Dipartimento di Matematica "F.Brioschi"
Politecnico di Milano
Piazza Leonardo da Vinci, 32
20133 Milan, ITALY
Lines and surfaces are boundary elements of objects and buildings: it is very important to give the students a mathematical approach to them. We think that linear algebra (by vectors and matrices) is an elegant and synthetic method, not only for the description but also for the virtual reconstruction of shapes.
Another important aspect of linear algebra to be pointed out
to the students is its application in graphics software packages
. All the programs used in city-planning work with transformations
that change the position, orientation and size of objects in
The activity developed in our courses is complementary to theoretical lessons and exercises, and offers students a collection of shapes that recur frequently in the contexts mentioned above.
We address the activity to first-year students who have an average mathematical background (they come from a wide spectrum of high schools), so we focus on peculiar, simple, but quite attractive examples. In Section 1 of this paper we describe the activity in details, briefly presenting essential mathematical tools, even if well-known to everybody working with Mathematics; in Section 2 we present a short collection of students' final projects. We conclude with brief comments.
1. DESCRIPTION OF THE ACTIVITY
1.01 Mathematical tools: vectors and matrices
The first step is to present the theory of matrices and vector calculus. The second step is to give the students geometrical applications in 2D and 3D Cartesian spaces. The third step is to apply matrices and vector calculus to significant examples in the artistic and architectural field.
Here we introduce briefly the elementary vectors and matrices calculus in 3D Cartesian space, exactly as we approach it in our courses. Naturally specialists can skip this section but those less familiar with the subject may be interested in reading it. We limit our mathematical illustration to 3D space (where we live!), but the definitions can be extended to any n-dimensional abstract space (for a larger theoretical description see for example , ).
The principal elements to be known are:
The elementary operations with matrices and vectors are:
1.02 Mathematical tools: transformations
The following notation
can formalize affine transformations in the Cartesian Oxyz space. In relation (1), and correspond to the starting point and to its transformation respectively; A is the (3,3) matrix related to the transformation, and is the translation vector.
In the following we give some simple cases:
We can also combine the transformations mentioned above by multiplying the corresponding matrices, but we have to remember that the order must be respected (the product of matrices normally is non-commutative). For example the scaling matrix with at least two non-equal factors does not commute with the matrices of other transformations.
1.03 Practical steps
The teacher invites the students:
Now the students are ready to apply the right matrices to the crucial vectors, realizing the sequential steps (that is, the plane transformations) necessary to rebuild the shape virtually so that the whole object gradually emerges, step by step.
It is essential to operate with a computer, to become familiar with a dedicated software and to create attractive graphic images to compare with the original object.
--i) The first simple exercise consists in realizing the homothety, evident in the famous painting Carré noir by K. Malevich (1929) (Fig.1):
The painting is centered in the xy-plane origin O, the plane scaling allowing the transition from the black square inside to the white frame is represented by the matrix applied to the internal square sides.
The same plane transformation is also evident in Fig.2, the virtual reconstruction of Itten's drawing entitled The white man's house (1920), shown in Fig.3. In fact, the pavement around the house comes from a homothetic transformation of the base.
In Fig.4 you can see the Am Horn house by G. Muche and A. Meyer (1923), inspired by the Itten's drawing. The parallelepiped on the bottom, P1, is first reduced, then translated along the vertical axis: the mathematical transformation (1) works with
where c is the height of P1.
--ii) The second exercise is conceived to illustrate rotation: the frieze in Fig.5, the decoration of a building in an area of Milan from the period of the Liberty style, suggests the use of the rotation matrix having .
The whole frieze is formed by n subsequently horizontal translations applied to the left basic motive, centered in the xy-plane origin O.
In the synthetic formula (1) we put
(a is related to the width of the basic motive).
--iii) - The third exercise, based on the choir stalls in the Cathedral of Pienza (1642, Tuscany) (Fig.6), collects all the transformations: the pattern is centered in the xy-plane origin O, it is possible to generate the internal "flower" starting from its fourth part (one petal) by subsequent rotations of . Reducing the "big flower" by a suitable scaling, the new flower undergoes a translation, e.g., in the direction of the x-axis. The wooden marquetry is completed by three rotations of the "small flower" around the z-axis .
In conclusion the teacher invites the students to discover a different way of reconstruction, involving the symmetry axes.
2. EXAMPLES OF STUDENTS' WORK
The following examples represent some of the proposals given by the students, which have now become part of the material we use as teaching supports, because of their validity.
