The relationship between geometry and architectural design are described and discussed along some examples. Geometry is the fundamental science of forms and their order. Geometric figures, forms and transformations build the material of architectural design. In the history of architecture geometric rules based on the ideas of proportions and symmetries formed fixed tools for architectural design. Proportions were analyzed in nature and found as general aesthetic categories across nature and art. Therefore proportions such as the golden section were seen as the power to create harmony in architecture as well as in art and music. According Pythagoras there were general principles for harmony. They were also applied in architecture and they found a further development especially in the renaissance. Leon Battista Alberti integrated such general harmonic proportion rules in his theory of architecture and realized them in his buildings. To find general principles of harmony in the world were the main research aims of Johannes Kepler in his "Harmonice mundi". These principles of harmony were based on geometry. Another important branch in the history of architectural design principles was the "golden section" or "divina proportione". "Modulor" of Le Corbusier is an example of an architectural design and formation concept based on the golden section. The concept of symmetry is combined with the idea of harmony and proportion. Symmetry operations are concerned with motions of figures and shapes. Geometry can be seen also as a structural science. The architectural design is based on geometric structures developed out of the idea of transformations. The symmetry transformations are visible as design concepts through history of architecture. In contemporary architecture there are no fixed rules about design concepts. But there are still relations to geometric space concepts. There is a need of new geometric background for architectural design. Examples of architecture and designing will be presented and dis- cussed in their relationship to geometry. The role of geometry in architectural design processes will be analyzed exemplarily through history of architecture and new fruitful approaches show actual and future perspectives.

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12TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2006 ISGG

6-10 AUGUST, 2006, SALVADOR, BRAZIL

Pa p e r #T35

GEOMETRY CONCEPTS IN ARCHITECTURAL DESIGN

Cornelie LEOPOLD

University of Kaiserslautern, Germany

ABSTRACT: The relationship between geometry and architectural design are described and dis-

cussed along some examples. Geometry is the fundamental science of forms and their order. Geo-

metric figures, forms and transformations build the material of architectural design. In the history of

architecture geometric rules based on the ideas of proportions and symmetries formed fixed tools

for architectural design. Proportions were analyzed in nature and found as general aesthetic catego-

ries across nature and art. Therefore proportions such as the golden section were seen as the power

to create harmony in architecture as well as in art and music. According Pythagoras there were gen-

eral principles for harmony. They were also applied in architecture and they found a further devel-

opment especially in the renaissance. Leon Battista Alberti integrated such general harmonic pro-

portion rules in his theory of architecture and realized them in his buildings. To find general princi-

ples of harmony in the world were the main research aims of Johannes Kepler in his "Harmonice

mundi". These principles of harmony were based on geometry. Another important branch in the

history of architectural design principles was the "golden section" or "divina proportione". "Modu-

lor" of Le Corbusier is an example of an architectural design and formation concept based on the

golden section. The concept of symmetry is combined with the idea of harmony and proportion.

Symmetry operations are concerned with motions of figures and shapes. Geometry can be seen also

as a structural science. The architectural design is based on geometric structures developed out of

the idea of transformations. The symmetry transformations are visible as design concepts through

history of architecture. In contemporary architecture there are no fixed rules about design concepts.

But there are still relations to geometric space concepts. There is a need of new geometric back-

ground for architectural design. Examples of architecture and designing will be presented and dis-

cussed in their relationship to geometry. The role of geometry in architectural design processes will

be analyzed exemplarily through history of architecture and new fruitful approaches show actual

and future perspectives.

Keywords: Geometric structures, harmony, proportions, architectural design.

2

1. INTRODUCTION

As the fundamental science of forms and their

order geometry contributes to the process of

composition and designing in architecture.

Composition in architecture starts with ele-

ments and their relations. Geometry is able to

make a contribution to this process by dealing

with geometric figures and forms as elements

as well as proportions, angles and transforma-

tions as relations between them. Structures

build the foundation of composing. Structures

indicate general systems of order in various

scientific disciplines, derived from the Latin

notion "structura" which means join together in

order. Mathematics can be seen as a general

science of structures by considering systems of

elements and their relations or operations. This

concept is for example the background for the

innovative approach to composition of Richard

Buckminster Fuller. "Mathematics is the sci-

ence of structure and pattern in general." [7]

Figure 1: R. B. Fuller with models, 1949

In his research he developed for example a sys-

tematic way to subdivide the sphere. His struc-

tural thinking, starting with the Platonic Solids,

led to the geodesic grids and finally his built

geodesic domes. Geometry can be seen as the

science to describe structures. Max Bill works

in his art with geometric structures as processes,

for example in his variations about a single

theme, the process from triangle to octagon.

