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Role of Geometry on Architecture

Role of Geometry on Architecture
Ar. Apoorva Ajmera
Assistant Professor, Buddha Institute of Architecture and Town Planning
Udaipur, Rajasthan
Man develops geometry, Geometry is the language of man,
which is used in different ways The basic definition of
geometry given in 'a visual dictionary of architecture" by
Francis D.K.Ching states that ' a branch of mathematics that
deals with the properties, measurement and relationship of
points, lines and angles, end solids, deduced from the defining
conditions by the means of certain assumed properties of
space " . But there are several other definitions by the
different use of geometry. It brings sense of continuity and
structural qualities to the building. Geometry is the means,
created by human beings whereby they perceive the external
world and express the world within them. Geometry is the
world use "common language'. Geometry is the foundation of
life; it is also the material basis on which we build those
symbols that represent us perfection and divine.
Keywords- Geometric forms, Building forms, curves and
angles of building
It is an overview of the basic elements, systems, and orders
that constitute a work of architecture. All of these
constitutes can be perceived and experienced. Some may
be readily apparent while others are more obscure to our
intellect and senses. Some may dominate while others play
a secondary role in a building’s organization. Some may
convey images and meaning while others serve as
qualifiers or modifiers of these messages.
In all cases, however, these elements and systems should
be interrelated to form an integrated whole having a
unifying or coherent structure. Architectural order is
created when the organization of parts makes visible their
relationships to each other and the structure as a whole.
When these relationships are perceived as mutually
reinforcing and contributing to the singular nature of the
whole, then a conceptual order exists- an order that may
well be more enduring than transient perceptual visions.
As this analysis is based on geometry, in various aspects, I
have tried to find relationship with geometry and so, all the
criteria are based on geometry. In various architects work,
geometry is manifested in form & shape. Geometry is
evident in plan as space sequence and in details. The whole
building is visualized in form of geometry and philosophies
are attached to them. All these criteria are discussed in later
part of this dissertation report.
The fundamental element of basic designing process is a
point. Point is the basis of all geometry. Point is the most
basic and prime geometrical element, it is not dimension
less but infinitely tiny elements. It can be said that its
smallness is always relative to a respective frame. A point
indicates position, and has no-length, width or depth. It is
just there and is therefore static centralized and
directionless, and does not occupy any area of space. It is
the beginning and end of a line and is where two lines meet
or intersect in the form of a center in various geometric
A point marks a position in space. Conceptually, it has no
length, width, or depth, and is therefore static, centralized
and directionless. In architecture a point is used as an
organizational element, and is the only organizational
element that gives’ constancy’ in relationship between the
point and various peripheral components. As the prime
element in the vocabulary of form, a point can serve to
The two ends of a line
The intersection of two lines
The meeting of lines at the corner of a plane or
The center of a field
3.2 LINE
As a point moves, its path becomes a line. Line is perhaps
the most fundamental elements of design, since civilization
began. People have been fascinated by lines. Lines fill
every part of ones world, from the endless of the horizon to
the delicate venis in a leaf, from the soaring to the vertical
drawing or painting. The natural world abounds with lines
of every description. If one studies the world around one
self one finds a wealth of natural configuration in line.
Sometimes the line is also called as a distance between two
points (axis). 'Line' is a vector scale which has length but
no width or depth, where as a point is by nature static. A
line in describing the path of a point in motion, is capable
of visually expressing direction, movement and growth. It
forms the border of a plane. It can serve to: Join, link,
support, surround or intersect other visual elements.
Describe the edges of and give shape to planes.
A form is organized as a line because of two reasons:
(a) Its breadth is extremely narrow and
(b) Its length is quite prominent.
