exploring systems of architectural proportions

Exploring systems of architectural proportions

Sumil Dutta, 10th sem B.arch, P.I.A.D.S

Introduction

Architecture, for the most part, may only be produced by a creative,

intellectual dialectic between imagination and reason. Judgments contained

within the broad limits of theory are a fundamental part of the architectural

creative process. It is a fundamental premise of this course that most

architectural propositions are, whether textual or formal, a reflection of some

overriding social, political, practical or aesthetic concern and as such are,

whether implicit or explicit, affected (to some extent) by theoretical concerns

that are informed by a broad spectrum of philosophical-ideological

assumptions and cultural values.

As it pertains to architecture, proportion may be a precise study that orders

entire buildings, or, on a more limited basis, a system used to design facades,

entrances or individual rooms. In the work of Vitruvius, Palladio and Le

Corbusier, for example, proportion addresses the requirement of architects for

a rational system of dimensional grammar. On a mystical plane, as in the work

of Frank Lloyd Wright and Louis Khan, proportion addresses transcendental

aesthetic-compositional matters.

Architectural practice has often used proportional systems to generate or

constrain the forms considered suitable for inclusion in a building. In almost

every building tradition there is a system of mathematical relations which

governs the relationships between aspects of the design. These systems of

proportion are often quite simple; whole number ratios or easily constructed

geometric shapes (the golden ratio).

Generally the goal of a proportional system is to produce a sense of

coherence and harmony among the elements of a building.

Objective of the study

The idea of proportion as specifically applied to architecture raises several

questions:

• Is proportion a prescient oracle for the development of form, or an ex post

facto justification for built work?

• How does, and to what extent, proportion affect the appearance of

architecture?

• What is its relationship to structure?

• Can one readily detect the application of proportions in buildings?

• Can (should) new systems of proportion be formulated?

• To investigate the parallels between measure in architecture and the

mathematics of nature is therefore the basic aim of the study?

Architectural

proportions

Structural

proportions

Material

proportions

Theories of

proportion

Golden section

Renaissance

theories

Modulor

theory

Anthropo-

metry

Visual scale

Classifying architectural proportion

Structural proportions

In construction of building structural elements

spaces and transmit their load through vertical supports to the

foundation system of a building.

• The size and proportion of structural elements are

structural tasks they perform

size and scale of the spaces they help enclose

• Beams for example transmit their loads horizontally across spaces their

vertical supports. If the span or load of a beam were doubled, its bending

stresses would likewise double, possibly

depth were doubled, its strength would increase fourfold.

• Depth is the critical dimension of a beam and its

a useful indicator of its structural role.

• In similar manner, columns become thicker

height increases. Together , beams and columns form a skeletal structural

framework that defines modules of space. By their size and proportion,

Structural proportions

of building structural elements are called upon to span


foundation system of a building.

The size and proportion of structural elements are directly related to the

structural tasks they perform and can therefore be visual indicators of the

size and scale of the spaces they help enclose

Beams for example transmit their loads horizontally across spaces their


stresses would likewise double, possibly causing it to collapse. But if its


is the critical dimension of a beam and its depth to span ratio

a useful indicator of its structural role.

olumns become thicker as their loads and unsupported

. Together , beams and columns form a skeletal structural


are called upon to span


directly related to the

ual indicators of the

Beams for example transmit their loads horizontally across spaces their


causing it to collapse. But if its


depth to span ratio can be

as their loads and unsupported

. Together , beams and columns form a skeletal structural


columns and beams articulate space and give it scale and hierarchical

structure.

• This can be seen in the way joists are supported by beams, which in turn are

supported by girders. Each element increases in depth as its load and span

increase in size.

E.g. a stone slab that is four inches thick and eight feet long can be

reasonably expected to support itself

But if its size were to increase fourfold to sixteen inches thick and two

feet long it would probably


This can be seen in the way joists are supported by beams, which in turn are



pected to support itself as bridge between two supports.


feet long it would probably collapse under its own weight


This can be seen in the way joists are supported by beams, which in turn are



as bridge between two supports.


collapse under its own weight.

Material proportions

• All building materials have distinct properties of elasticity, hardness and

durability. Thus all materials have rational proportions which are dictated

by these inherent strengths and weakness.

