Elena MarchettiLuisa Rossi Costa Dipartimento di Matematica del Politecnico di Milano Piazza Leonardo da Vinci,32 - 20133 Milan, Italiy
BAB 1999].An intriguing example of such an approach is the project for
The aim of this paper is to describe the shape of
The project of
Such an exceptional project is undoubtedly the result of re-elaborated
ideas and familiar architecture, as well as the artistic mood
of the period. The buildings of the villages near Berne where
Itten spent many months both as a child and adult, gave him his
initial inspiration. Thun castle, for example, seems to have
been very important, with its timber interior formed of elements
laid one on the other around the central axis. A spiral movement
is also often reflected in the exterior stairs of quite a few
houses in the Swiss Alps. The elegantly constructed wood bonfires
lit on the first of August in the Swiss Alps, probably suggested
both the shape for the project and the name chosen by Itten,
Itten's mathematical knowledge undoubtedly played an important role in the choice of the basic component of the tower. In the different Bauhaus courses it was quite common for teachers and students to work with elementary geometric figures and their re-elaborations, with more or less rationality and more or less fantasy. Itten's aim was, as he wrote:
The following works, mainly from the twenties, underline how the square and the cube are the elements mostly used by Itten, as well as by the students and artists of the Bauhaus. The list could be longer, but it is limited to the works dated in the same period as the ideation and the construction of the tower model. In all these works the mathematician will recognize plane and space transformations in the different re-elaborations of geometric objects, and can describe them with the instruments of linear algebra: - Lothar Scheyer -
*Carte Postale*(1921) and Wolfgang Molnàr -*Carte Postale*(1922), [Droste 1994: 39]; - Rudolf Lutz -
*Plaster reliefs with characteristics of quadratic and rectangular shapes*(1920/21) (**Figure 2**); - Else Mögelin -
*Composition with Cubes*(1921) (**Figure 3**); - Johannes Itten -
*Composition with Dice*(Würfel Komposition) - (1919) (**Figure 4**); - Johannes Itten -
*The White Man's House*(1920) (**Figure 5**).
Before giving a mathematical description of the prototype
for the tower, it is significant to remember the numerous sketches
of the project made in 1919-1920, as is apparent in the It is also interesting to read the comments accompanying the
tower plans, which give precise building details and help reveal
the symbolism connected with philosophic-religious beliefs of
the artist. The twelve cubes composing the tower are assembled
in three groups. The four bottom cubes, designed to be clay or
stone, stand for minerals, plants, animals and humans. The central
four cubes are described as being made of hollow metal with bells
inside but no symbolic interpretation is offered. The upper four
glass cubes represent the four elements (earth, water, air, fire)
and above them there is supposed to be a yellow light (logos-sun).
The number twelve is not coincidental. In some of Itten's notes
it is connected to the twelve zodiacal signs, in others to the
different graduations of colours as they are visible in In mentioning the importance of Itten's studies in colour
theory, we need to remember that he had been considering the
problem of the relation between colour and sound since his stay
in Vienna. The artist had also introduced his interest for sounds
and colours in the Weimar school, where teachers (including Gertrud
Grunow [ The combination of cubic shapes is accompanied by spiral forms
in Itten's work. The spiral, often intended as a symbol for "ascent",
has always intrigued artists and architects. Two examples of
the use of the spiral appear in The turn-of-the-century fashion of employing the spiral motif
in projects for towers and monuments should also be remembered.
as for example, in Rodin's Tower of the Work [Ray 1987], Tatlin's
Monument of the Third International and Obrist's Project for
a Monument [Altamira 1997]. Itten also used the spiral motif
in his pictures, in works such as
First linear algebra is applied to the plane, to underline the sequence of the in plan transformations, and then to the three-dimensional space, in order to build the front elevation of the tower.
Let l be the length of the square
side having its centre in Following Itten's descriptions, the square is rotated and
reduced so that the vertices Introducing the usual notations of linear algebra, column
vectors (with two components) give the points of the plane; after
the transformation, the vectors corresponding to the vertices
The roto-homothety representing these transformations adds
the counterclockwise rotation around the origin The two transformations represented by a linear transformation , are realised by the matrix M=SR, where
Each point of the first square, where changes to More generally, named Let l=l The relation between them is n=1,2,
…, 11,that is,.[4] The S where with and J = arctg 1/3. It must be underlined that, starting from the vertices The four polygonals S Increasing The tower has two super-imposed supports with square bases
(see
The cubes In such a way the cube
where The points The cubes The part of the tower formed by the twelve cubes has a height
Four conic spirals contain the vertices This cone has the vertex in that is, a=(8+2Ö10)/3. The vectorial equation of a spiral line G Naturally each sequence of the vertices It is evident that when ,
the spirals converge to the same point, the vertex Looking the photograph of the tower, it is possible to pick out other spirals or, more precisely, other polygonal lines having vertices belonging to spirals. Particularly evident are those created from the decorative part of glass described in the following section.
_{0}
and l_{-1} of the sides of
the respective base-squares is the same homothety that links
the vertices of the cubes K_{n}, while
as regards the heights, there are two different constants of
reduction, one for each support.Beginning with cube K_{1}, the base P_{0}
on which K_{1} is supported can be generated,
transforming K_{1} by means of the matrixWhere ( The height The inferior support The height of the virtual reconstruction is which leads us to the supposition that . - the four parts connected with the faces of the cube
*K*_{1}and fixed to the first support are obtained one from the other by rotation around the*z*-axis; - all the others , which decorate the sides of the superior
cubes, starting from
*K*_{2}, are realised (in different cubes) with the same transformation generating the*K*_{n}sequence.
Trying to stick to the model, the cones connected with The tower reconstruction modules can be seen in
return to text[2] In der nacht sollten die Glaswände von innen
erleuchtet (als Merkzeichen einer Stadt für Flieger!)...Zuoberst
wäre ein Leuchtfeuer, das sich dreht [Bogner 1994: 87].
[3] For other examples of Itten's work, see ArchitetturaModerna.com. [4] It must be underlined that the constant k of reduction
of the sides in De Michelis and Kohlmeyer [1996] is erroneously
indicated as 1/3, because it was confused with the quantity r=1/3,
representing the ratio between the two parts in which the sides
are divided after the roto-homothety. Reading the Tagebücher
[Badura-Triska 1990] it is evident that Itten looked for different
solutions; in the bottom left drawing of fig.8 the ratio is r
=1 and consequently k = Ö2/2. The ratio r=1/2 was chosen for the tower reconstruction
realised in 1971 for the Kunsthalle of Nürnberg [De Michelis
and Kohlmeyer 1996] and this choice corresponds to k=Ö5/3.
Il Secolo Sconosciuto.
Milan: Rossellabigi EditoriBadura-Triska, E., ed. 1990.
Bogner, D., ed. 1994-5. Cook, T.A. 1979. De Michelis M. and A.Kohlmeyer, eds. 1996.
Droste, M. 1994. Ghyka, M. 1977. Itten, J. 1921. "Analyses of Old Masters".
In Manara, C.F. 1967. Ray, S. 1987.
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