Elena Marchetti and Luisa
Rossi CostaDipartimento 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
[3]. All the programs used in city-planning work with transformations
that change the position, orientation and size of objects in
a drawing. 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.
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 [1], [2]). The principal elements to be known are: - Kinds of matrices and vectors:
*matrix operations*illustrated for square matrices and vectors;*bijection*between points P=(x,y,z) in 3D Cartesian space*Oxyz*and column vectors (sometimes written in the*transposed*row-form); in describing transformations in*xy*-plane we will use vectors having the third component equal to zero;*linear transformations*in 3D realized by appropriate matrix operations.
The elementary operations with matrices and vectors are: *multiplication by a scalar*k
*addition*
_{T}*product of a square matrix*(3,3)*by a column vector*(3,1) ( the result is a column vector (3,1))
c = A b =
= ;*product of two square matrices*(3,3), known also as "row-column product", non-commutative and natural extension of the previous matrix-vector product:
D =AB
The following notation
can formalize affine transformations in the Cartesian 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.
The teacher invites the students: - to analyse one of the chosen forms;
- to fix it in a suitable Cartesian coordinate system;
- to pick out the essential part;
- to identify in it some crucial points, basic lines or surfaces with vectors;
- to discover the correct transformations necessary to rebuild the object;
- to associate the corresponding matrices.
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.
Fig. 1 The painting is centered in the The same plane transformation is also evident in Fig.2, the
virtual reconstruction of Itten's drawing entitled
In Fig.4 you can see the where c is the height of
Fig. 5 The whole frieze is formed by n subsequently horizontal translations
applied to the left basic motive, centered in the In the synthetic formula (1) we put (a is related to the width of the basic motive). -- Fig. 6 In conclusion the teacher invites the students to discover a different way of reconstruction, involving the symmetry axes.
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.
Fig. 7 Fig. 8 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).
Another interesting plan because of the evident symmetries,
scaling and translations is
Fig. 9
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.
[2] G.Strang, [3] J.Foley, A. vanDam, S.Feiner, and J.Hughes,
[4] H.Weyl, 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".)
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