Luisa Rossi Costa
Dipartimento di Matematica del Politecnico di Milano
Piazza Leonardo da Vinci,32 - 20133 Milan, Italiy
An intriguing example of such an approach is the project for The Fire Tower by Johannes Itten, one of the most important exponents of the movement (Figure 1). The project, frequently mentioned in his Tagebücher (Diaries) [Badura-Triska 1990], was realised as a prototype. Unfortunately only a few photographs, dated around 1919-20, are left; the photographs take on great importance, given that the prototype has long since been lost. Since then many researchers have taken an interest in it and have tried different reconstructions: the most recent reconstruction was commissioned by Fondazione Mazzotta  of Milan in 1995/1996, on the occasion of the seventy-fifth anniversary of the foundation of the Bauhaus, and is still located at the Kunstsammlungen zu Weimar.
The aim of this paper is to describe the shape of The Fire Tower with the language of linear algebra and give a virtual reconstruction, in order to understand how Itten managed to concretise his strong mathematical intuition in an artistic form, even though he was unable to formalise it entirely with adequate instruments.
JOHANNES ITTEN (1888-1967)
The project of The Fire Tower developed in the last years of his stay in Vienna and the prototype was realised in the first months of his activity in Weimar. The photograph of the Tower model that appears in Figure 1 was taken in that period in front of Itten's studio, which was known as Tempelherrenhaus (Templars' house), a neo-Gothic building that had already conceived by Goethe during his stay in Weimar.
GENESIS, DESCRIPTION AND MOTIVATION BEHIND
THE FIRE TOWER
The Fire Tower was probably designed to be a public monument, perhaps a bell-tower for a church or a beacon for the expansion of the Weimar Airport. The latter hypothesis is supported by one the author's observations: "During the night the glass walls must be lightened from inside (as a signal for aviators!) and a turning light must be positioned on the top." The project came to nothing, because the plans to enlarge Weimar Airport had to be abandoned for lack of funds.
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, Turm des Feuers.
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:
The White Man's house determined the architectonic evolution of the Bauhaus school under Gropius's direction (post-1923) and its influence can be seen in many Bauhaus-inspired buildings. Among other buildings worth mentioning are the house Am Horn (Figure 6), projected by Georg Muche and realized (plan and execution) by Adolf Meyer in 1923, and the teachers' and students' houses for the new Bauhaus school in Dessau (1925/26).
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 Tagebücher. Many of them concern the tower, even if the motif of the superimposed cubes, turned around a vertical axis and eventually reduced, is repeated in projects for other buildings. It is, used, for example, in the realization of a church bell-tower [Badura-Triska 1990: II, 330], or is proposed for houses (Figure 7). In significant children's portraits of the same period, Itten also keeps in mind the cube and its transformations. In the Mazzotta Catalogue, for example, the cube is shown as underscored as it mixes with figurative elements [De Michelis and Kohlmeyer 1996: 156].
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 The man in the colours circle (1919) or in Sphere of colours in seven bright graduations and twelve tonalities (1921) [De Michelis and Kohlmeyer 1996], in others to the musical scales (dodecaphonic music).
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 [BAB 1999]) and students tried to give "colour" to sounds.
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 Tower of Babel by Bruegel the Elder (now in the Kunsthistorisches Museum, Vienna) and in the cupola of S. Ivo alla Sapienza in Rome by Borromini.
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 Die Begegnung (1916) [Badura-Triska 1990: I, 45]. In the Tagebücher, drawings of spiral motifs are often accompanied by specific studies on different way to represent this curve; of great interest is the description of a spiral obtained by using the golden ratio [Badura-Triska 1990: II, 282].
THE MATHEMATICAL INTERPRETATION OF THE
In plan analysis
Let l be the length of the square side having its centre in O, origin of the orthogonal co-ordinates of the plane Oxy, and sites parallel to the axes.
Following Itten's descriptions, the square is rotated and reduced so that the vertices A1 (l/2, -l/2), B1 (l/2, l/2), C1 (-l/2, l/2), D1 (-l/2, -l/2) change respectively into A2 (l/2, -l/4), B2 (l/2, l/4), C2 (-l/2, l/4), D2 (-l/2, -l/4) (see Itten's drawings in Figure 8, Figure 9 and Figure 10).
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 A1B1C1D1 of the first square change into those giving the vertices A2B2C2D2 of the second square.
The roto-homothety representing these transformations adds the counterclockwise rotation around the origin O of an angle J = arctg 1/3 to a scale (scaling) with factor k = Ö10/4.