With his variations he clarified his methods for

generating artworks.

Figure 2: Max Bill, 1935-1938, Variations

Max Bill thought about the relationship be-

tween structures and art. In his opinion rhyth-

mical order as the creative act of the artist pro-

duces an artwork starting with a general struc-

ture. Through history of geometry and archi-

tecture there were developed some rules based

on geometry which formed the basis for archi-

tectural composition. In the following we will

analyze the role of geometry in the architectural

design processes through several examples

along history of architecture.

2. HARMONY AS A PRINCIPLE OF

COMPOSITION

The notion of harmony is seen as a fundamental

principle of composition in history of architec-

ture. Composition is based on harmony and or-

der as aesthetic categories. The understanding

of harmony is based on the mythological person

"Harmonia", the goddess of harmony, who was

seen as the daughter of Ares, the god of war,

and Aphrodite, the goddess of love and beauty.

Harmonia is the symbol of the union of antago-

nisms. Harmony means the connection of dif-

ferent or opposed things to an arranged whole.

The antiquity science itself is conducted by

principles of harmony and order.

2.1 Pythagoras

In the Pythagorean approach all occurrences are

seen under a general principle. This principle

wants to be a principle of composition by un-

derstanding all processes in mathematical or-

ders. Arithmetic, geometry, astronomy and mu-

sic, the sciences of Quadrivium are all based on

this general principle. Pythagoras was con-

vinced that harmony, all things and principles

of being can be grasped by integers and

mathematical regularities. He discovered that

3

the music intervals form simple relations ac-

cording the division of the string and the num-

ber of oscillations. The Tetraktys: numbers 1 to

4 (4 elements, 4 cardinal points) form the

foundation according Pythagoras. The idea of

harmonic proportions is a general principle for

all sciences and applications.

2.2 Alberti

In reference to this antique understanding of

harmony as the union of antagonisms Leon

Battista Alberti (1404-1472) developed his

principles of architecture. "De Re Aedificato-

ria" [1] is subdivided into ten books and de-

scribes how to achieve harmony in architecture.

Beauty was for Alberti "the harmony of all

parts in relation to one another," and subse-

quently based on the Pythagorean ideas "this

concord is realized in a particular number,

proportion, and arrangement demanded by

harmony". Alberti's ideas remained the classic

treatise on architecture from the sixteenth until

the eighteenth century and even longer.

2.3 Kepler

Harmony as a concept for all sciences and the

whole world is also expressed in Johannes Ke-

pler's "Harmonices mundi". Johannes Kepler

(1571-1630) well known as scientist, astrono-

mer and mathematician based his harmony

concept on geometry, especially the Platonic

Solids. He was a Pythagorean mystic and con-

sidered mathematical relationships to be the

fundament of all nature and creations. Geomet-

rical concepts are in his theory the fundament

of nature and science as well as art and music.

Figure 3: Kepler's "Harmonices mundi"

Therefore also creation and design is based on

the geometric world concept.

2.4 Golden Section

Such a fundamental principle of harmony de-

rived from nature, applied in art, architecture

and music can be seen in the golden section.

The idea of the golden section shows the co-

herence of composition and geometry. This

idea steps longtime through history of architec-

ture. Hippasos of Metapont (450 B.C.) found it

in his research about the pentagon and the rela-

tion of its edge length and the diagonal. Euclid

(325-270 B.C.) was the first who described the

golden section precisely also as a continuous

division. In the following time golden section

was seen as the ideal proportion and the epit-

ome of esthetics and harmony. Especially in the

renaissance, harmonic proportions were based

on the geometric relations according the golden

section in art, architecture as well as in music.

Filippo Brunelleschi built Santa Maria del Fiore

in Florence 1296 based on the golden section

and the Fibonacci numbers.

The "Modulor" of Le Corbusier [5] is an ex-

ample of an architectonic concept of designing

and creating according geometric rules in mod-

ern architecture, but it remains bound to the

classical conception of harmony.

