Mathland: The role of Mathematics in Virtual Architecture Michele Emmer
(English version) (Autumn 2005)
This paper is dedicated to some arguments that could be of interest both for students and practicing architects. A short adventure in the reign of mathematics and culture.The example that I have chosen is that of the idea of space, how this idea and the perception of space around us has changed up to the point where it has arrived to the form of virtual architecture.

Mathland: Il ruolo della cultura matematica nell'architettura virtuale Michele Emmer
(versione italiana) (Autumn 2005)
L'articolo parla di argomenti che possono interessare sia gli studenti sia gli architetti. Una breve avventura nel regno dell matematica e cultura. L'esempio che ho scelto è quello dell'idea dello spazio, come quest'idea e la percezione dello spazio intorno a noi è cambiata fino ad arrivare alle forme della architettura virtuale.

Università Roma Tre 1995-2005: Architecture and Mathematics Andrea Pagano and Laura Tedeschini Lalli
(English version) (Autumn 2005)
Andrea Pagano and Laura Tedeschini Lalli present a brief note on what has to be considered a teaching success story. They tell of ten years of teaching mathematics at the Faculty of Architecture of the Università di Roma Tre, describing some new course content that has been introduced, the methods used and, above all, the spirit that has driven the ideas about teaching mathematics to future architects. The experience covers the full curriculum of the student: from first year courses to individual projects.

Università Roma Tre 1995-2005: Architettura e Matematica Andrea Pagano e Laura Tedeschini Lalli
(versione italiana) (Autumn 2005)
Nel presente lavoro Andrea Pagano e Laura Tedeschini Lalli presentano una breve nota su quella che è da considerare una storia di successo. Racconta di dieci anni di insegnamento presso la Facoltà di Architettura dell'Università' di Roma Tre. Si descrivono i nuovi contenuti introdotti nei corsi, le metodologie di insegnamento e soprattutto lo spirito che ha guidato le nostre idee su cosa e come insegnare la matematica a futuri architetti. L'esperienza descritta copre l'intero percorso degli studenti: dai corsi di primo anno ai progetti individuali.

Architecture for Mathematics: The School of Mathematics at Città Universitaria in Rome Claudio Presta
(English version) (Autumn 2005)
One obvious aspect of the relationship between architecture and mathematics is the nature of the actual places designed by architects for the mathematical community. The School of Mathematics of the Città Universitaria designed by Giò Ponti. Claudio Presta examines Ponti's design to see how architecture can create an appropriate space for doing mathematics. What is the meaning, if there is one? What are the motives that drove the architect to create a building expressly for the School of Mathematics and not a simple/complex building in which the School could simply be housed? Ponti provides an answer in following the shape of the Greek theater for one part of the building, while the dimensions of the other part tend towards the Golden Section.

Architettura per la matematica: la Scuola di Matematica a Città Universitaria di Roma Claudio Presta
(versione italiana) (Autumn 2005)
Nel rapporto tra Architettura e Matematica c'è anche, ovviamente, la concretezza di luoghi progettati per la vita della comunità matematica. La Scuola di Matematica di Città Universitaria è stato progettato da Gio' Ponti. Claudio Presta prende in esaminazione il progetto di Ponti per vedere come l'architettura crea degli spazi addatti alla matematica. Quale è il senso se ce n'è uno? Quali sono le spinte su cui lavora l'architetto nella finalità di creare l'edificio di Matematica alla Sapienza, e non una semplice/complessa costruzione in cui poi "sistemare" la Scuola? Una delle risposte di Ponti si trova nella forma del teatro greco. Un'altra risposta si trova nelle dimensione che tendono alla sezione aurea.

The archKIDecture Build IT! Exhibit Project Julie Cowan (Spring 2005)
archKIDecture is an independent architecture education project that encourages children to explore and participate in the built environment. The archKIDecture Build IT! exhibit teaches children the vocabulary of building so that they can build and interact with their built environment in a personal way that reflects and empowers them as individuals. It is also an accessible and practical context for exploring mathematical concepts such as tessellations, ornament design, symmetry, scale, proportion, and composition in a tangible and stimulating way.