• e.g. the load carrying capacity of a 6 feet long and 1” thick stone slab will be

different as compared to a wooden plank of same dimensions.

• Similarly the load carrying capacity of wooden truss work will be different as

compared to a steel truss of same dimensions.

Golden section

At least since the Renaissance, many artists and architects have proportioned

their works to approximate the golden ratio

golden rectangle, in which the ratio of the longer side to the shorter is the

golden ratio—believing this proportion to be

Mathematicians have studied the golden ratio because of its unique and

interesting properties.

The golden ratio is often denoted by the Greek letter

golden section illustrates the geometric relationship that

Expressed algebraically:

The golden section is a line segment sectioned into two according to the

golden ratio. The total length a + b is to the longer segment a as a is to the

shorter segment b.

Some of the greatest mathematical minds of all ages, from Pythagoras and

Euclid in ancient Greece, through the medieval Italian mathematician Leonardo

of Pisa (Fibonacci) and the Renaissance astronomer Johannes Kepler, to

present-day scientific figures such as Oxford

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. In

fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all

disciplines like no other number in the history of mathematics.

Renaissance, many artists and architects have proportioned

their works to approximate the golden ratio—especially in the form of the


believing this proportion to be aesthetically pleasing.


The golden ratio is often denoted by the Greek letter ϕ (phi). The figure of a

golden section illustrates the geometric relationship that defines this constant.



greatest mathematical minds of all ages, from Pythagoras and



day scientific figures such as Oxford physicist Roger Penrose, have




ystics have pondered and debated the basis of its ubiquity and appeal. In



Renaissance, many artists and architects have proportioned

especially in the form of the


aesthetically pleasing.


(phi). The figure of a

defines this constant.



greatest mathematical minds of all ages, from Pythagoras and



physicist Roger Penrose, have




ystics have pondered and debated the basis of its ubiquity and appeal. In



Some studies of the Acropolis,

including the Parthenon,

conclude that many of its

proportions approximate the

golden ratio.

facade as well as elements of its

facade and elsewhere can be

circumscribed by golden

rectangles.

Some studies of the Acropolis,

including the Parthenon,

conclude that many of its

proportions approximate the

golden ratio. The Parthenon's

facade as well as elements of its

facade and elsewhere can be

circumscribed by golden

rectangles.

Classical orders

The classical orders here illustrated by the Temple of Hephaestus in Athens,

showing columns with Doric capitals are largely known through the writings of

Vitruvius, particularly De Archetura (The Ten Books of Architecture) and studies

of classical architecture by Renaissance architects and historians. Within a

classical order elements from the positioning of triglyphs to the overall height

and width of the building were controlled by principles of proportionality

based on column diameters.

By way of contrast to the elongated Ionic order, Doric orders never became so

slender as to require a base but do have entasis as the column shaft tapered

upwards like a degree of the earth's surface. The column shafts of the Doric

order are always fluted

capital has an abacus square in plan and a rounded echinus which supports it.




classical architecture by Renaissance architects and historians. Within a



based on column diameters.




order are always fluted and twenty flutes is the usual number. The column





classical architecture by Renaissance architects and historians. Within a






and twenty flutes is the usual number. The column


Pentagram

A pentagram colored to distinguish its line segments of different lengths. The

four lengths are in golden ratio to

The golden ratio plays an important role in regular pentagons and pentagrams.

Each intersection of edges sections other edges in the golden ratio. 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 center) is φ, as

the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and five obtuse

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

is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are

golden gnomon.

Pyramids

A regular square pyramid is determined by its medial right triangle, whose

edges are the pyramid's apothem (a), semi

inclination angle is also marked. Mathematical proportions b:h:a of and and

are of particular interest in relation to Egyptian pyramids.

Both Egyptian pyramids and those mathematical regular square pyramids that

resemble them can be

ratios.


four lengths are in golden ratio to one another.




g edges (a side of the pentagon in the pentagram's center) is φ, as

color illustration shows.


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



edges are the pyramid's apothem (a), semi-base (b), and height (h); the face


are of particular interest in relation to Egyptian pyramids.


analyzed with respect to the golden ratio and other





g edges (a side of the pentagon in the pentagram's center) is φ, as


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



height (h); the face



analyzed with respect to the golden ratio and other

A pyramid in which the apothem (slant heig

equal to φ times the semi

golden pyramid. The isosceles triangle that is the face of such a pyramid can be

constructed from the two halves of a diagonally split golden rec

semi-base by apothem), joining the medium

apothem. The height of this pyramid is times the semi

of the face is ); the square of the height is equal to the area of a face, φ times

the square of the semi-base.

