Mathematics for Architecture: Some European Experiences
Dipartimento di Scienze per l'Architettura
Stradone S. Agostino, 32 - 16123 Genoa, Italy
Which and how much mathematics for architecture?
What kind of teaching? Is it better to have information on many
aspects, or a deeper insight into only a few? Is it better to
privilege a historical-philosophical overview, or to focus aspects
of application that are current today? These are some of the
questions that are raised by whoever deals with teaching mathematics
in a faculty or school of architecture. The problem is made more
acute by a final question: what do we mean by "architect"?
Rivers of words and paper have poured out in the attempt to define
this term. The European Economic Community (EEC) directive EEC
384/85 (see Appendix 1) has provided
some indications in this direction, but the means to achieve
the objectives set forth there can be variously interpreted and
understood; the debate continues.
What is meant by architecture? What are the possible future
"professions" of an architectural graduate? What effect
will the new world scene currently being shaped have? And further,
is it better to prepare for one profession (which is revealing
itself to be multifacted, as has been shown by studies on the
occupations of graduates), or to provide a methodological-cultural
background that allows for future flexibility? The problem is
further aggravated by the restructuring of university courses
of study that are taking place on the European level and in the
Italian universities as well.
It is from these observations that arose my own study on the
teaching of mathematics in European faculties or schools of architecture;
a study that I have not yet terminated but that, for now, has
taken into consideration only Belgium, France, Portugal, Spain
and Switzerland. The paper that is follows does not therefore
pretend to be exhaustive, not even in terms of the individual
countries presented. It is only the fruit of some considerations,
on the basis of conversations with educators and the consultation
of articles, texts, dossier, course outlines, student handbooks
and webpages of some of the schools or faculties of the countries
mentioned above. Of special merit has been the consultation of
Architetti in Europa. Formazione e professione.
The formation of architects is undertaken
by both the ISAI, Instituts Superieurs d'Architetture Intercommunals,
(considered since 1991, in compliance with EEC directive 384/85,
"institutes of superior instruction of long duration"
at the university level, dependents of the Ministry of Education,
Scientific Research and Formation) and by the Technical Universities.
The course of study lasts for five years, divided into two cycles.
The first, of two years, provides the qualification for the title
of "Kandidaat architect" (architect candidate), which
allows an initial insertion into the professional world. The
second, three-year, cycle leads to the title of "architecte/architect"
or "burgerlijk ingenieur". In all architectural schools
the first cycle is is primarily constituted of obligatory basic
subjects; the second aims at specializations, with each school
offering different options. There is no limit on the number of
entering students and for inscription it is necessary to possess
a diploma from a secondary school.
The Institut Superieur d'Architecture de la Communité
Française La Cambre differs from other schools in that
it is less technical and more focussed on design (and thus closer
to the French schools). In any case the first cycle offers a
course in mathematics in which the first part teaches elements
of differential and integral calculus and differential equations.
In the second part "laws of harmony" are taken into
consideration so that beginning with the history of mathematics
the close relationship between architecture and mathematics are
evidenced, through the science of proportion. The first cycle
also includes a course in descriptive geometry and applications
that comprise studies of solids and surfaces. A great deal of
attention is given to computer science and its applications,
one course being offered in the first cycle and two courses in
the second. These courses not only concentrate on design and
modelling, but also on the study of the properties of complex
In other ISAI are found, in varying degrees, teaching modules
of differential and integral calculus, probability and statistics,
geometry of curves and topology, computer science (for example,
at ISAI-Horta there is a total of 150 hours of these, while at
the ISAI-Mons there are 240 hours) and descriptive geometry (150
or 120 hours). These courses are generally localized in the first
cycle, but sometimes there are modules of computer science and
statistics in the second cycle as well (as for example in the
In the universities the bienniumis usually common to the course
of study of engineering. For the trieenium there is the option
to choose between a more technical formation or one more strictly
tied to architecture. Inscription is free, but it is necessary
to pass an entrance examination in which the exams in mathematics,
regarding trigonometry, algegra, analysis, synthetic and analytic
geometry are decisive. In order to pass the examination, usually
it is necessary to have some reinforcement of the mathematical
studies of the last year of secondary school. In the formation
of the architect by the university, the teaching of mathematics
and computer science is notably present with advanced courses
even in the second cycle.