Figure 4: "Jeux de panneaux" and "Unité

d'Habitation", Le Corbusier

The structuring of the windows in Unité

d'Habitation, Marseille, 1947 (Figure 4) shows

various kinds of formations by maintaining the

same structure principle subdividing according

the golden section. A structural equivalence

between music and architecture is obvious in

4

"Ondulatoires" in the music composition "Me-

tastasis" by Xenakis and the façade of "La

Tourette" by Le Corbusier/Xenakis 1952 [11] .

In his book "Architektur und Harmonie" Paul v.

Naredi-Rainer [13] characterizes the relation

between architecture and geometry: geometry

has an important role for architecture in the

process of form finding and form development

without determining architecture exclusively.

3. SYMMETRY AND TRANSFORMA-

TIONS

Another fundamental notion in the history of

architecture is the concept of symmetry closely

connected with the idea of harmony. "Symme-

try", derived from the Greek "syn" which

means together and "metron" which means

measure, is understood as the harmony between

the parts of an object and the way of the com-

bination of several parts. Vitruv [15] described

in this comprehension "Symmetria" as an im-

portant category of architecture. In this early

understanding, symmetry describes the combi-

nation of the parts in a general way not in the

mathematical meaning.

The mathematical symmetry concept of today

was developed in the context of crystallography.

Only in the middle of the 19th century mathe-

maticians became interested in the concept of

symmetry. Later by introducing the concept of

transformation in mathematics the concept of

symmetry was changed and expanded. The

"Erlangen Programme", 1872, by Felix Klein

introduces geometry as the discipline of the in-

variants of transformation groups. Symmetry

now is understood as the invariance under a

certain kind of transformation [14].

Figure 5: Two ornaments based on the same

sequence of congruence transformations

Figure 5 shows a student work (Markus Weis-

senmayer) of creating patterns by congruence

transformations. The students had to choose a

start element and to develop a sequence of con-

gruence transformations in a storyboard. Then

the same storyboard is applied to another start

element. This simple task tries to achieve an

understanding of structural thinking.

The process of congruence transformations can

be expressed in a clear way by the operations of

folding and cutting. Folding gets more and

more important for creating processes in archi-

tecture as well as in industry. A study about the

geometric symmetry conditions of folding by a

student of architecture (Eric Pigat) [12] and an

example of industrial origami [17] are shown in

Figure 6. The mathematical comprehension of

symmetry as transformation processes only

starts nowadays to influence the creating proc-

esses in architecture.

Figure 6: Folding study and industrial origami

4. GEOMETRIC AND ARCHITECTURAL

SPACE CONCEPTS

The architectural space is based on a geometric

space concept. Especially in the creation proc-

ess architecture is thought in relation to a geo-

metric space. Robin Evans [6] analyzes the re-

lationship between geometry and architecture:

"The first place anyone looks to find the ge-

ometry in architecture is in the shape of build-

ings, then perhaps the shape of the drawings of

the buildings. These are the locations where

geometry has been, on the whole, stolid and

dormant. But geometry has been active in the

space between and the space at either end." [6,

p.xxxi].

According Evans, in history of architecture you

find this misunderstanding of the role of ge-

ometry. In his historical study he refers to the

5

relations between Gaspard Monge's Descrip-

tive Geometry and Jean-Nicolas Louis Du-

rand's theory of architecture. Durand taught

architecture at l'École Polytechnique in Paris at

the same time as Monge around 1800. Durand

developed an universal planning grid for archi-

tecture. Evans describes that Durand's grid ar-

chitecture (Figure 7) is based on the misunder-

standing of the spatial coordinate system. In-

stead of understanding the coordinate system in

an abstract way, he transformed the coordinate

planes directly in architecture as floor and

walls.

Figure 7: Durand's grid architecture, courtyards

Figure 8: Intersection of two cones by Monge

At the same time the mathematicians were able

to create curves and curved surfaces with the

help of the coordinate system.

"… while descriptive geometry encouraged the

free orientation of forms in relation to one an-

other, Durand's orthographic projection was

used to enforce the frontal and rectilinear. So

whatever the defects in Durand's teaching,

Monge's geometry can hardly be held to ac-

count for them." [6, p.327]. Also today you find

this misunderstanding on the role of geometry

in architecture if the strong forms are equated

with geometry and the organic forms are seen

in contrary to geometry. Evans states in his

book appropriately: "When architects attempt

to escape from the tyranny of geometry, mean-

ing by the tyranny of the box, where can they

escape to? Either they must give up geometry

altogether (which would be exceptionally dif-

ficult), or they escape to another, always more

complex and demanding geometry, or they do

the last while giving the impression of having

done the first, …" [6, p.331].