If a line moves in a direction other than the intrinsic
direction, the trace of its path describes a plane. On a
two dimensional surface, all flat forms that are not
commonly recognized as points or lines are forms as
planes. Plane has length and breadth but no depth
(Third dimension). A planar form is bound by
International Journal of Engineering Research & Technology (IJERT)
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conceptual lines which constitute the edges of the
form. The characteristics of these conceptual lines and
their interrelationships determine the shape of planar
Vertical and horizontal are the most basic planes of
spatial unit. They are the primary areas of interest in
architecture. In the composition of a visual
construction a plane serves to define the limits or
boundaries of a volume. If architecture as a visual art
deals specifically with the formation of three
dimensional volumes of mass and space, then the plane
should be regarded as key element in the vocabulary of
architectural design. Basically the horizontal is the
plane for human activities and movement. The vertical
plane is the limit for each space and there by the limit
for horizontal movement.
Horizontal and vertical are most basic two planes.
A plane extended in a direction other than its intrinsic
direction becomes a volume. Conceptually, a volume
has three dimensions: length, width and depth.
All volumes can be analyzed and are understood to
consist of:
Points or vertices where several planes come
Lines or edges where the planes meet
Planes or surfaces which define two limits or
boundaries of a volume
3.5 FORM
Form is an inclusive term that has several meanings. It
may refer to an external appearance that can be
recognized, as that of a chair or the human body that
sits in it. It may also allude to a particular condition in
which something acts or manifests itself, as when we
speak of water in the form of ice or steam. In art and
design, we often use the term to denote the formal
structure of a work the manner of arranging and
coordinating the elements and parts of a composition
so as as to produce a coherent image.
In the context of this study, form suggests reference to
both internal structure and external outline and the
principle that gives unity to the whole. While form
often includes a sense of three-dimensional mass or
volume, shape refers more specifically to the essential
aspect of form that governs its appearancethe
configuration or relative disposition of the lines or
contours that delimit a figure or form.
Proportion is an ordered relationship between two
comparable entities, visible or invisible. It can be
manipulated and experienced by geometry; arithmetic
ratios and visual perception. Certain proportion have been
found to be generally satisfying to the human senses.
Proportional system is evolved out of geometrical
principles. This system sets interrelationship of parts to
form harmonized composition. It helps in achieving
ordered relationship in design. It is a system used to relate
parts in an overall composition. This system divides larger
areas into parts and forms an integrated as a whole. It
establishes a system of ratios to interrelated elements of
different sizes and shapes within one composition. It
generates an order and harmonizes elements of different
sizes and shapes within one composition. It generates an
order and harmonizes elements of different scales.
Proportion is a means by which elements are composed to
achieve the qualities of unity, balance, rhythm etc.
Organization is the systematic arrangement, connection and
co-ordination of parts to form a coherent totality which is
not only conceivable but also perceivable and which serves
a desired purpose for the vital functions and processes.
Organization brings about a harmonized composition
interrelating to the various parts, leading units of purpose
to the whole composition. It establishes sequence by
ordering and structuring relationship between solids-voids,
solids-solids and voids-voids.
Organization is a combination of an order of linkage
between the number of elements, which coordinate the
parts for vital functions, and make a unified as whole.
The path taken in getting from a starting point to the
destination can be termed as a “Route of Movement" A
route without a destination is difficult to traverse and hence
to an observer the route linking him to his destination is
prime importance.
The path of our movement can be conceived as the
perceptual thread that links the spaces of a building, or any
series of interior or exterior spaces, together.
Since we move in Time through Sequence of Spaces, We
experience a space in relation to where we've been and
where we anticipate going.
'Geometry' is related to form and space. So a try has been
made to inquire the depth of geometry in different forms
and spaces as building. Form is inclusive term that has
several meanings, which may refer to an external
appearance that can be recognized. An approach to
architecture is based on an individual understanding of an
architect. The idea of form and space is always as the
starting point and at the center while beginning any new
Form suggests reference to both internal structure and
external outline and the principle that gives unity to whole.
But most of projects, concentrated on external
configuration of form. While form often includes a sense of
volume, shape refers more specifically to the essential
aspects of form that governs its appearance (characteristic
outline). In addition to spaces (shape) and forms have
visual properties of size, color, texture. Forms also have
relational properties which govern the pattern and
composition of elements, position, orientation, visual
inertia etc.