From Natural Forms to Models Mari Alati, Liliana Curcio, Roberto Di Martino, Lino Gerosa, Cinzia Tresoldi
(English version) (Spring 2005)
The course presented at the Istituto Statale d'arte, a high school for visual and design arts, is a guided tour of the world of forms of seashells. The main goal of this course is to present to the student prevalently scientific methods of interpretation of forms.

Dalle forme naturali ai modelli Mari Alati, Liliana Curcio, Roberto Di Martino, Lino Gerosa, Cinzia Tresoldi
(versione italiana) (Spring 2005)
Il percorso didattico di cui tratteremo ci propone un viaggio guidato nel mondo della forma delle conchiglie, condotto all'interno della didattica di una scuola di design. La finalità primaria di questo lavoro è quella di proporre agli studenti metodi di lettura della forma a carattere prevalentemente scientifico.

Designing a Problem-Based Learning Course of Mathematics for Architects Francisco Javier Delgado Cepeda (Spring 2005)
In the past nine years, the teaching model of the Instituto Tecnológico y de Estudios Superiores de Monterrey has rapidly evolved, taking into account the development of abilities, attitudes and values without forgetting the development of knowledge. The mathematics for architecture course was redesigned, using problem-based learning and an intensive application of computer technology to overcoming those difficulties. Now, the main purpose is to develop a mathematical, physical and technological culture in students of architecture to allow them to analyze and solve complex problems related to mathematics in architecture and design. The course was planned and implemented for the first semester of the architecture program and is actually related (through curriculum integration) to future courses which require specific mathematical applications.

The Geometry of Frank Lloyd Wright Linda Keane and Mark Keane (Spring 2005)
Mark and Linda Keane describe a seminar that seeks to answer these questions with evidence of a renaissance of work in the twenty-first century that emanates or owes allegiance to mathematical explorations configured in Wright's body of work. This seminar, The Geometry of Wright, offers students in the state of Wisconsin the opportunity to learn about Wright's life, those who influenced him, and those whom he influenced. The combination of history, theory, mathematics, and design activities in this seminar offer students an opportunity to become aware of Wright's use of geometry, understand its roots and precedents, and apply them to a project of their own. This whole language approach to learning embeds appreciation of mathematic principles and encourages students to apply geometric relationships in their own search for proportion and form.

Reconstruction of Forms through Linear Algebra Elena Marchetti and Luisa Rossi Costa (Spring 2005)
Lines and surfaces are boundary elements of objects and buildings: it is very important to give the students a mathematical approach to them. Elena Marchetti and Luisa Rossi Costa present linear algebra (by vectors and matrices) as an elegant and synthetic method, not only for the description but also for the virtual reconstruction of shapes.The aim of our activity is to facilitate -- and, at the same time, to develop -- the comprehension of crucial mathematical tools involved in the realization of forms and shapes in arts, architecture and industrial design and in computer graphics. Another important aspect of linear algebra to be pointed out to the students is its application in graphics software packages, which work with transformations that change the position, orientation and size of objects in a drawing.

Methods for Evaluation in Mathematics for Architecture and Design Hernán Nottoli (Spring 2005)
In mathematical teaching, there exists a dichotomy between two entities. On the one hand, there are the methodologies for imparting mathematical knowledge; on the other hand there are different mechanisms for evaluating students. Hernán Nottoli analyses some aspects that we consider relevant with respect to how knowledge is verified and ranked in evaluation tests in the case of mathematics in architecture and design schools, provides statistical evidence of experiences in the Faculty of Architecture, Urbanism and Design at the University of Buenos Aires, and provides an example of the kind of exerise that has been used with success with his own students.

The Education of the Classical Architect from Plato to Vitruvius Graham Pont (Spring 2005)
Plato divided science (episteme) into 'science of action' (praktike) and 'science of mere knowing' (gnostike). His argument is the first known attempt to distinguish what is now recognised as technology, as distinct from more purely rational science. Aristotle coined the compound term technologia and thereby established this new department of science within the general system of knowledge. Plato did not develop his novel characterisation of the architect any further, for the ancient Greeks did not consider architecture a fine or estimable art. The best available source of Greek architectural pedagogy is the Roman Vitruvius. Graham Pont discusses Vitruvius's distinction between the 'practical' side of architecture (fabrica) and the 'theoretical' (ratiocinatio), and examines the mathematical preparation of ancient architects.