One Egyptian pyramid is remarkably close to a "golden pyramid"

Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51°

52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π

based pyramid inclination of 51° 51';

A pyramid in which the apothem (slant height along the bisector of a face) is

equal to φ times the semi-base (half the base width) is sometimes called a


constructed from the two halves of a diagonally split golden rec

base by apothem), joining the medium-length edges to make the

apothem. The height of this pyramid is times the semi-base (that is, the slope


base.

In the mid nineteenth century,

Röber studied various Egyptian

pyramids including Khafre,

Menkaure and some of the

Gizeh, Sakkara and Abusir

groups, and was interpreted as

saying that half the

side of the pyramid is the

middle mean of the side,

forming what other authors

identified as the Kepler triangle;

many other mathematical

theories of the shape of the

pyramids have also been

One Egyptian pyramid is remarkably close to a "golden pyramid"

(also known as the Pyramid of Cheops or Khufu). Its slope of 51°

52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π

based pyramid inclination of 51° 51';

ht along the bisector of a face) is

base (half the base width) is sometimes called a


constructed from the two halves of a diagonally split golden rectangle (of size

length edges to make the

base (that is, the slope


In the mid nineteenth century,

Röber studied various Egyptian

pyramids including Khafre,

Menkaure and some of the

Gizeh, Sakkara and Abusir

groups, and was interpreted as

saying that half the base of the

side of the pyramid is the

middle mean of the side,

forming what other authors

identified as the Kepler triangle;

many other mathematical

theories of the shape of the

pyramids have also been

One Egyptian pyramid is remarkably close to a "golden pyramid" – the Great

(also known as the Pyramid of Cheops or Khufu). Its slope of 51°

52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-

Adding fuel to controversy over the architectural authorship of the Great

Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that

Egyptian mathematics would not have supported the ability to calculate the

slant height of the pyramids, or the ratio to the height, except in the case of

the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to

the Egyptians and they did not know the Pythagorean theorem nor any way to

reason about irrationals such as π or φ

To the extent that classical buildings or their elements are proportioned

according to the golden ratio, this might indicate that their architects were

aware of the golden ratio and consciously employed it in their designs.

Alternatively, it is possible that the architects used their own sense of good

proportion, and that this led to some proportions that closely approximate the

golden ratio.

Vitruvian proportion

Vitruvius described as the principal source of proportion among the orders the

proportion of the human figure. .

Leonardo is clearly illustrating Vitruvius' De architectura 3.1.3 which reads: The

navel is naturally placed in the centre of the human body

with his face upward, and his hands and feet extended, from his navel as the

centre, a circle be described, it will touch his fingers and toes. It is

a circle, that the human body is thus circumscribed, as may be seen

it within a square. For measuring from the feet to the crown of the head, and

then across the arms fully extended, we find the latter measure equal to the

former; so that lines at right angles to each other, enclosing the figure, will

form a square.

Vitruvian proportion


portion of the human figure. .


navel is naturally placed in the centre of the human body, and, if in a man lying


centre, a circle be described, it will touch his fingers and toes. It is

a circle, that the human body is thus circumscribed, as may be seen

. For measuring from the feet to the crown of the head, and





, and, if in a man lying


centre, a circle be described, it will touch his fingers and toes. It is not alone by

a circle, that the human body is thus circumscribed, as may be seen by placing

. For measuring from the feet to the crown of the head, and



According to Leonardo's notes in the accompanying text, written in mirror

writing, it was made as a study of the proportions of the (male) human body as

described in a treatise by the Ancient Roman architect Vitruvius, who wrote

that in the human body:

