping golden rectangles form the main body. Inset

between the water table and the base of the cornice are two perfect

squares. The windows are golden rectangles, as well as the space

between the columns. One often wonders just how much of this was

planned and how much was designers intuition. Either way it is beautiful.

Figure 73 : The front facade of Candoro marble works showroom



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17. DESIGN PROCESS

The aim of the thesis is to exhibit a design process in the end that

showcases the use Le Modulor as a design aid. Various steps are

involved in achieving it. Case study, literature study and site selection

form an integral part of the effort.

17.1 CASE STUDY

Le Corbusier worked on his philosophy regarding golden ratio, human

anatomy and building design and as a result produced Le Modulor in

1950s. The buildings he designed after it are claimed to be following the

concept by him. A case study was conducted in the Capitol Complex of

Chandigarh to understand the implications. Capitol Complex of

Chandigarh houses three buildings namely, Secretariat, High Court and

State Assembly which functions both for Punjab and Haryana. The three

buildings were studied in the light of the concept of Modulor.

Various measurements from the Blue and the Red series of the Modulor

scale can be identified in the Secretariat right from the structural grid to

the elements of the dominating balcony. Similarly, a set of 15

measurements were made use of while designing Unite DHabitation at

Mersailles.



89 | P a g e

Figure 74 Section and plan of Secretariat at Chandigarh

17.2 INFERENCES

The site demands that the building be built using Corbusiers

philosophies and principles. Along with taking care of the faade control

and the materials, special attention would be given to the use of Modulor

to make the building aesthetically pleasing as well as functionally sound.

After understanding the application of Modulor in the various works of Le

Corbusier, the 11 storey at sector 17, will be designed to exhibit the

application of the Modulor scale.

The site and the building thus selected should be designed with utmost

care to be in harmony with the already existing sector and should be able

to justify being the highest building in Chandigarh. Moreover, it should be



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kept in mind that the 11 storey once constructed, would act as the focal

point of not just the piazza but the whole sector 17.

17.3 CONCLUSION

The 11 storeyed office cum shopping complex at Sector 17 (Chandigarh)

was designed with the aid of Le Modulor. The resultant of the process

was a methodology for designing a building using Modulor scale as a

design tool. The methodology is discussed below:

Select a form

Sketch a few options for the form of the building. Select the most

appropriate one depending on the clients choice, climate, zoning

regulations, and site constraints.

Choose a structural grid

Simultaneously, Depending on size of the structure and functional

requirements decide a structural grid taking measurements from either of

the Modulor series.

Take the chosen form and the grid decided and compare them by

overlaying one on the other.

Refine and Freeze grid

Check the form and the grid for compatibility. In case the structure is

unable to sustain the form, revise the structure according to the form. Do

changes in either the grid, or the form, or both of them depending on the

priority of the owner.

Zoning

Once the structural grid and the form are frozen, planning can be started.

Zoning is the first step to planning, it should be done along the grid lines

as far as possible

Plan services

Services are the most rigid part of planning. Adhering strictly to the

norms and requirements, plan the services.



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Plan areas on grid

Once the services are planned and placed, plan the served spaces

adhering to the grid as far as possible.

Work on levels

Modulor measurements are derived from anthropometry and thus the

levels and heights of a human-scale building coincides largely with

Modulor dimensions. Select heights of siting, countertop, sill level, lintel

level, ceiling level and any other height which is of consequence to the

building and/or the user from the two series (prefer red series for

furnitures and blue series for buildings levels).

Merge the service layout, space layout, final grid and compare with the

levels.

Deviations from the grid

Check for the need of any kind of deviations from the grid in the plan.

Usability and norms

Check for the usability of the spaces. Using Modulor while compromising

the usability is not aimed at. Building norms should be respected.

Changes should be made to accommodate all relevant norms and

usability of the spaces.

Tackle the deviations

Take note of wherever the plan is deviating from the grid on account of

planning, usability or norms. And then plan in a way that the deviations

are also a measure from the Modulor series. This is aided by the additive

property of the Modulor series. While doing so, take into account the fact

that not all spaces are perceivable by the user and that the most frequent

perception of the space by the user might be from a particular area. Thus

use Modulor depending on this knowledge.

Fix all dimensions

Once all deviations are taken care of, fix all the dimensions in the design.

Fine tune measurements

Depending on the market practice, the skill of the labours and the



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construction techniques all dimensions may be fine-tuned or rounded off

for convenience. The tolerance shall depend on the scale of the project

and the architects discretion, but it should generally be maintained at

less than or equal to (+-) 400 mm.

Execution

The design is ready for the preparation of construction drawings.



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18. SHEETS



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19. REFERENCES

19.1 ARTICLES

1. Thesis: An Objective Search And A Subject Analysis, TH-0874BHAR,

15897, Asit Bhatt, School Of Architecture, CEPT, A copy of which is

available at CEPT library (Ahmedabad).

2. Review: Tom Riedel (Regis University), A copy of which is available

with the author

3. Golden Ratio in Art and Architecture, Samuel Obara, University of

Georgia, Department of mathematics education,

(http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat669

0/Golden%20Ratio/golden.html), A copy of which is available with the

author

4. A New Class of Geometrical Constructions, Janusz Kapusta,

Brooklyn, NY, A copy of which is available with the author

5. Seminar: Geometrical principles in nature, Harsh S. Anjaria, Student

of post Graduation, Department of Landscape Architecture, CEPT, A

copy of which is available at CEPT library (Ahmedabad); and

Simulation Modelling Of Plant And Plan Ecosystem, Communication

of ACM, July 2003, Vol 43.