Teaching Geometry to Artists J.M. Rees (Spring 2005)
JM Rees discusses his experience teaching geometry to artists. The aim is to introduce scientific ideas to arts students through the visualizations that are such an important part of discourse in science. I describe the intellectual context, define selected concepts using geometry (classically, a liberal art) and introduce elementary mathematical formulae--all relying on graphic visualizations to make fundamental ideas clear. My goal is to provide a means by which visually sophisticated persons may think with geometry about culture.

Euler's Theorem as the Path towards Geometry Emil Saucan (Spring 2005)
The course in Mathematics for Architecture Students at the Technion - Israel Institute of Technology needed to encapsulate as much formative knowledge as possible. Above and beyond the absolute importance of Euler's Formula relative to the corpus of classical mathematics and the role it played in its development (as in Betti numbers and Homology in general on one hand and the Global Gauss-Bonnet Theorem on the other hand), its simplicity, yet potency (in the sense of representing an jumping board, an opening towards a variety of subjects belonging to the fields of Topology and Geometry) recommend Euler's Theorem as natural candidate for a cornerstone, a red thread running along and directing the whole course.

Teaching Mathematics in Architecture Arzu Gonenc Sorguc (Spring 2005)
The Department of Architecture of Middle East Technical University offers a course entitled 'Mathematics in Architecture' for the third year students. In the beginning of the term, students are forced to imagine themselves as 2-dimensional creatures living in a 2-dimensional space. At this point, fundamentals of architectural geometry are introduced first in the plane, simply by employing the set concept; mapping as a general tool is then introduced and students are asked to use mapping in their design to correlate the project requirements and geometry. Following that, the principles of isometries and isometric constructions are introduced. In the second part of the term, students are allowed to think in terms of 3-dimensional space and topics related with the principles of similarities and proportions and symmetry are presented. In the final part of the course, students are forced to think themselves as 3-Dimensional creatures living in a 4-Dimensional space and this fourth dimension is sought. The last topic of the course is related with biomimicry in architecture and mathematics inherent in bio-forms and man-made structures.

Leo: a Multimedia Tale of Structural Mechanics Nicola Luigi Rizzi and Valerio Varano (Spring 2005)
The authors have devised a method for teaching structural mechanics that articulated in the following three phases: observations (the description of mechanical phenomena, increasingly complex, selected with regards to their pertinence of the problem that one wants to affront, and their efficiency); modeling (the construction of a physical-mathematical model that takes into account its formal content and stresses its importance as an instrument and has the potential for other applications); design (suggestion of cues for applications stimulate the student to exercise his creative imitation). What is proposed to the student is not so much a set of notions, as a method and set of instruments for selecting experiences (for example previous design solutions) to the end of evaluating their repeatability in diverse situations, by means of a physical-mechanical reading which comes from phenomena which one finds in daily life. "Leo" was created as a teaching instrument which is presented as a tale in the form of a hypertext.

Number is Form and Form is Number Anne Tyng. Interview by Robert Kirkbride (Spring 2005)
Robert Kirkbride interviews Anne Tyng, Fellow of the American Institute of Architects and a member of the National Academy of the Arts on the potentials of geometry and number in architectural practice. Through such examples as Pascal's Triangle and her "Super Pythagorean Theorem," Dr. Tyng asserts that geometry is not only a metaphor for thought and the creative process, it is a spatial demonstration of how the mind generates associations by the combination, or layering, of pattern and chance.

Teaching Mathematics through Brick Patterns David Reid (Autumn 2004)
There are many elements of architecture that provide teachers and students useful opportunities for mathematical explorations. In this article educator David Reid examines a few aspects of what is possible with only one structure, the brick wall. Mathematics can make us more aware of aspects of the world we might normally ignore. This allows students to develop different view of mathematics, richer than the image of rules and facts that they often have. In the activities Reid describes here, the study of the patterns found in brick walls and pavements makes his students more aware of symmetry as a way of seeing.

The Effect of Integrating Design Problems on Learning Mathematics in an Architecture College Igor M. Verner and Sarah Maor (Autumn 2003)
A number of universities and colleges have developed mathematics courses based on the relationship between architecture and mathematics. Igor Verner and Sarah Maor report on a study of learning mathematics in professional context in one of the architecture colleges in Israel, with a focus on assessment and educational research. This paper consider in detail applied contents and learning activities in the course, and our way forward in order to discuss them with the NEXUS community.

