POINT AND LINE TO PLANE : MATHEMATICAL PROBLEMS
RELATED TO KANDINSKY’S CONCEPTION OF ART
Musso, Emilio, (IT)
Abstract. We analyze the interrelations between Mathematics and the Theory of abstract
Art elaborated by W.Kandisnky in the essay ”Punkt und Linie zu Fläche”.
Key words and phrases. W.Kandinsky, Abstract Art, Curvature, Serret-Frénet equations, Bessel and Special Functions.
Mathematics Subject Classification. Primary 00A08, 53A04; Secondary 65D17, 65D18,
65D20, 33C10.
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First Section
This paper analyzes the interplay between Mathematics and abstract Art, examined by W.
Kandinsky in the second chapter of the essay Punkt und Linie zu Fläche, published in 1926. The
book has the rather restrictive subtitle: Contribution to the analysis of the pictorial elements.
However, it contains many hints that go over pictorial facts, and face general problems of the
creative activity and of the fundamental elements of the form. Our aim is the clarification of the
role that Kandinsky assigns to Mathematics in a hypothetical exact theory of the composition,
which is the ultimate goal of the essay. Secondarily, we will show that the mathematical
problems set by Kandisnky, even though elementary, hide non trivial subtleties. Let begin by
putting Punkt und Linie zu Fläche in his historical context. In June 1922, Wassily Kandinsky
joined the faculty of the Staatlisches Bauhaus in Weimar. He gave courses on Colors, Analytical
Drawing and Abstract Form Elements, the treatise is an outgrowth of these courses. In this
book Kandisnky analyzes the geometrical elements which compose every painting, namely the
point, the line, the physical support and the material surface on which the artist draws or
paints. The declared objective of the book was to establish the foundation of a science of the
art: ”this brief book has the only intent to point out in general, and in principle, the ”graphic”
primary elements, both
1. in an abstract way, that is isolated from the real environment of the material form;
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2. on the material surface - the manifestation of the fundamental properties of this surface.
But this can be realized in this book only in a rather summary fashion - as an attempt to find
the method for the science of the art.”
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A brief Summary of ”Punkt und Linie zu Fläche”
There are three primary graphical elements in any artwork : the point, the line and the plane.
The point is a small stain of color put on the canvas. So the point is not a geometric point, it
possesses a certain extension, a form and a color. This form can be a square, a triangle, a circle,
a star or even more complex. According to its placement it will take a different tonality. It can
be alone and isolated or put in resonance with other points or lines. The point is essentially a
”statical” element.
The line is the product of a force : ”the geometric line is the trajectory of the point in
movement, therefore it is a product of the point. It arises from the movement....Here (in the
passage from point to line) happens the transition from stasis to dynamics. Thus, the line is a
secondary dynamical ”graphical” element. The line is produced by a point on which a living
force has been applied in a given direction, the force applied on the pencil or on the paint brush
by the hand of the artist. The produced forms can be of several types: a straight line which
results from a unique force applied in a single direction, an angular line which results from the
alternation of two forces with different directions, or a curved line produced by the effect of a
field of forces. The essence of the line resides in the force that determines the movement. The
subjective effect produced by a line depends on its orientation: the horizontal line corresponds
to flatness, it possesses a dark and cold affective tonality (black or blue). The vertical line
corresponds to height which offers no support, it possesses a luminous and warm tonality (from
white and yellow). A diagonal possesses a warm or cold tonality according to its inclination.
A force producing an unbounded curve corresponds to lyricism, while a forces that determines
a closed curve form a drama. The angle too possesses an inner sonority: warm and close to
yellow for an acute angle, cold and close to blue for an obtuse angle and similar to red for a
right angle.
The pictorial plane is in general rectangular or square, thus it is composed of delimiting
horizontal and verticals lines. The tonality is determined by the relative importance of horizontal and vertical lines. The horizontal lines give a calm and cold tonality to the plane, while
the verticals give a calm and warm tonality. The artist develop the intuition of this inner
effect, which he chooses according to the tonality he wants to give to his work. Every part of
the plane possesses a proper coloration which influences the global tonality of the composition.
The above of the plane corresponds to the looseness and to lightness, while the below evokes
the condensation and heaviness. The work of the painter is to understand these effects in order
to produce paintings which are not the effect of a random process, but the fruit of an authentic
work and the result of an effort toward the inner beauty.
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Mathematics and ”Punkt und Linie zu Fläche”
The interrelations between Mathematics and Arts are discussed in the third section of the
second chapter of the book. Here we quote the words of Kandinsky and we postpone the
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Figure 1: Wassily Kandinski : Kompositio8, July 1923. Solomon R.Guggenheim Museum.2007
Artists Right Society, New York/ADAGP Paris.
comments :
”aside from the differences of the characters determined by the inner tensions and from the
specific processes of formation, the native source of all the lines is the force. The collaboration
of the force with the material produces the vital element, which is expressed in the tensions.
The tensions express the internal characteristics of the element. The element is the result of
the force on the material. The line is the clearest and simplest case of this work of formation.
Such work proceeds in an exact-normative way and therefore it makes possible and deserves an
exact and normative use. Thus, the composition is an exact-normative organization of the alive
forces contained in the elements as tensions.
Every force finds its expression in the number, and this is called numerical expression. Today, this is a theoretical assertion since the possibility of the measurements is not attainable.