The formation and research in the field
of architecture in France appertain to the Ministry of Culture
and Communication. The title of architect in France is conferred
by either the "diplome d'architecte diplomé par le
gouvernament" (DPLG) awarded by the twenty-two Ecoles d'Archiecture,
or by the "diplome d'architecte de l'ENSAIS" awarded
by the Ecole National Superieur d'Architecture of Strasburg or
by the "diplome d'architecte DESA", awarded by the
Ecole Speciale d'Architecture of Paris, a private institution.
The Ecoles d'Architecture originated from the Ecoles des Beaux
Arts and some of the French schools are still tied to the method
and the type of education of the Beaux Arts, in the sense that
the greatest emphasis is placed on the artistic formation of
the student. Reform was begun in the academic year 98/99 that
provides for a course of six years' duration subdivided into
three biennial cycles. At the end of the second cycle a first
diploma is conferred, the DEUG, which allows for admission to
the university. The reform confirms the centrality of architectural
design, and to a lesser degree (in the first two cycles), urban
design; in the third cycle courses specializing in urban design
and computer science for architecture are offered. In the first
two cycles, teaching is articulated through interdisciplinary
modules and is based on the relations between "architecture
and knowledges for architecture".
The French situation regarding mathematics education is, among
all the countries in this study, perhaps the most distant from
the Italian. The reason for this much be sought in the origin
of the schools from Beaux Arts and therefore in their having
privileged the artistic formation of the students up until recent
years. It can be seen then how the teaching of differential or
integral calculus would be reduced to a minimum if not completely
nonexistent, while placing more emphasis on descriptive and projective
geometry, the study of solids and surfaces, topology and on their
use for morphogenetric studies and as a theoretical basis for
the construction of models (see, for instance,  and  in
There are integrated courses in geometry, computer science, construction
of models, which may be accounted for by the fact that the study
of the relationships between architecture and the computer sciences
is a central subject for some of the French research centers.
The debate on what should be taught in a school of architecture
is still a very lively one in France (see ), but it is agreed
that in order to maintain architectural design as a central and
fundamental discipline, it is necessary to integrate studies
of "knowing and knowing how" through ateliers either
within the school or coordinated between various schools (Grands
Ateliers de l'Isle-d'Abeau); in these cases as well mathematics
has something to contribute (, ).
With the activation of the new reforms projects for urban design
has been given more attention, and therefore courses of statistics
and geographical information systems have been introduced.
It is only since 1983 that the education
of architects is no longer entrusted to the Academies of Fine
Arts, but rather to the faculties or departments specifically
constituted for that purpose. Nevertheless, there has been a
split between the technical-constructive emphasis of some faculties
and the artistic-humanistic emphasis of others. The debate has
been quite heated, arriving finally at the European Consulting
Committee. Actually, thanks to a more balanced composition of
a program of study, the conflict seems to have been resolved.
In order to for inscription in a faculty of architecture, it
is necessary to have a diploma from a secondary school and to
pass a national examination for admission to the university in
the specific subjects that are required by individual faculties.
Generally, for architecture these subjects are mathematics, descriptive
geometry and art history. The courses last six years at the faculty
of Lisbon and Porto, and five years at Coimbra and the private
universities of Lusiada. These courses of study are not subdivided
into cycles and it is only in the fifth year that various options
are made available.
The profession of architect is only open
to those who have a university diploma of a superior degree conferred
by the "Escuela Tecnica Superior de Arquitecture".
The five-year courses, subdivided into two cyles, that lead to
award of the title of architect are supplemented, in almost all
universities, by shorter three-year courses that lead to the
qualification of "Arquitecto tecnico".