6. Thesis: Strategic Variations Attempted In Systems Of Proportion,

Ashwin Milisia, RATH-0064MIL, 1972-73, School Of Architecture,

CEPT, A copy of which is available at CEPT library (Ahmedabad), p.

72-73



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19.2 BOOKS

7. Effels Engg Work And Millarts Bridges Ch- 03a

8. Architecture : Form Space Order, Francis D. K. Ching, John Willy and

Sons Inc, USA, 1943, A copy of which is available with the author,

p.286-

9. 3rd Book, Ten Books On Architecture By Vitruvius, A 720VIT,

A16097, A copy of which is available at CEPT library (Ahmedabad)

10. 4th Book, Ten Books On Architecture By Vitruvius, A 720VIT,

A16097, A copy of which is available at CEPT library (Ahmedabad)

11. Francisco Giorgio architectural principles in the age of humanism,

Rudolf Wittkower

12. The Modulor, Le Corbusier, Faber and Faber, London, 1954, A copy

of which is available with the author; and Modulor 2, Le Corbusier,

Faber and Faber, London, 1958, A copy of which is available with the

author

13. Six Houses, Le Corbusier: Architect of the Century, Tim Benton,

Hayward Gallery, London, 1987, p.61

14. Banham, Reyner, The New Brutalism: Ethic or Aesthtic?, Reinhold

Publishing Company, New York, 1966, p. 16.



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19.3 WEB PAGES

15. Marquadat Beauty Analysis, Aesthetics Research & Diagnostic

Analysis (http://www.beautyanalysis.com); last referred on 18th

dec11, at 10:00pm.

16. GAP System, School of Mathematics and Statistics, University of St.

Andrews, Scotland, (http://www.gap-

system.org/~history/Biographies/Hemchandra.html) and

(http://www.gap-system.org/~history/Biographies/Fibonacci.html); last

referred on 12th dec11, at 6:00pm.

17. Merriam-Webster Dictionary; last referred on 16th dec11, at 12:00pm.

18. "Aesthetic Judgment", Stanford Encyclopaedia of Philosophy, Nick

Zangwill, 02-28-2003/10-22-2007.

19. www.wikipedia.org; (http://en.wikipedia.org/wiki/Pentagram);

(http://en.wikipedia.org/wiki/golden number);

(http://en.wikipedia.org/wiki/phyllotaxi)last referred on 17th dec11, at

6:00am.

20. http://milan.milanovic.org/math/english/golden/golden4.html; last

referred on 16th dec11, at 5:00pm.

21. Phi and the Golden Section in Architecture;

http://www.goldennumber.net/architecture.htm; last referred on 15th

dec11, at 8:00pm.

22. Harmony and Home Where Balance, Serenity And Living Come

Together. http://harmonyandhome.blogspot.com/2008/12/golden-

mean-and-modern-design.html; last referred on 10th dec11, at

3:00am.

23. Marseille's Cit Radieuse: photos and hotel review

(http://www.tripadvisor.com); last referred on 6th dec11, at 6:00am.

24. Fibonacci Numbers and Nature-part 2, Why is the Golden section the

"best" arrangement?, Ron Knott,



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(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/); last

referred on 5th dec11, at 10:00am.

25. http://archgeom.blogspot.com/2010/03/golden-section-in-taj-

mahal.html; last referred on 28th nov11, at 5:00pm.

26. http://www.flickr.com/photos/ad_symphoniam/3926961973/lightbox/;

last referred on 10th dec11, at 10:00am.

27. http://books.google.co.in/books?id=1KI0JVuWYGkC&printsec=frontco

ver&source=gbs_atb#v=onepage&q&f=false; last referred on 1st

dec11, at 5:00pm.

28. http://books.google.co.in/books?id=exnMTOE_t-

EC&pg=PA159&dq=The+modulor:+a+harmonious+measure+to+the+

human+scale,+universally+applicable+to+architecture+and+mechani

cs&source=gbs_selected_pages&cad=3#v=onepage&q&f=false; last

referred on 1st dec11, at 5:00pm.

29. http://www.marseille-citeradieuse.org/cor-

cite.php?zotable=tabcmsv1_cms&zotype=accue&zopage=cor-

site&zogra=Galerie%20photos&zogrb=&zogrc=&zopcles=&zohaut=8

00&zolar=800&zocols=1&zocarti=ffffff&zofonti=b51a13&zopafond=&

PHPSESSID=f943c3da07df6455a9659abbcc5094d8; last referred on

4th dec11, at 4:00pm.

30. http://books.google.co.in/books?id=5ja-

3GavJssC&pg=PA130&lpg=PA130&dq=The+modulor:+a+harmoniou

s+measure+to+the+human+scale,+universally+applicable+to+archite

cture+and+mechanics&source=bl&ots=HMV_5DNg9z&sig=HkWX_D

ojvn6Wm9UDi-

hLhbu4qEc&hl=en&ei=tnyDTuCJJpCIrAfk0aivDg&sa=X&oi=book_res

ult&ct=result&resnum=9&ved=0CE8Q6AEwCDgK#v=onepage&q&f=f

alse; last referred on 1st dec11, at 2:00pm.

31. http://www.intmath.com/numbers/beauty.

Fibonacci Series and Golden Ratio in Architecture

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