It is not the challenge of geometry to provide a

catalogue of eidetic forms for architecture.

Geometry rather provides the geometrical un-

derstanding of space as a background for archi-

tecture. Whereas the Euclidean geometry has its

roots in measurements and therefore corre-

sponds with the tactile space, the projective

geometry corresponds with the visual space and

refers to the perception. Evans pleads in in-

volving projective geometry in architecture. He

states that the indication of the two different

kinds of geometry "enable us to see why archi-

tectural composition is such a peculiar enter-

prise: a metric organization judged optically, it

mixes one kind of geometry with the other kind

of assessment. Perhaps this is reason enough

for the confusion surrounding it." [6, p.xxxiii]

The development of projective geometry got its

impetus from architectural demands. The ar-

chitect Brunellschi developed constructive

principles for perspectives by using geometrical

projection methods. Alberti summarized the

results of the research in perspective to a

6

teaching concept. Projective geometry origi-

nated from generalizing the use of vanishing

points and constructing perspective drawings.

Now according Evans there could be new im-

pulses for architectural design by looking for

the relationship between projection and archi-

tecture itself. The projections are operating

between things and are seen as transitive rela-

tions. The diagram in Figure 9 shows four types

of targets: designed object, orthographic pro-

jection, perspective and imagination combined

with the perception of an observer. The dia-

gram is thought by Evans as a tetrahedron, so

that the center disappears and all relations are

equitable. The routes between the targets can be

traveled in either direction.

Figure 9: Diagram of projective transactions

The diagram of Evans can be interpreted as the

sign model of the architectural design processes.

Geometry is located in the mind, architecture in

perceptible materialized reality. The diagram

classifies the sign relations between the de-

signed architectural object, the imagination and

the drawings. "Between geometry and archi-

tecture we have somehow hopped from inside

the mind to outside. So when dealing with ar-

chitectural geometry, we seem to be dealing

with this route or doorway between mental and

real." [6, p.354].

Such interactions between the described fields

will be illustrated by some examples. Perspec-

tive for example affects back to the designed

object. The classical example of an arcaded

courtyard in "Palazzo Spada" in Rome, de-

signed by Francesco Borromini 1635 (Figure

10), is an architectural trompe-l'œil in which

diminishing rows of columns and a rising floor

create the optical illusion of a long gallery.

Figure 10: Arcaded courtyard of "Palazzo

Spada", Rome, F. Borromini 1635 [18]

An actual example can be seen in the building

"Phaeno Science Center" in Wolfsburg, Ger-

many, built by Zaha Hadid in 2005 (Figure 11).

An interacting between perspective and de-

signed object can be often noticed in Zaha

Hadid's work.

Figure 11: Phaeno Science Center, Wolfsburg,

Germany, built by Zaha Hadid, 2005 [19]

7

Preston Scott Cohen [4] works in his architec-

tural design projects directly with the method of

projection. He studied along historical exam-

ples that according principles of harmony

symmetrical designed buildings get distorted by

perspective projections in order to create a vis-

ual reality. Cohen develops a method using a

"perspective apparatus" in the designing proc-

ess. "The process of projection is reversible;

perspectives may serve as objects and vice

versa." [16]. He starts for example with a per-

spective drawing of an object. This perspective

is then assumed to be an orthographic projec-

tion from which other views are later derived to

produce a third dimension and finally to create

the object.

Figure 12: Patterns for Head Start Facilities,

drawing and model, Cohen, 1994 [16]

Cohen uses the method of perspective projec-

tion to create architecture. His ideas can be lo-

cated in Evans diagram (Figure 9) between the

fields 3, 4 and 5. It is an interesting new exam-

ple of architectural design where geometry

leads the design process according the idea of

projection.

Another example for such a doorway between

mental and real may be seen in some works of

Ben van Berkel and Caroline Bos when they

apply topological thinking to architecture. In

"Move 1" they write: "Blob or box - it doesn't

matter anymore. (...) Do you like boxes? No

problem, nowadays your shoes can be packed

in a box or a bag; you can put the box in the

bag too - it's up to you." [2, p.221]. The transfer

from box to blob may be realized by topologi-

cal symmetry where the connectedness of an

object remains. In the Moebius house for ex-

ample the idea of the Moebius band leads the

design process, not in a figurative but in a con-

ceptualized sense, as they write in Move 2:

"The mathematical model of the Moebius band

is not literally transferred to the building, but is

conceptualised or thematised and can be found

in architectural ingredients, such as the light,

the staircase and the way in which people move

through the house." [2, p.43].