• a palm is the width of four fingers or three inches

• a foot is the width of four palms and is 36 fingers or 12 inches

• a cubit is the width of six palms

• a man's height is four cubits and 24 palms

• a pace is four cubits or five feet

• the length of a man's outspread arms is equal to his height

• the distance from the hairline to the bottom of the chin is one-

tenth of a man's height

• the distance from the top of the head to the bottom of the chin is

one-eighth of a man's height

• the maximum width of the shoulders is a quarter of a man's height

• the distance from the elbow to the tip of the hand is one-fifth of a

man's height

• the distance from the elbow to the armpit is one-eighth of a man's

height

• the length of the hand is one-tenth of a man's height

• the distance from the bottom of the chin to the nose is one-third

of the length of the head

• the distance from the hairline to the eyebrows is one-third of the

length of the face

• the length of the ear is one-third of the length of the face

The Modulor theory

The Swiss architect Le Corbusier, famous for his contributions to the

modern international style, centered his design philosophy on systems of

harmony and proportion. Le Corbusier's faith in the mathematical order of the

universe was closely bound to the golden ratio and the Fibonacci series, which

he described as "rhythms apparent to the eye and clear in their relations with

one another. And these rhythms are at the very root of human activities. They

resound in man by an organic inevitability, the same fine inevitability which

causes the tracing out of the Golden Section by children, old men, savages and the

learned."

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale

of architectural proportion. He saw this system as a continuation of the long

tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon

Battista Alberti, and others who used the proportions of the human body to

improve the appearance and function of architecture. In addition to the golden

ratio, Le Corbusier based the system on human measurements, Fibonacci

numbers, and the double unit.

He took Leonardo's suggestion of the golden ratio in human proportions to an

extreme: he sectioned his model human body's height at the navel with the

two sections in golden ratio, then subdivided those sections in golden ratio at

the knees and throat; he used these golden ratio proportions in the Modulor

system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor

system's application. The villa's rectangular ground plan, elevation, and inner

structure closely approximate golden rectangles

The basic grid consists of three measures 113, 70 and 43 centimeters

proportioned according to

43 + 70 = 113

113 + 70 = 183

113 + 70 + 43 = 226 or (2x 113)

• 113, 183 and 226 define the space occupied by the human figure. From

113 and 226 Le Corbusier developed

scales of dimensions that were

figure.


proportioned according to the Golden section

113 + 70 + 43 = 226 or (2x 113)

113, 183 and 226 define the space occupied by the human figure. From

113 and 226 Le Corbusier developed red and blue series, diminishing

scales of dimensions that were related to the stature of the human

Le Corbusier made his own tool based on

both (aesthetic dimensions of the Golden

section and Fibonacci series), and the

proportions of the human body (functional

dimensions).


113, 183 and 226 define the space occupied by the human figure. From

, diminishing

related to the stature of the human

his own tool based on

both (aesthetic dimensions of the Golden

section and Fibonacci series), and the

proportions of the human body (functional

Visual proportion

The Tajmahal at Agra

Bibi ka maqbara at Aurangabad

The variation in the proportions

of a structure can result in change

in the visual appreciation of the

structure

• In the Tajmahal the minarets are used to prevent the structure from merging into the

horizon. Whereas the Bibi ka maqbara with its minarets of wider diameter acts as a

framing element for the structure.

• The height of the dome being greater from the minarets gives it domination while

in the other case the relation is opposite.

• The absence of regular octagonal sides in Bibi ka maqbara makes it look smaller and

disproportionate.

The change in proportion of spaces affects

the way we perceive and utilize the space.

The different dimensions of height and

width regulate the axis of our vision

The change in proportion of spaces affects

the way we perceive and utilize the space.

ifferent dimensions of height and

regulate the axis of our vision.

Proportions in Indian architecture

According to the Sthapatya Veda (the Indian tradition of architecture), the

temple and the town should mirror the cosmos. The temple architecture and

the city plan are, therefore, related in their conception.

The Harappan cities have a grid plan, just as is recommended in the Vedic

manuals. The square shape represents the heavens, with the four directions

representing the cardinal directions as well as the two solstices and the

equinoxes of the sun’s orbit.

An assumed equivalence between the outer and the inner cosmos is central to

the conception of the temple. It is because of this equivalence that numbers

such as 108 and 360 are important in the temple design.

The number 108 represents the distance from the earth to the sun and the

moon in sun and moon diameters, respectively. The diameter of the sun is also

108 times the diameter of the earth, but that fact is not likely to have been

known to the Vedic rishis. This number of dance poses (karanas) given in the

Natya Shastra is also 108, as is the number of beads in a rosary (japamala).

