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About this book

The volume reports on interdisciplinary discussions and interactions between theoretical research and practical studies on geometric structures and their applications in architecture, the arts, design, education, engineering, and mathematics. These related fields of research can enrich each other and renew their mutual interest in these topics through networks of shared inspiration, and can ultimately enhance the quality of geometry and graphics education. Particular attention is dedicated to the contributions that women have made to the scientific community and especially mathematics. The book introduces engineers, architects and designers interested in computer applications, graphics and geometry to the latest advances in the field, with a particular focus on science, the arts and mathematics education.

Table of Contents


Unexpected Geometries Exploring the Design of the Gothic City

Geometry and architecture have always collaborated in the project, experimenting in the second the achievements of the first. In every age great mathematicians have worked alongside the great architects, although we do not always have direct information about them. History has recognized this relationship in many cases where the literary tradition provided news for its evidence: Pyramids, Greek temples, the Pantheon, the architecture of Humanism etc. Can historical research discover less evident forms of this relationship by working exclusively with geometry, in examples where the literary tradition has not directly given the news? The research was focused on the design of the gothic city, which historiography relates by finding in the past the characters considered necessary for the development of the subsequent history rather than the objective requirements linked to the knowledge and intentions of the historical moment.
Maria Teresa Bartoli

Observation, Drawing, Modeling. Elements of a Cognitive Process Between Analogic and Digital for Design Learning

Geometry accompanies the knowledge of space and places both intuitively and through cultural education. Some innate elements are at the base the instantaneous perception and processes of visual recognition of forms. Moreover, the theorical teaching and practical activities allow to structure visual and tactile and knowledge of primary geometric shapes, as well as the capacity for the mental modelling of space. Following a similar procedure in our Design Laboratory, both analogue and digital modeling methods are coherently explored through the assigned project. The results of some case studies, recently concluded, are presented here, oriented to the composition of elementary geometric elements for the construction of reticular architectural structures, and for the radio centric enquiry of vegetal elements. The aim of these experiences concerns the possibility of experimenting visual understanding, on the bases of drawing, modeled in an analogue way by hand, and then verified through digital procedures by means of modeling and rendering software. A further example concerns a modeling exercise, based on the Made in Italy Design collection of Triennale di Milano. These specific training courses took place within the framework of the new training methods defined in Italy in the recent Alternanza Scuola Lavoro guidelines (in collaboration with Parco Nord Milano and the Accademia—Fondazione Fiera Milano), where the training focus on soft skills is integrated with the learning in specific and ordinary didactic disciplines.
Federico Alberto Brunetti

Computational Process and Code-Form Definition in Design

In design process, drawing has always preceded the construction phase. The act of drawing, based on basic geometric elements such as lines, curves, surfaces and solid, allows to organize one’s ideas, manage resources and predict results.
Giorgio Buratti

To Observe, to Deduce, to Reconstruct, to Know

This paper deals with a didactic experience gained in courses that are part of the first year of studies for the Schools of Architecture and Design at Politecnico di Milano. One of the themes of these courses concerns geometric problems, in order to introduce students to 3D space. The peculiarity of the didactic method used is, first of all, to induce the student to observe the real object by identifying its geometric characteristics (symmetries, proportions, contours, and surfaces enveloping it).
Franca Caliò, Elena Marchetti

The Role of Geometry in the Architecture of Louis Kahn and Anne Tyng

Among the salient features of the architecture of Louis I. Kahn, one recognizes the ability to reconcile the aesthetics of Modernism with monumentality and one of the elements used to implement this programmatic attitude is Geometry. In this regard, we cite the projects for the Trenton Bath House (1955–1956) and the unrealized Philadelphia City Tower (1952–1957), which are today attributed in the geometric conception to Kahn’s collaborator, Anne Griswold Tyng. Her production of concepts of unrealized projects is less known, among which the one for the General Motors Exhibit 1964 (1960–1961), which adopted particular geometric figures, showing not only her knowledge, but also her ability to manipulate regular and semi-regular polyhedra. Tyng was also interested in women role in the culture and she expressed in her most appropriate form—the geometrical schemes in her texts written in 1989 and 1997—her own theory about the evolution of the woman’s role towards an autonomous creative expression.
Cristina Càndito

Thinking Architecture in Four Dimensions

In mathematics, it is quite easy to define four-dimensional geometry. With their equations, in fact, mathematicians work without any difficulty with any “n” dimension. From this point of view, it is also quite easy to describe what shape in our 3d word a hypercube, for example, can assume, taking advantage of projections of the geometric figure in the lower dimension. We have to say that architects are accustomed to draw the space they imagine through orthogonal projections and therefore to see the 3d space through its 2d projection in plan and section.
Alessandra Capanna

Reuleaux Triangle in Architecture and Applications

The Reuleaux triangle is a figure with the remarkable property of having constant width, a typical property of the circle. It takes its name from Franz Reuleaux, a 19th century German engineer, who studied its properties, in particular the ones related to applications to mechanics. However, this figure was previously known: actually, we find it in the shape of the windows and in the ornaments of some Gothic architecture. Furthermore, Leonardo da Vinci, to represent the terrestrial globe, used eight Reuleaux triangles, each one corresponding to an octant of the spherical surface. Even the mathematician Euler encountered this figure in his study of geometric forms with constant width.
Giuseppe Conti, Raffaella Paoletti

Interplays of Geometry and Music: How to Use Geometry to Analyze an Artwork in Order to Compose a Musical Piece