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Let us consider what a vista, which is absent in one type of
garden and available at all points in the other, means. A
French garden is not only extremely artificial but
constructed. Its flowers and trees do not appeal to ones
feelings about nature but rather constitute material
subordinating the entire plan. Form and material are clearly
divorced and the former clearly dominates the latter.
Although trees are continuously changing and being
created, the consequences of such activity are not
acknowledged, the trees are identified as conical a cubic
forms. All plants as plants have been banished from the
garden and it is only rectangles and circles that are placed
and arranged on the extensive, level site paved with gravel,
creating a symmetrical, geometrical pattern. There is only a
completely static surface that will remain forever
unchanged. To create a unified garden by compelling
materials whether they may be trees or water, to take
unchanging forms with no concern for diversity or
generative transformations represents the complete
dominance of the intellect. The result is a highly
constructed space.
Making the object insubstantial confers on it a transparency
and endows vision with a privileged character. Visual
transparency is a quality or state allowing one to penetrate
all parts of the object. It is as if the source of light and
power of vision residing within the self were situated God-
like at the vertex of a cone-shaped world and the self could
see into the furthest depths of the cone. No doubt,
transparency according to the law of perspective
approaches this conical situation.
As soon as them is a disfigurement or distortion for some
reason, it must be discovered and eliminated, each time
new leaves appear on plants and grow unchecked,
threatening to distort conical forms and other geometrical
patterns, the gardener must wield his shears with utmost
seventy to correct even the slightest aberration. The
planting in conical or cubic form constitutes an enclosure
that confines meaning and prevent it form strring,
dissolving or straying. Consequently, a French garden
where the spatial order converges on one point can be said
to have only one landscape. The diversity or polysyllabic
of landscape and vision is to he completely ignored. That is
the basic principle behind a constructed garden.
But what happens when, unlike a French garden, the
garden is outcropped and the landscape is left to sway and
stir as it may? What would happen if, instead of
heightening perception, enlarging the cone of vision, and
thereby achieving integration of the landscape through
transparency at a larger scale, vision traveled from one
distorted fragment to another in accordance with shifts in
the point of view or the actions of the buy? Japanese
garden might be the result.
As has already been pointed out, the absence of vista in
characteristic of a Japanese garden. This absence of vista is
a consequence of the multi-layered quality of the
landscape, evident in the planting, the rock arrangement
and the trees.
Rock obstructs rock and the landscape is always
fragmented. Vision is not allowed in the whole of this
garden either. As in the garden of Tenryuji or Nanzenji, the
landscape becomes a set of overlapping and contiguous
landscapes that deviate slightly from each other. Multiple
landscapes coexist. As a result of Overlapping, there are
always parts that are hidden. A hidden part, if one changes
one's view Point, becomes patent, and what had been
visible now becomes latent. This mechanism makes it
difficult to apprehend the entire garden. For the viewer, the
landscape at each moment certainly exists in front of his
eyes, but the plan of the whole is difficult to structure. It is
difficult to him to measure his creation relative to the entire
garden or the distance to a given plant or rock, and without
a clear order, multiple landscapes are generated as one’s
point of view shifts. The absence of a transcendent center
or vision that can give order to space robs a Japanese
garden of perspective and transforms it into un- constructed
However, the Japanese garden is after all an object
deliberately created by man, and negative terms such as
‘without order’ or ‘unconstructed’ are surely inadequate
descriptions. A Deconstructionist principle that actively
promotes irregularity must be behind what appears at first
glance to be without order. The multilayered, contiguous
landscapes constitute parts, but these parts do not simply
and willfully link themselves. Perhaps then there is
something to be discovered in the way the parts are
Geometry is the foundation; it brings with it the noble joy
of mathematics. Machinery is the result of geometry. The
age in which we live is essentially a geometrical one. All
the ideas are oriented in the direction of geometry. Modem
art and thought, after a century of analysis, are now seeking
beyond what is merely accidental. Geometry leads them to
mathematical forms, a more and more geometrical attitude.
geometry is used as a form of an organizational tool.