The “distance” between the body and the inner sun is also taken to be 108, and

the number of marmas in Ayurveda is 107. The total number of syllables in the

Rigveda is taken to be 432,000, a number related to 108.

The solar and lunar numbers that show up in the design of the Angkor Wat

temple are the number of nakshatras, the number of months in the year, the

days in the lunar month, the days of the solar month, and so on. Lunar

observations appear to have been made from the causeway.

The division of the year into the two halves of 189 and 176.37 was recently

explained as being derived from the Shatapatha Brahmana. In layer 5 of the

altar described in the Shatapatha, a division of the year into the two halves in

the proportion 15:14 is given (Kak, 1998, 2000). This proportion corresponds to

the numbers 189 and 176.4 used at Angkor Wat, where in the central tower the

topmost elevation has dimensions of 189 east-west and 176.37 north-south.

It is possible that the period from the spring equinox to the fall equinox was

taken to be about 189 days by doubling the period of the spring season; 176

days became the period of the reverse circuit.

The equinoctial half-years are currently about 186 and 179, respectively; and

were not much different when Angkor Wat temple was constructed. Given that

the length of the year was known to considerable precision there is no reason

to assume that these counts were not known. But it appears that a `normative'

division according to the ancient proportion was used.

As it was known that the solar year was about 365.25 days, the old proportion

of 15:14 would give the distribution 188.92 and 176.33, and that is very much the

Angkor Wat numbers of 189 and 176.37 within human error. In other words, the

choice of these `constants' may have been determined by the use of the

ancient proportion of 15:14.

In the Shilpa Prakasha, a 9th-12th century Orissan temple architecture text,

Ramachandra Kaulachara describes the Yogini Yantra for the layout of the

goddess temple:

[The Devi temples] represent the creative expanding forces, and therefore

could not be logically be represented by a square, which is an eminently static

form. While the immanent supreme principle is represented by the number

ONE, the first stir of creation initiates duality, which is the number TWO, and

is the producer of THREE and FOUR and all subsequent numbers up to the

infinite.

The dynamism is expressed by a doubling of the square to a rectangle or the

ratio 1:2, where the garbhagriha is now built in the geometrical centre. For a

three-dimensional structure, the basic symmetry-breaking ratio is 1:2:4, which

can be continued further to another doubling.

The constructions of the Harappan period (2600-1900 BC) appear to be

according to the same principles. The dynamic ratio of 1:2:4 is the most

commonly encountered size of rooms of houses, in the overall plan of houses

and the construction of large public buildings. This ratio is also reflected in the

overall plan of the large walled sector at Mohenjo-Daro called the citadel

mound. It is even the most commonly encountered brick size.

Concluding remarks:

This report shows how the proportional systems and the sense of

proportioning of the human being together contribute in the creation of

aesthetically appealing architecture. Also it can be said without any doubts that

the use of proportioning systems developed in classical gives us some of the

best architectural marvels of the past and further study of the systems will

open new avenues of aesthetics in future.

Interesting non-architectural facts:

Some specific proportions in the bodies of many animals (including humans and parts of the

shells of mollusks and cephalopods are often claimed to be in the golden ratio.

The proportions of different plant components (numbers of leaves to branches, diameters

of geometrical figures inside flowers) are often claimed to show the golden ratio proportion

in several species. In practice, there are significant variations between individuals, seasonal

variations, and age variations in these species. While the golden ratio may be found in some

proportions in some individuals at particular times in their life cycles, there is no consistent

ratio in their proportions.

Bibliography:

Books

• Form , space and order

• Proportion – science, philosophy and architecture

• Space and cosmology in Hindu temple (Presentation by Subhash Kak)

Internet

• Wikipedia encyclopedia

PRIYADARSHINI INSTITUTE OF

ARCHITECTURE AND DESIGN STUDIES

Electronic Zone, MIDC, Hingna Road Nagpur- 440016(M.S)

Certificate

This is to certify that Mr. Sumil Dutta has submitted the 10th semester (B.arch)

seminar as per the curriculum requirement of Rashtrasant Tukdoji Maharaj

Nagpur University as a part of course work in Degree of Bachelor in Architecture

under the guidance of Mrs.Vaidehi Pathak.

Prof.A.V.Purohit Mrs.Vaidehi Pathak Prof.R.K.Bhargav

Director Seminar guide Seminar coordinator

exploring systems of architectural proportions

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