Figure 13: Moebius band concept

Figure 14: Moebius house, Berkel, 1993-98

The examples show various ways to refer to

geometry in the architectural processes also in

present time. It seems that there exists a fruitful

relationship between geometry and architecture

in the past as well as today although the role of

geometry is sometimes only seen as a past his-

torical one. It takes some time until geometric

8

new developments are picked up by architec-

ture even though architecture leads the way for

mathematics in the renaissance time. It starts

now that the mathematical ideas of transforma-

tion geometry, projective geometry, non-

Euclidean geometry or topology of the 19th

century find their correspondent in architecture.

Perhaps a new powerful relationship of geome-

try and architectural design begins.

5. CONCLUSIONS

The relationship between architectural design

and geometry starts with the notion of harmony

as the principle for all sciences and creations.

The analysis of the antique comprehension of

harmony shows the geometrical root and the

superior idea of this concept for all sciences

and designing disciplines. Today the various

sciences and arts are in most cases strongly

separated. Therefore there is the risk that the

powerful relationship between geometry and

architecture gets lost. Steven Holl, who refers

in his architectural work to geometry and other

sciences, noticed: "For example Johannes Ke-

pler's Mysterium Cosmographicum united art,

science, and cosmology. Today, specialization

segregates the fields; yawning gaps prohibit

potential cross-fertilization." [8].

By remembering the historical relations be-

tween geometry and architectural design we

help to keep the background of our culture but

also to understand the fruitful combination be-

tween geometrical thinking and architectural

designing. By integrating experiments on using

geometric structures for designing in the archi-

tecture curriculum we should reflect this rela-

tionship and try to develop new impulses for

geometrical based designing in architecture.

Only few examples were shown here in an

overview. There are more efforts necessary in

the future to work out this relationship in detail,

historical and theoretical, from an architectural

and a geometrical point of view as well as to

experience and apply it in the practice of archi-

tectural design.

REFERENCES

[1] Alberti, Leon Battista: The Ten Books of

Architecture. Dover Publications, New

York, 1987.

[2] Berkel, Ben van, Caroline Bos: Move. UN

Studio Amsterdam, 1999.

[3] Bill, Max: Struktur als Kunst? Kunst als

Stuktur? In: Georg Braziller: Struktur in

Kunst und Wisssenschaft. Éditions de la

Connaissance, Brüssel, 1967.

[4] Cohen, Preston Scott: Contested Symme-

tries and other predicaments in architecture.

Princeton Architectural Press, New York,

2001.

[5] Corbusier, Le: Der Modulor. Deutsche

Verlags-Anstalt, Stuttgart 1956, 7th ed.

1998.

[6] Evans, Robin: The Projective Cast. Archi-

tecture and Its Three Geometries. The MIT

Press, Cambridge, Massachusetts, 1995.

[7] Fuller, R. B..: Synergetics. 606.01.

http://www.rwgrayprojects.com/synergetics

http://www.boontwerpt.nl/images/jpg/220b

uckmfuller.jpg

[8] Holl, Steven: Parallax. Birkhaeuser Basel,

Boston, Berlin, 2000.

[9] Ivins, William M.: Art and Geometry. A

Study in Space Intuitions. Dover Publica-

tions, New York, 1964 (Reprint of 1946).

[10] Kepler, Johannes: Weltharmonik (Har-

monices mundi, 1619). R. Oldenburg Ver-

lag, München, 1997.

[11] Leopold, Cornelie: Experiments on Rela-

tions between Geometry, Architecture and

Music. Journal for Geometry and Graphics.

Volume 9, Number 2, 2005 p.169-176.

[12] Leopold, Cornelie: Geometrische

Strukturen. Exhibition of student's works at

University of Kaiserslautern, 2005.

[13] Naredi-Rainer, Paul v.: Architektur und

Harmonie. Zahl, Maß und Proportion in der

abendländischen Baukunst. Du Mont

Buchverlag, Köln, 1982.