It is possible to vary the education through the selection of
optional courses that correspond to approximately 10% of the
total hours. The rest of the hours are divided between fundamental
obligatory subjects that are set by directives of the Ministry,
and other obligatory subjects that are set by the each school
on the basis of its own specific characteristics.
In Spain, the authority attributed to architects goes decidedly
beyond that set by the European directives, and because of this
some universities lament the six-year course, offering and preparing
a continuing education that is superior, from a scientific point
of view as well as that of duration, to that of other countries.
For admission to the university, once secondary school has been
completed it is necessary to attend a one-year Course in University
Orientation with a final examination (Selectividad); the score
obtained determines admission to a university course of study
that accepts only a limited number of students, such as the School
As far as mathematics education is concerned, the situation is
rather similar to the Italian. Mathematics is present in the
first cycle, but in the third year as well in some schools. In
schools in which lesser emphasis is placed on mathematics, greater
emphasis is placed on information sciences and applications,
with ample space given to the geometric aspects of application.
The arguments presented, in various proportion, are: infinitesimal
calculus, integral calculus, differential equations, numeric
methods, problems of maximum and minimum constraints, elements
of statistics, geometry, linear algebra and descriptive geometry.
Within courses of geometry and algebra are sometimes included
proportional systems applied to architecture, isometrics and
tiling. The subjects cited are certainly not new; the problem
is the degree to which they are examined, their connections to
other courses in architecture, and an engaging presentation.
In this sense the formulation given in L'art de calcular en l'arquitectura
 seems to me to be particularly interesting. Appendix
2 presents the outline of the book.
The situation in Switzerland is unique:
the eduction traditionally provided for architects is strong
technically. In order to design and construct buildings, only
a diploma from a secondary technical school is necessary; but
only those who have been awarded a university degree have the
right to the title of architect. In any case, a reform is underway
that seeks to adjust the architect's professional preparation
in conformation with EEC 384/85, even though Switzerland is not
part of the European Union. Thus, for example, the Swiss Fachhochschulen
are modifying their systems of instruction so that they are comparable
to the German Fachhochschulen.
Besides the superior technical schools, in modification, the
degree in architecture can be conferred by the universities or
the polytechnical institutions. In all but the University of
Italian Switzerland, courses last four years, subdivided into
two biennial cycles, with an extra obligatory year of apprenticeship.
At the Polytecnical of Zürich (ETH), the institution of
a year-long course of study of fundamental subjects that are
either not presently taught or not sufficiently taught at the
secondary level is being considered. The University of Geneva
offers only the second cycle, and the first cycle must be taken
either at the École Polytecnique Féd. De Lausanne
(EPFL) or athe the ETH. Geneva and Lausanne have sought, in the
second cycle, to vary their specializations, Lausanne maintaining
a technical/engineering specialty (besides a specialization in
history). At the EPFL mathematics education is given both during
both semesters of the first year and both semesters of the second
year, with special attention paid to geometry (descriptive, the
study of curves and surfaces, proportions, the Golden Section,
the Modulor and tiling) and graph theory. Also fundamental is
a semester of computer science and two semesters of "informatique
et dessin", preparatory to a course of "modelisation
informatique" of the second cycle.
The University of Italian Switzerland merits a discussion of
its own, where the diploma is awarded by the Academy established
in 1995; the emphasis is purposely humanistic and less technical
with respect to the other Swiss degree programs. Particular attention
is given to interdisciplinarianism, to the fundamentality of
design (intended in the broadest sense) and to the necessity
of finding an common approach to other historic and scientific
disciplines. The organization is unique (see ). The course
lasts 6 years.
As regards mathematics education there is an introductory course
in the first year, followed by these advanced courses following
years: "Representation of forms", "Geometric forms
for visualization", "Mathematic structures in architectural
design", "Mathematics in the history of architecture",
"Ecology", "Structures", etc. These are conceived
as differentiated areas of study of a single field that are expected
to "nourish" the complex process of design; the requirement
for interdisciplinarianism is very strong.
The experiment is underway; the program is very stimulating.