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 x' = Mx, that is,
More generally, named x1=x, the points
of the squares generated successively by the transformation L are given as ,
Let l=l1 be the length of the side of the first square and ln of the n-square with vertices AnBnCnDn, n=1,2, , 12.
The relation between them is
The SA vertices belong to a logarithmic spiral that has the following vectorial equation:
with and J = arctg 1/3.
It must be underlined that, starting from the vertices B1C1D1 of the first square, three other polygonal lines SA SB SC SD are described and, at the same time, other three spirals.
The four polygonals SA , SB , SC , SD are connected to one other by a rotation of an angle p/2 or mp / 2 (m natural integer), as are the corresponding spirals (see Figure 10).
Increasing n, or for , the length of the side (tends to 0) and at the same time .
The tower has two super-imposed supports with square bases (see Figure 1); these bases are rotated with the same rule, that is the same linear transformation L. It is easy to understand that an adequate extension of the spirals also connects the vertices of the base of the two supports.
Analysis of the front elevation
The cubes Kn grow one from the other through a roto-homothety and a translation, that is, by means by an affine transformation. The homogeneous co-ordinates are convenient for an adequate description of the affine transformation; therefore a point of the three-dimensional space is represented by a vector v = [x, y, z, y]T.
In such a way the cube K1, with one face laying on the plane Oxy, with the edges parallel to the axes and the centre of the base in the origin O, can be described by
K1 is rotated counterclockwise by an angle J = arctg 1/3 around the z-axis, reduced of a factor k=Ö10/4 and translated a quantity l1 in the positive direction of the z-axis, by the linear transformation represented with the matrix:
where R*, S*, T1* are respectively the corresponding matrices of rotation, scaling and translation
The points v of K1 are transformed in the points belonging to K2, that is
The cubes Kn are generated one on another, applying the matrix successively, or, equivalently, applying to the points of K1 the matrices , (n = 1,2, , 11).
The part of the tower formed by the twelve cubes has a height h obtained by adding the first twelve terms of a geometrical progression with the initial element l1 and factor Ö10/4, that is,
Four conic spirals contain the vertices AnBnCnDn of the inferior base of the cubes Kn and line on the cone written in homogeneous co-ordinate (x, y, z, u) with the following equation:
This cone has the vertex in V(0,0, al1, 1) and the circle section centred in O, with radius , lying on the Oxy plane, as directrix . The value al1 gives the height of an ideal structure obtained by the superposing infinite cubes Kn,
that is, a=(8+2Ö10)/3.
The vectorial equation of a spiral line GA (passing through the points An) has the following form
Naturally each sequence of the vertices Bn , Cn , Dn, belongs respectively to other three spiral-lines GB , GC , GD, obtained from GA by rotations around the z-axis of multiples of p / 2 (Figure 11).
It is evident that when , the spirals converge to the same point, the vertex V of the cone, and their projection on the plane Oxy are the spirals represented in Figure 10.
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.
Bases and glass-decorations
Analysis of the base. The two supports are square-based
parallelepipeds: the law that furnishes length l0
and l-1 of the sides of
the respective base-squares is the same homothety that links
the vertices of the cubes Kn, while
as regards the heights, there are two different constants of
reduction, one for each support.
Where (R*)-1 is the inverse of the rotation-matrix R* defined in the paragraph above and
The height h0 = l0/4 = l1/Ö10 is deduced from Figure 1, and is in accordance with the reconstruction commissioned by the Mazzotta Foundation.
The inferior support P-1, with height h-1, is generated from P0 following the same process and changing conveniently the matrix . From the examination of the same photograph it is presumed that h0 and h-1 are in the ratio 7/6. With h0 = l1/Ö10 and h-1 = 6/7 h0, it is possible to estimate the height of the entire tower as a function of l1 and understand the artist's choice for having a model with the height of approximately 3.6 meters.
The height of the virtual reconstruction is
which leads us to the supposition that .
Trying to stick to the model, the cones connected with K1 are imagined as having the vertex on the vertical edges of the cube, and an adequate arch of ellipses as directrix. All the other conic parts have the indubitable position of the vertices; the arch of ellipse chosen as directrix lies on an adequate plane containing one edge of the cube.
The tower reconstruction modules can be seen in Figure 12.
 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]. return to text
 It must be underlined that the constant k of reduction of the sides in De Michelis and Kohlmeyer  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. return to text
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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.
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.
Copyright ©2002 Kim Williams