This paper, which summarizes a collaboration with D. Amodio (Conservatorio Benedetto Marcello, Venezia), describes how geometric techniques can be used to analyze an artwork and to obtain parameters employed for the composition of a musical score; in particular these approach was applied to a painting by Jackson Pollock and a poem by Giacomo Leopardi. In the first case, the initial task is the study of the graphic structure of the canvas, looking for forms and their spatial organization; this work is followed by the choice of the mathematical techniques used to examine the different classes of objects previously singled out; the last step is the computation of the parameters which will be used by the composer to orchestrate the score.
Chiara de Fabritiis

Harmony in Space

In their simplest and most elementary form, the structures used in design science are the three-dimensional version of the planar interlacing (biaxial and triaxial) that has always been used for the construction of gratings, baskets weavings and textures. The reciprocal frames can be considered as a premise to the tensegrity structures, which in turn can be considered as a premise to the geodesic structures. Geodesic structures arise from the correct subdivision of polyhedral shapes. The nascent reciprocal joint as a simple, natural and economic form, can be reworked towards the starred joint where the rods contribute towards a single junction point. The structural stability of natural structures is guaranteed by the presence of the triangle. A triangulated structure, optimized for use, does not require additional materials to ensure its resistance.
Biagio Di Carlo

Caterina Marcenaro + Franco Albini for the Love of Art

I was immediately smitten when I visited Il Tesoro di San Lorenzo in Genoa many years ago (Fig. 1). I happened to discover the buried treasure that holds the sacraments and remnants of the crusades beneath the Duomo, and I was baffled by the fact that I was completely unfamiliar with it. I knew immediately that this modern architecture suited me, and I began a quest to learn more.
Kay Bea Jones

How to Solve Second Degree Algebraic Equations Using Geometry

One of the most complicated problems faced by mathematicians was to calculate the solutions of the algebraic equations of each degree. First examples of first-degree equation solutions are reported in an Egyptian papyrus dating back to 1650 BC. In some Babylonian tablets, we find methods of resolution of some second-degree equations by geometric construction. Euclid, around 300 BC, described a geometric method for solving equations.
Paola Magnaghi-Delfino, Tullia Norando

Teatro Comunale, Ferrara: The Question of the Curve. From the Debate to the Geometric Analysis

The Teatro Comunale was built in Ferrara at the end of the 18th century, at a time when modern theatre was gradually leaving the space of the Duke’s Court and Academy to become part of the urban fabric, shifting from representing the elite to turning towards wider communities. The models of court theatre and public theatre with several levels of boxes coexisted for a long time, until the complete codification of the “teatro all’italiana”, of which the Comunale represents one of the clearest examples. Over time there have been several renovations.
Giampiero Mele, Susanna Clemente

Women and Descriptive Geometry in Italian University

This paper aims to analyse the women’s contribution to the teaching of Descriptive Geometry in the Italian Faculty of Architecture and Engineering.
Barbara Messina

Witch of Agnesi: The True Story

The recent celebration of the three hundredth anniversary of the birth of Maria Gaetana Agnesi, offers an opportunity to reflect on how we have understood and misunderstood her legacy to the history of mathematics. Maria Gaetana was the author of an important vernacular textbook, Instituzioni analitiche ad uso della gioventú italiana (Milan, 1748), the first book dedicated to learners of mathematics and one of the best-known women natural philosophers and mathematicians of her generation.
Tullia Norando, Paola Magnaghi-Delfino

The Dividing of the Sphere in Domes of Medieval Anatolia

The stylistic language of art and architecture in medieval Anatolia largely consists of geometric features with various levels of mathematical complexity. Whereas the two-dimensional graphic designs employ certain geometric relations and rules, their making, in three-dimensional space, relies on the spatial material qualities and the overall architectural form more than just visual transformations. For understanding how their architectonic harmony was implemented, it is crucial to consider not only the geometric design but also other parameters such as the surface geometry, the physical properties of the material, and the crafting technique. Under the patronage of Seljuks in Anatolia, the rigorous application of the decoration program on historical buildings manifests a collaboration coordinated by a master builder between mathematicians, designers, and craftsmen. Geometric patterns were applied to all kinds of building surfaces. Dome decorations particularly addressed challenges of building with spherical geometry. We investigate the historical ways to construct continuous patterns on dome surfaces and how each simultaneously handles aspects of geometrical calculation, the design, and construction processes.
Sibel Yasemin Özgan, Mine Özkar

Organic Reference in Design. The Shape Between Invention and Imitation

Historical treatises pursued the search for a rule in geometric laws that envisage the relationship between numbers and forms, fixing the articulation and the measure of architecture. Despite an inevitable inertia, architectural research always showed in formal and structural canons the concepts expressed by geometry, which like any science evolves in an attempt to explain increasingly complex facts, as the man’s ability to observe the nature’s world progresses. New geometries coincide with new space-structural conceptions that refer to inspirational models, which are based on the commitment of nature: on one hand it asks questions to explain, on the other hand it offers solutions to design problems.
Michela Rossi

Inverse Formulas: From Elementary Geometry to Differential Calculus

The aim of our contribution is to give a didactical perspective to the subject of the meeting. In particular I would like to illustrate a didactic path that starting from the inverse formulas of elementary geometry, in continuity through school of different degree, reaches the differential equations.
Anna Salvadori, Primo Brandi

A Concrete Approach to Geometry

A concrete approach to geometry is barely accepted in primary school but is not usually used and proposed in higher school levels. I present some examples to show how the activity of touching, building, manipulating mathematical objects can stimulate thoughts, observations and questions that are not at all elementary, at every level.
Emanuela Ughi


In this contribution we discuss tessellation. We analyze basic tessellation, types of tessellation, geometric approach and applications of tessellation in geometry as well as in architecture and art. We study as well as groups of tessellation used in Spanish Alhambra [13]. Finally we open possibilities how to use tessellation for aggregations, aggregations functions and aggregate tessellation.
Mária Ždímalová
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