It is worthwhile to follow its development carefully, although
the logistical situation of the Academy is so favorable that
could be followed only with difficulty in our own schools and
would be complicated to implement.
From this study I have formed the following
-- that descriptive geometry, analyzed in its historical development
and linked to stereometry and stereotomy, is a good point of
intersection between the history of architecture, mathematics
and the science of construction (see ). If it is united with
the construction of models (for the preliminary design of which
is indispensable a certain geometric capacity), the ability of
the student to invent forms may be developed.
-- the space dedicated to the interaction of computer sciences
(geometry of forms) and architecture must be enlarged, as it
is already in some other countries, and that, given the strong
urban design component of our system of study, courses of geographical
information systems need to be provided.
It is important that the students acquire a use, or better,
a "design" for intelligent use of the methods and instruments
of computer science.
The central problem remains: what kind of education should
an architect have? I share the opinion expressed by the director
of the Academy of Meudrino during the presentation of a course
of study, that the future architect must "know how to design
a project, but a project of ideas, spaces, materials, forces,
light, etc.; all designs, but not only drawings".
Design is, however, complex and requires knowing how to conduct
an interdisciplinary activity, understanding in some cases the
insufficiency of one's own knowledge and knowing therefore how
to maintain a dialogue with experts. This does not mean merely
asking an expert for the answer to a problem, as often happens,
but knowing how to participate in formulating an answer, that
is, knowing how to exchange ideas. It is therefore towards the
construction of a common language that our attention ought to
be focussed, bringing to light the possible connections between
the knowledge of our own discipline and that of others that are
useful for creating a design, but the participation must be complex,
not unilateral. It is a difficult problem; to me it still seems
an uphill battle.
1: EEC Directive 384/85
According to EEC Directive 384/85, studies
at the university level that lead to the title of architect must
be divided in a balanced manner between theoretical and practical
aspects of the education of the architect and assure that following
objectives are achieved:
1. The capacity to create architectural designs
that satisfy aesthetic and technical requirements;
2. An adequate knowledge of the history and
theory of architecture, in addition to the arts, technologies
and human sciences pertinent to it;
3. A knowledge of the fine arts in as much
as they influence the quality of the architectural conception;
4. An adequate knowledge of urban design and
planning and the techniques involved in the planning process;
5. The capacity to grasp the relationships
between man and architectural creations and between architectural
creations and their environments, as well as the capacity to
grasp the necessity of revising architectural creations and spaces
in accordance with human needs;
6. The capacity to understand the importance
of the profession and of the function of the architect in society,
particularly in elaborating designs that respond to social factors;
7. The knowledge of methods of enquiry and
the preparation of a construction project;
8. The knowledge of problems of a structural
nature, and of construction and civil engineering connected to
the design of buildings;
9. An adequate knowledge of physical and technological
problems, as well as of building functions, in order to render
them comfortable and to protect them for climatic factors;
10. A technical capacity that permits the
design of buildings that respond to the needs of the building
users within the constraints imposed by factors of cost and construction
11. An adequate knowledge of the industries,
organizations, regulations and procedures necessary to realize
designs of builds and for the integrations of planning.
2: Preface to L'art de calcular en l'arquitectura 
A grans trets, podríem classificar
els càlculs dels arquitectes en els tipus següents:
a) Càlculs constructius
Són els càlculs inherents a l'edificació
en sentit estricte: la representació topogràfica
del terreny, l'estudi de la mecànica del sòl, fonamentacions,
moviments de terres, etc. fins arribar a la construcció
efectiva de l'obra i el seu control de qualitat.
b) Càlculs estructurals
Són els propis de l'estructura de l'edificació
i asseguren per sobre de tot la rigidesa de l'obra. En una subtil
combinació de conceptes de mecànica, resistència
de materials, equacions diferencials, càlculs de moments,
torsions, flexions, etc., és possible crear aquesta estructura
que sovint apareix disimulada i maquillada per altres elements,
però sense la qual res no romandria dret.
c) Càlculs de condicionament i serveis
Integrats a l'edifici, hom troba un món complex d'elements
elèctrics, mecànics, acústics, lumínics,
calorífics, etc. Cal fer càlculs relatius a la
integració en la construcció i càlculs sobre
el funcionament específic dels elements en qüestió.