Geometry is being related to different mythological
theories of a form.
As man developed geometry, they attached various theories
to geometrical forms. Different symbols have different
meanings according to Hindu philosophy: the 'square'
represents the 'earth' in conformity with the cardinal Points.
It is also thought of as an abode of deity. It is mystical,
absolute and a basic geometric element. It permits no
changes during the process of transformation due to its
religious origin. The ‘Triangle' Symbolizes the Indian
‘Trinity' and is the principal governing element. Based on
natural form, people developed basic geometric forms.
In Islamic architecture, the 'circle' is thought as a symbol of
eternity, a form without a beginning or an and. It is the
most perfect and an expression of justice. Circle is the most
basic and parent figure of all the geometric polygons. The
'Dome', a highly geometric element has become' an
increasingly a characteristic feature of Islamic buildings. It
is thought of as an 'image of heaven'. The 'square' is
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thought as the most externalized form of creation and a
symbol representing the 'earth'. These various basic
geometric forms are used in religious, public buildings in
many ways. The courtyards Of King's tomb and queen's
tomb are square in shape. The domes of both tombs are
built in traditional Islamic ways. Like wise 'square' is being
used as courtyards in most of the Mosques & 'Dome' is
used to represent Tombs. The development of mosque was
done with combination of square and circles.
The Romans Wished to create a Symbolic form of a new
Kind of systematic world order, which embraced a unified
religion, a unified body of laws, and a unified civilization.
This organizational spirit is revealed in their manner or
grouping buildings as done in the Forums, in the
organization Of business activities in common centers, in
combination of three orders on the exterior of the building
as in the colloseum (Tuscan Doric on first, Ionic on second
and Corinthian on third and fourth); in the combination of
Ionic and Corinthian to form the composite order, in the
development of the multi family apartment house; various
forms of recreation in baths and finally in the erection of a
super temple for the principal Gods; as in the Pantheon. An
organization of expansion of interior space is reflected in
the great halls of the baths, Basilicas and the Partheon 'ref.2
(as written by Fleming in 1974). Romans are supposed to
be main Inventors of geometry', ancient civilization.
Its geometry is based on the union of a cylinder (rotunda)
and a hemisphere (dome) over a circular ground plan. The
design of Pantheon displays a harmonious disposition of a
circle and a rectangle. The rectangle contains two squares
and a diagonal of each square is equal to the diameter of
the circle. Thus, the 'latent geometry' clearly establishes a
relation between the width of the portico and rotunda of the
Pantheon. It unifies a celestial dome and a longitudinal axis
with the vertical rising freely to the heavens through the
opening zenith into a meaningful whole. The Point to be
stressed was the center of the floor and not the center of the
sphere. The Pantheon with its utter simplicity laid the
foundation of the idea of the centralized building. The form
was revitalized in different periods, each in its own
manner. Even the lighting of a dome gained an almost
mystic connotation in the later centuries. By creating
Pantheon, Romans gave a very good example of form as
geometrical organizational tool.
Greeks were able to find some of use of geometrical forms.
Greek took first step to the conception of independents
forms detached from any unitary word view.
In medieval and Gothic architecture, basic form was
syntheses of rectangular planes with conical domical roof
over it. Most of the buildings were centric forms. Besides
the single space building there were examples of the
extension of the centric from Into rich and complex spatial
compositions, a high centric space surrounded by a ring of
lower space. In S.Constanza, Rome, in 359 A.D., principle
was introduced - the modestly dimensioned dome was
surrounded by a barrel vaulted ambulatory.
The pure circular form was broken in S. Lorenzo, Milan,
by an ambulatory and three axial chapels. The octagonal
central space of S. Vitale in Ravenna, gave rise to the
sixteen sided water shell. The form was steadily becoming
richer and more complex.
During the medieval ages, centric form and linear form
were developed together and had been juxtaposed on each
other to develop the interior spaces and the exterior forms
of Turkish Mosque, Gothic and renaissance Cathedrals.