Example 1. One of the most interesting and well-presented applications of our method is the virtual reconstruction of Villa Bianca-Seveso (Milan, Italy), designed by G. Terragni in 1936-37 (Fig.7). The young student did it starting from the rectangular frame R of the façade, using scaling matrices and translation vectors; he re-drew windows, basement decoration and top. Finally he completed the virtual front with the main door, drawn by scaling, rotation (angle ) and translation of R. The result is shown in Fig.8, as it appears in the computer screen using a software familiar to students.
Along the same lines, other students proposed the German Pavilion in Barcelona (1929) and the Convention Hall in Chicago (1953), both by L.Mies van der Rohe (we don't provide images for these and other buildings mentioned, but these are easily found in texts about architecture).
Example 2. Many students were interested in plans of historical buildings, where symmetries and rotations are very often recognisable. For example, you can immediately discover an axial symmetry in the Kedleston Hall plan (1759), projected by Robert Adam for the elegant manor in Derbyshire, England.
Another interesting plan because of the evident symmetries, scaling and translations is Le Phalanstère by the sociologist C.Fourier (1772-1837). At the beginning of nineteenth century Fourier studied the quality of life in the countryside and wrote several treatises on it, complete with projects.
Example 3. Looking for patterns and tessellations, among different works done by the students, we find also a pretty presentation of the pavements of S.Maria in Cosmedin (Rome), where different transformations of the equilateral triangle are recognisable: scaling, translation and rotation. Other students were inspired by the decoration tiles on the façade of S. Maria Collemaggio (L'Aquila, Italy, fourteenth century).
Example 4. Concerning classic English Architecture, students did not overlook the Royal Crescent in Bath. Designed by the son of the architect John Wood, it comprises thirty houses; the basic vertical module (one column and one adjacent small façade) is rotated along the Bath Circus (inspired by the roman Coliseum). The idea of blocks of town houses like this has another previous famous example in Place des Vosges (1605) in Paris, where the houses are translated along the sides of a rectangle and rotated (by an angle in the vertices.
Example 5. We now mention, last but not least, one peculiar example in 3D: the cubic houses in Rotterdam, by Piet Blom (1984). These are an intriguing collection of row-houses, obtained by starting from a cube centered in the origin, with faces parallel to the coordinate planes. Appropriate rotations around each axis place one vertex as if it were the support (Fig.9). Having built up the first module, superimposing the rotated cube on a basement, the others are arranged by translation along a line.
As teachers, we are very satisfied at the end of the courses
because of the feedback from the students. We hope to have been
sufficiently clear in our descriptions of the virtual reconstructions,
as well as emphasizing the significant interdisciplinary work.
 G.Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, 1998.
 J.Foley, A. vanDam, S.Feiner, and J.Hughes, Computer Graphics: Principles and Practice, Addison-Wesley Publishing Company, 1992.
 H.Weyl, Symmetry, Princeton University Press, 1982.
We mention few mathematical books, examples
among many others, where you can find technical information on
Linear Algebra and Transformations. Every book on the same subject
is adequate to learn the necessary mathematical tools.
In working with personal computers, we used MATLAB®, Maple® and didactic software realized on purpose for our aims, but you can work with any other suitable software. (The software we developed is available free at the address : http://web.mate.polimi.it/viste/studenti/main.php; choose the teacher's surname "Marchetti" or "Rossi", and the course "Matematica per l'Architettura".)
Luisa Rossi Costa earned her doctorate in Mathematics in 1970 at Milan University and she attended lectures and courses at Scuola Normale Superiore in Pisa and at Istituto di Alta Matematica in Rome. Since October 1970 she has taught at the Engineering Faculty of the Politecnico of Milan, where she is associate professor of Mathematical Analysis. She first developed her research in Numerical Analysis, on variational problems and on calculating complex eigenvalues. Her interest then changed to Functional Analysis and to solving problems connected with partial differential equations of a parabolic type. She also studied inverse problems in order to determinate an unknown surface, an unknown coefficient in the heat equation and a metric in geophysics, with the purpose to find stable solutions in a suitable functional space. She published several papers on these subjects. She took part in the creation of lessons for a first-level degree in Engineering via the Internet. She also researches subjects regarding teaching methods and the formation of high school students; she collaborates on the e-learning platform M@thonline. Following a continuing interest in art and architecture, and believing that mathematics contains a strong component of beauty, she tries to connect these apparently different fields. She published papers connected to this aim.
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