Matemàtica, física i enginyeria troben aquí
un bon camp per fer-hi aportacions.
d) Càlculs projectuals
El projecto, com a element vertebrador de l'obra, ha de tenir
en compte necessàriament la integració de totes
les components i d'axiò deriven sovint càlculs
específics: pilars, canonades, esteses de cables, envans,
endolls, ascensors, etc. podrien esdevenir una barreja esperpèntica
si no hi hagués un disseny global de l'obra.
e) Càlculs gràfics
Les tècniques d'expressió gràfica contenen,
de fet, un bon gruix de càlculs que acaben permetent la
resolució gràfica dels problemes. Marcar un punt
de fuga, distingir les escales convenients per presentar els
diferents elements o fer palesa la forma d'una volta o d'una
escala de cargol pressuposen un joc geomètric fi, no mancat
ni de mesura ni d'altres components matemàtiques.
f) Càlculs legals
Les obres són realitzades en un lloc precís tenint
en compte una normativa legal que en fixa limitacions molt diverses.
Calcular fondàries edificables, alçàries,
patis de llum, ventilacions mínimes, plans d'evacuació,
resistències al foc, etc. són problemes difícils
de resoldre, però inexcusables.
g) Càlcils de planificació
La realització efectiva d'un projecte sempre porta aparellada
una bona planificació respecte dels diferents equips i
professionals que hi intervenen, una regulació temporal
imprescindible, i un càlcul econòmic acurat que
faci l'obra viable i, si pot ser, rendible (!). Organigrames,
grafs, sistemes d'organització, càlculs financers,
càlculs actuarials, etc., són el nostre pa de cada
dia; i càlculs d'assegurances per allò del «per
si de cas».
 Claudi ALSINA i CATALÀ, L'art de calcular
en l'arquitectura, Edicions UPC, Universitat Politecnica
de Catalunya, 1993.
 Jean-Marie DELARUE, "Structures
gonflables", Bilan de l'Atelier à l'Isle d'Abeau
en octobre 1997, EAPV Bulletin d'information de l'Ecole d'Architecture
Paris-Villemin, n. 29, 1998.
 Jean-Marie DELARUE, Morphogénèse, Paris,
UPA, n. 1.
 Roberto MASIERO, Michela MAGUOLO e Vittoria POLESE, (editors),
Architetti in Europa. Formazione e professione, Dossier
di una ricerca finanziata dal "Jacques Delors Research Grant
within the European Culture", dell'Accademia di Yuste con
il sostegno della Comm. Europea (in print).
 Joël SAKAROVITCH, "Architecture et représentation.
Géométrie descriptives et Stéréotomie",
EAPV, Bulletin d'information de l'Ecole d'Architecture Paris-Villemin,
n. 29, 1998.
 Joël SAKAROVITCH, Épures d'architecture,
Birkhäuser Verlag, 1998.
 Jean-Louis VIOLEAU (ed.), Quel enseignement pour l'architecture?,
Editions Recherches - École d'architecture Paris-Belleville,
ON THE WWW
ABOUT THE AUTHOR
Associate Professor, teaches mathematics in the Faculty of Architecture
of the University of Genoa and at the Scuola di Specializzazione
in Restauro dei Monumenti (School for Specialization in Restoration
of Monuments). Her past research interests were tied to functional
analysis, while presently her research is in a) the relationships
between mathematics, art and architecture in their historical
development; b) mathematical methods in urban design and geographical
information systems. She is also in didactic and educational
research. She is a member of the Italian Commission of UNESCO.
The correct citation for
this article is:
Pedemonte, "Mathematics for Architecture: Some European
Experiences", Nexus Network Journal, vol. 3, no.
1 (Winter 2001), http://www.nexusjournal.com/Didactics-Pedemonte-en.html
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