A Proposed Two-Semester Programme for Mathematics in the Architecture Curriculum Luisa Consiglieri and Victor Consiglieri (Spring 2003)
Luisa Consiglieri and Victor Consiglieri propose a one-year mathematics course for architecture students. The aim of this work is to examine the relevance of mathematics in contemporary architecture, namely its most representative forms of cultural or sport buildings. Because today the architectural object has a great exuberance, as it did in the Gothic age with its ogival forms, or the Baroque with its vaults and spherical calottes, some notions of topology are required; the classic linear algebra and analytical geometry are becoming inadequate for the purpose. For the education of an architect, with a modern vision of the utility of technology, the academic staff must understand what students lack, and promote quality in their professional work. Indeed, it is important that mathematics do not fall into neglect, and students might profit from mathematics and topological geometry as previous requisites for their imagination and poetic ability. Nevertheless, harmony, expression, or quality of the actual worth of architectonic messages are not explained rationally by mathematics, but by appealing to sentiment or sensibility.

Ertha Diggs and the Ancient Stone Arch Mystery Michael Serra (Autumn 2002)

Michael Serra describes a class project for constructing arches and examining their properties. The objective was for students to review and apply the properties of isosceles triangles, trapezoids, regular polygons, and of interior and exterior angle sums. They were to practice communicating mathematically and modeling in two and three dimensions. It is a fun two-day activity of hands-on mathematics and problem solving.

In the Palm of Leonardo's Hand: Modeling Polyhedra George Hart (Spring 2002)
George W. Hart presents three examples of new computer-based "3D printing" techniques for recreating the historically important polyhedral models of Leonardo da Vinci and Luca Pacioli. It is hoped that such models will inspire students and the public to appreciate the history and beauty of polyhedra for architectural and other applications.

Math-Kitecture at PS 88 Charles Bender (Autumn 2001)
Charles Bender explains "Math-Kitecture", a program for integrating, computer, mathematics and architecture into the elementary level curriculum. Math-Kitecture is put to use by fourth- and fifth-grade students in New York's Public School 88.

Didactics: Proportions in the Architecture Curriculum Roger Herz-Fischler (Summer 2001)
Roger Herz-Fischler presents a revised version of a chapter entitled "Proportions" that appeared in the problems part of his book, Space, Shape and Form /An Algorithmic Approach, developed for a mathematics course he taught in the School of Architecture at Carleton University from 1973-1984.

CD: Piero Matematico di Daniela Gentilin e Ennio Bettanello
Orietta Pedemonte (versione italiana) Spring 2001

CD: Piero Matematico by Daniela Gentilin e Ennio Bettanello Orietta Pedemonte (English version) Spring 2001

Osservazioni sul Piero Matematico Daniela Gentilin e Ennio Bettanello (versione italiana) (Spring 2001)

About Piero Matematico Daniela Gentilin e Ennio Bettanello (English version) (Spring 2001).

Mathematica per architettura: alcune esperienze europee Orietta Pedemonte (versione italiana) (Winter 2001)

Mathematics for Architecture: Some European Experiences Orietta Pedemonte (English version) (Winter 2001)

Esperienze di un laboratori dei modelli. Mostra didattica a cura dell'Istituto Statale Sperimentale d'Arte di Monza con la collaborazione del Liceo Artistico di Busto Arsizio. Liliana Curcio e Roberto Di Martino (versione italiana) (July 2000)

Experiences in a Model-Making Laboratory. Didactic exhibit curated by the Istituto Statale Sperimentale d'Arte di Monza in collaboration with the Liceo Artistico di Busto Arsizio. Liliana Curcio and Robert Di Martino (English version) (July 2000). This paper appears in the Nexus Network Journal 2 (2000): 147-154.

A Term Project: Creating a Geometry Cathedral Michael Serra (April 2000). This paper appears in the Nexus Network Journal 2 (2000): 159-161.

Il Rinascimento, geometria e architettura Pierangela Rinaldi (versione italiana). January 2000.

The Renaissance, Geometry and Architecture Pierangela Rinaldi. January 2000. This paper appears in the Nexus Network Journal 2 (2000): 155-158.

Copyright ©2005 Kim Williams Books