9

[14] Scriba, Christoph J., Peter Schreiber: 5000

Jahre Geometrie. Springer Verlag, Berlin,

Heidelberg, New York, 2000.

[15] Vitruv: Zehn Bücher über Architektur. 3rd

edition, Wissenschaftliche

Buchgesellschaft, Darmstadt, 1981.

[16] www.appendx.org/issue3/cohen/index.htm

[17] www.industrialorigami.com

[18] www.wikipedia.org

[19] www.wolfsburg-citytour.de/Museen/Phaen

o_Museum_1/phaeno_museum_1.html

ABOUT THE AUTHOR

Cornelie Leopold studied Mathematics and

Philosophy. She is a lecturer and head of the

section Descriptive Geometry and Perspective

in the Department of Architecture, Regional

and Environmental Planning, Civil Engineering,

University of Kaiserslautern, Germany. Her re-

search interests are: descriptive geometry and

new media, geometrical space conceptions and

architecture, visualization of architecture, de-

velopment of spatial visualization abilities,

geometry and architectural design. She can be

reached by e-mail: leopold@rhrk.uni-kl.de, by

phone: +49-631-205-2941, by fax:

+49-631-205-4510, or through postal address:

University of Kaiserslautern / P.O. Box 3049 /

D-67653 Kaiserslautern / Germany. Website:

http://www.uni-kl.de/AG-Leopold

... Symmetry is a sense of balance, defined as a state of equilibrium of the visual weights in a composition (Leopold 2006). Symmetry (bilateral/linear and radial) and asymmetry are both equally important design tools. ...

... Rhythm in design is defined as a regular and harmonious repetition of specific patterns (Leopold 2006). Rhythm could be repetitive or progressive (see Fig. 3.4). ...

... Proportions are the relations between different dimensional elements of a form: lengths, areas, or volumes ( Fig. 3.6). According to Euclid, a ratio refers to the quantitative comparison of two similar things, while proportion refers to the equality of ratios (Leopold 2006). Thus, a proportioning system establishes a consistent set of visual relationships between the parts of a building, as well as between the building elements and entire structure. ...

This chapter presents several successful examples of biomaterial façade design. It discusses façade function from aesthetical, functional, and safety perspectives. Special focus is directed on novel concepts for adaptation and special functionalities of façades. Analysis of the structure morphologies and aesthetic impressions related to the bio-based building façades is supported with photographs collected by authors in various locations. Finally, particular adaptations and special functionalities of bio-based façades going beyond traditional building envelope concept are supported by selected case studies.

... Korelasi antara geometri arsitektur dan desain dimulai dengan gagasan harmoni sebagai prinsip untuk semua ilmu dan ciptaan. Analisis pemahaman harmoni menunjukkan akar geometris dan gagasan unggul konsep ini untuk desain dan ilmu pengetahuan (Leopold, 2006). Estetika arsitektur adalah sistem kriteria koheren yang bersifat formal dan simbolis pada saat yang sama dan hal-hal formal berkaitan dengan pertanyaan proporsi, harmoni dan kontras, dan lain lain (Sotoudeh dan Abdullah, 2012). ...

Malang is well-known for colonial buildings. Visual quality of historic buildings in the Kayutangan corridor makes it an icon of Malang City. Assessment of visual quality is affected by daytime conditions. Day lighting are factors that influenced the visual quality assessment of historic buildings. This study meant to assess the visual quality of historic buildings and aspects that influence by society during the day. This study used a descriptive quantitative method explaining public perception about the visual quality of historical buildings in Kayutangan street corridors during the day. Semantic Differential Scale (SD) was the instrument to describe the respondents' perceptions (positive and negative ones). From the result showed that visual quality of 1 of 10 historic buildings in Kayutangan was below the average scores and the most influential variables by society with day lighting in the historic building.

  • Prafulla Parlewar Prafulla Parlewar

Geometric patterns in interior and exterior of building provides an appealing aesthetic. The chapter here illustrates a project for rehabilitation of the cafeteria in an administrative building. In this project the rehabilitation of old interior was undertaken through use of innovative aesthetic consideration of design. The research here investigates how geometric patterns can be applied in the architectural design process? What is the process of rehabilitation in small spaces in buildings? How materials are designed in rehabilitation of interior spaces? What are the architectural considerations for design of interiors? How structural criteria are assessed before rehabilitation of old interiors? Indeed, it is important to combine aesthetic and structural knowledge in rehabilitation. So, the research here illustrates an application of hexagonal patterns in the architectural design process for rehabilitation of old interiors.