The linear path passing through a centric space illustrates
the crossing of the two fundamental ideas. Overlaying the
cross onto the circle creates a powerful symbolic pattern, a
form of mandala, where passage and destination are fused
in a powerful story of faith and exhalation.
A few fundamental principles based on geometric elements
were manifested by inherited spatial form in early Christian
architecture. From here, the aspect of 'Third Dimension'
distinguishes Byzantine architecture from Early Christian
architecture. From here the basic principles remained
constant with development in the third dimension. The
skeletal structure and an additional tower are main
character of Romanesque architecture. After Byzantine,
Gothic architecture was characterized by use of pointed
arch, the rational system of vaulting & buttressing.
In renaissance architecture, geometry played a key role.
The spatial geometry was determined by functions. Larger
organizations were achieved by adding different geometric
units retaining an identity, yet responding to the entire
configuration. Functions were pre supposed by perfection
form as the circle being the most perfect and divine form,
as it is evident in his work.
The building that marks development of the renaissance is
st. Peters in Vatican. Papal service in the sixteenth century
witnessed geniuses of this time, Bramante, Vinci,
Michelangelo and Raphael, in action in Rome. Bramante in
1503 was first to draw two plans for the church. He
interpreted the ‘pentrigion’, as the central plan with four
towers at the corners. The first alternative modified the
Greek cross by widening the crossing, making the dome as
a domical centre. This was a completely centralized church
with the inner arms forming a square ambulatory around
the centre. The second attempt was at general strengthening
of load bearing piers and the addition of four semi-circular
north axis around the four apses of the main space. The
central space was crowned by a semi-spherical dome which
exaggerated the meaningful centre for not only Rome but
to the Christians as a whole.
In the 18 century industrialization transformed the goals
and Methods of human creativity. The specific forms
employed in Roman Byzantine, Romanesque, Gothic,
Renaissance and baroque periods have become symbols of
this epoch, with new dimensions, new character and new
In Italy, futurism was born of a demand for the cultural
rejuvenation of the country. Putting together the similar
kinds of forms created a unity. The new paintings
denies the cube and wants the dynamic arabesque
such as the sphere, the ellipse, the spiral and all other
dynamic forms which artists can discover.
In 1919 the Bauhaus was founded on the remains of the
Arts and Crafts Movement in Germany, It resulted in an
unprecedented success in influencing architectural
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education all over the world. Bauhaus initiated the study of
architecture by manipulating abstract shapes without any
reference to building functions or the ultimate strength of
materials. It was solely with a view of achieving
ornamental appeal, in terms of a magnificent form’. The art
of architecture turned into a kind of large-scale abstract
sculpture. The art of utilizing the vocabulary of the newly
fashionable cultural shapes to serve the functional and
structural requirements of a specific architectural program.
The extent to which severe geometrical planes were used as
a form of ornamentation is evident. The comment of Hans
Mayer, Director of Bauhaus from 1928 to 1929: Despite
Gropies declarations that architecture has built sociology;
theirs was an artificial world of forms whose limits ended
in a cube whose sides were yellow, red, blue. White, grey
and black within which people sat and slept on furniture
like colored geometry and lived in houses like colored
In Holland in 1917, the ‘De Stijl’ moment developed. The
use of vertical and horizontal lines at right angles offered a
balanced relationship of unequal parts. The architect’s task
in the movement of De Stijl, was to produce metaphorical
images which foreshadow the dissolution of art into the
New Harmony do, the from was not really used as
organizational tool for geometry. Rietveld’s famous
‘Schrodler House’ in Utrecht, built 1924, is a good
example of De Stijl. It seems to demonstrate that De Stijl’s
contemporaneous manifestation on architecture could in
fact be translated into actual building.
In 1918, another movement was started in France called
'Purism'. It was to purify the application of Cubist forms to
machine iconography. Andee Ozenfant (188E1966) and
Charles Edourd Jeanneret (18871985): published Apre's le
Cubism' (after Cubism) in the same year. Cubism has
degenerated into a form of elaborate decoration so the
stress was on a pure form of art.