  • Paola Casu Paola Casu

The Library of the Department of Architecture of the University of Cagliari houses the drawings of 41 graduation theses of Architecture that had been debated during the second half of the 19th century. These drawings are the results of a three-year long course of "Architecture, Drawing and Ornate" that was taught in Cagliari by Prof. Arch. Gaetano Cima (1805-1878). From the analysis of these drawings it is possible to make some remarks both of historic and methodological aspects. It is very interesting how the students carried out the architectural composition using geometric constructions. The marks of the geometric constructions are yet visible on many of the drawings. In the work presented here, these tiny signs are used as instruments for reconstructing, using the accuracy of CAD, the outlining process. Drawings were sorted according to the rules of construction found for each architectonic composition: golden section, dynamic symmetry (root 2 rectangle), squares and triangles, etc. The classification of the graphic analysis was then related to the note of Prof. Cima and the books he made use of to teach architecture, in order to value affinity and differences. The aim of the work is to regain Cima's teaching method from the graphic results of his students and to establish a method that could be applied to similar cases where architectural drawings are the only documents to regain the outline of a building.

  • Robin Evans

Winner of the 1997 Alice Davis Hitchcock MedallionAnyone reviewing thehistory of architectural theory, Robin Evans observes, would have to conclude thatarchitects do not produce geometry, but rather consume it. In this long-awaitedbook, completed shortly before its author's death, Evans recasts the idea of therelationship between geometry and architecture, drawing on mathematics, engineering, art history, and aesthetics to uncover processes in the imagining and realizing ofarchitectural form. He shows that geometry does not always play a stolid and dormantrole but, in fact, may be an active agent in the links between thinking andimagination, imagination and drawing, drawing and building. He suggests a theory ofarchitecture that is based on the many transactions between architecture andgeometry as evidenced in individual buildings, largely in Europe, from the fifteenthto the twentieth century.From the Henry VII chapel at Westminster Abbey to LeCorbusier's Ronchamp, from Raphael's S. Eligio and the work of Piero della Francescaand Philibert Delorme to Guarino Guarini and the painters of cubism, Evans exploresthe geometries involved, asking whether they are in fact the stable underpinnings ofthe creative, intuitive, or rhetorical aspects of architecture. In particular heconcentrates on the history of architectural projection, the geometry of vision thathas become an internalized and pervasive pictorial method of construction and that, until now, has played only a small part in the development of architecturaltheory.Evans describes the ambivalent role that pictures play in architecture andurges resistance to the idea that pictures provide all that architects need, suggesting that there is much more within the scope of the architect's vision of aproject than what can be drawn. He defines the different fields of projectivetransmission that concern architecture, and investigates the ambiguities ofprojection and the interaction of imagination with projection and itsmetaphors.

  • Cornelie Leopold Cornelie Leopold

This paper presents some results of an interdisciplinary project where we brought together geometry, architecture and music. This combination enables a visual and audible approach to the formal thinking of sciences. Geometry has the role of formalization and mediation of the relations between architecture and music. Pythagoras' ideas about harmony and proportion impressed the forma-tion processes in music over many centuries. In architecture Le Corbusier and Xenakis for example stand for corresponding creations in architecture and music compositions. In the interdisciplinary project we worked together with students of architecture and mathematics of the University of Kaiserslautern and with stu-dents of music composition of the Music Academy of Cologne in Germany. The theoretical and historical analyses formed the basis for their own creative works in interdisciplinary groups. The results of the students' creative works were pre-sented in a concert with visual presentations and geometrical objects with the title "Sound-Sights. Seeing Music – Hearing Geometry". Some examples of the student projects will give an impression of the interdisciplinary experiments.

Der Modulor. Deutsche Verlags-Anstalt

  • Le Corbusier

Corbusier, Le: Der Modulor. Deutsche Verlags-Anstalt, Stuttgart 1956, 7 th ed. 1998.

The Ten Books of Architecture

  • Leon Alberti
  • Battista

Alberti, Leon Battista: The Ten Books of Architecture. Dover Publications, New York, 1987.

Weltharmonik (Harmonices mundi, 1619). R. Oldenburg Verlag

  • Johannes Kepler

Kepler, Johannes: Weltharmonik (Harmonices mundi, 1619). R. Oldenburg Verlag, München, 1997.