Le Corbusier developed a vocabulary of Purist aesthetics.
Taking his clues, painting was developed. His idealization
of platonic solids with notion of mechanization and
modernity was highly influential. Either for studying to a
general rationalization of building design processes, or to
the regulation of surface is a charesteric feature of Avant
Garde architecture. Often the general rationalization into
rectilinear geometry of the plan and section resulted in
surface regulation. It was usually beyond purely utilitarian
needs. The works of Le Corbusier once again lead the form
as the basic organizational tool for geometry.
Along with Le Corbusier and after him many architects
developed architecture with different criteria of geometry.
Aldo Rossi was the most fascinating figure of Neo
Rationalism. His forms were drawn from geometry and
Passed through the filter of history. It was an attempt of
identification, which can occur only by appealing to
consciousness and to mental image. Rossi's composition
method is based on the atonal and unexpected assemblage
of archetypes, which can be rearranged as form, which
have flowered from memory and can be used to interpret
new architectural possibilities. For him analogy is more
than a tool for pervading to images with mnemonic
reference and unconscious meanings.
This study, throughout this presentation of the
geometrical elements of form and space, has been
concerned primarily with the visual aspects of
their physical reality in architecture.
Points, moving through space and defining lines,
lines defining planes, planes defining volumes of
form and space.
Beyond these visual functions, these elements, by
their relationships to one another and the nature of
their organization, also communicate notions of
domain and place, entry and path of movement,
hierarchy and order.
Architectural forms and spaces also have
connotative meanings: associative values and
symbolic content that are subject to personal and
cultural interpretation, which can change with
Geometry, in combining forms and space in to a
single essence, not only facilitates purpose but
communicates meaningful organization of
elements can be achieved with conscious use of
geometrical ordering principles.
Geometry is used as a form as on ordering
Geometry makes our existence not only visible
but meaningful.
I had a myth that de-construction does not follow
any geometrical principle, through the case studies
of different buildings like Guggenheim museum
by frank o Gehry and House projects of Eisenman.
Through this dissertation, I have come to a
conclusion that De-construction follows hidden
geometry in their buildings.
Deconstruction is relatively a new concept, so I
was keen to know about it, through this
dissertation I have come to a conclusion that de-
construction led to the enhancement of geometry
and has taken geometry to a new level.
If we compare constructivist and de-constructivist
design we can find that only constructivist designs
follows geometric principles.
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“ARCHITECTURE Form Space and Order” by
Francis D. K. Ching.
“Architecture for the Future” by Sheila De Vallee.
“The Master Architects Series” Eisenman
“New Forms in Architecture” by Phiup Jodidio,
Publisher: Taschen.
“A + D Magazines.
“Inside Outside” Magazines.
“The Early Work of Great Architect” by Frank
Lloyed Wright.
“Le Corbusier the Creative Search” by Geoffrey
H. Baker.
“De-Construction in Architecture” by Lavinas
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... We can identify mathematical concepts (e.g., symmetries, algorithms, patterns or shapes) in various architectural constructions worldwide [1,2]. Moreover, typologies and their relationship to geometries are considered architecture features [3]. ...
Full-text available
In this paper we propose STEAM practices that would foster mathematics learning through modelling architecture while connecting to culture and history. The architectural modelling process is applied by the teachers as participants of these practices from different countries allowing a broad cultural and historical connection to mathematics education. The modelling is implemented in GeoGebra platform as it is an open-source platform to allow teachers to model on a mathematics basis. The architectural modelling process does not provide participants with steps to follow but rather allows them to explore the architectural models’ components and construct them with various approaches which may foster problem solving techniques. We aim to investigate how different phases of this approach (such as motivation, modeling, and printing process) reflect on opportunities of learning in STEAM education, with a particular lens in mathematical development from open tasks. This paper will show two use cases that took place in Upper Austria and the MENA region.
ResearchGate has not been able to resolve any references for this publication.