However sooner or later they could be discovered and not remain an utopia. As a consequence,
every composition will have its numerical expression. At a first time, this will be possible only
for the scheme and for the great complexes. ..... The exact theory of the composition will be
completely realizable only after the conquest of the numerical expression. The simpler relationships have been used, together with their numerical expression, in architecture, in music and,
partly, in poetry already from millennia, while the most complex relationships have not found
any numerical expression. It is very seductive to operate with simple numerical relationships,
and this justly corresponds to the actual artistic inclinations. But, after having overcome this
step, it will appear more seductive, the always greater complexity of the numerical relationships.
The interest for the numerical expression follows two directions : theoretical and practical. In
the first direction, the normative part is more important while, in the second direction, the
functional aspect plays the central role.
These words are plenty of premonitions and anticipate many of the problems inherent to the
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digital production of images 1 . The demand of a normative-numerical elaboration of the artistic
elements is, now, to the agenda. The new technologies of numerical elaboration of systems of
ordinary differential equations and the graphic potentialities of the modern softwares make
concretely realizable the theoretical intuitions of Punkt und Linie zu Fläche. It is interesting to
remark that the viewpoint of Kandisnky has many analogies with the mathematical approach to
object recognition and symmetry detection in computer vision recently developed by E.Calabi,
S.Haker, P.Olver, C.Shakiban, A.Tannenbaum in [1] and [8]. Moreover, Kandinshy’s ideas also
have remarkable analogies with the themes analyzed by D’Arcy W.Thompson in the celebrated
book On Growth and Form [10] and with the mathematical approach to the concept of ”shape”
outlined by R.Thom in the essay Structural Stability and Morphogenesis [9].
Figure 2: Wassily Kandinsky : Various Actions, August 1941. Solomon R.Guggenheim Museum.2007 Artists Right Society, New York/ADAGP Paris.
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Conclusions
The definition of a line as a product of a force acting on a point coincides with the classical
one formulated by the French geometers J.A. Serret and F-J Frénet ([6]) in 1851 : the shape of
a planar curve is determined by the curvature and torsion (i.e. the forces acting on a moving
point that describes the trajectory with constant speed). This theorem is readily proven on the
basis of the Serret-Frénet formulas. In the planar case the reconstruction of the curve from the
curvature requires two quadratures : consider a function k : R → R of class C h−2 , h ≥ 2 and
define its angular function by
Z
s
θ(s) =
k(u)du.
0
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We refer to the treatise Toward a New History of the Visual by M.Kemp for a detailed analysis of the new
perspective of the visual art.
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If we set
Z
s
γ(s) =
eiθ(u) du,
0
we then get a unit-speed parametrization of a C h curve Γ ⊂ R2 with curvature k and it
satisfies the normalization γ(0) = O.
Remark 4.1 The integrations can be performed numerically.However, these are oscillatory
integrals and it is more efficient to solve numerically the linear Serret-Frenet system.
A much less trivial matter is the control of the shape from the knowledge of the curvature.
For instance, is not at all clear how to produce simple closed curves without symmetries starting
from a given periodic function (see. [2] and [3]). It is a rather surprising fact that a so simple
and natural problem is not faced in the mathematical literature. Let us consider the simplest
possible case : start with the periodic function
k(t) = a cos(t) + b,
t ∈ R,
where a, b are real constants. The problem is to determine the relationships between the
parameters a and b so that the function k produces a closed curve. We then have:
• if b is non rational, the corresponding curve is not closed and its trajectory is dense in an
annular region;
• if b = p/q is a rational, non integer number, then the curve is closed, with length 2πq,
total curvature p and possesses a non-trivial symmetry group generated by a rotation of
an angle of 2π/q;
• for an integral value of b, the curve is closed if and only if Jb (a) = 0, where Jb denotes
the Bessel function of the first kind and of order b (cfr. [7] as a basic reference for special
functions).
This fact suggest the following (quite optimistic) conjecture : given two integers m, p then
there exist an explicit special function 2
Φp : R2m → R
such that any periodic function, of minimal period 2π of the form
k(t) = am cos(mt) + ... + a1 cos(t) + p + b1 sin(t) + .... + bm sin(t)
generates a closed curve without Euclidean symmetries if and only if
Φp (a1 , ..., am , b1 , ..., bm ) = 0.
This example shows that the mathematical problems posed by the artistic work of W.Kandinsky
are in fact of a subtle and difficult nature.
2
Possibly of hypergeometric type.
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5
An elementary example
Let illustrate the Kandisnsy’s assertion ”every composition will have its numerical expression”
with a simple graphical example : the taiji (tai-chi) emblem.The unity of a dual pair of opposites
is symbolized by the Yin-Yang figure : a circle equally divided into a dark, female half and
bright, male half, each with a small dot of the opposite color. The taiji emblem is composed by
the Yin-Yang figure put in the center of an octagon which is the image of the totality. The taiji
image represents the balanced dynamism, symbolizing the interdependence of contrary forces
and principles (see [11]).
Figure 3: The taiji emblem.
The mathematical expression of the internal forces of the Ying-Yang symbol and of the
octagon are the step functions represented in figure 4 and 5.
Figure 4: The curvature of the Yin-Yang symbol.
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Figure 5: The curvature of the octagon.
The numerical solution of the Serret-Frénet equations for a fixed choice of the curvature and
the subsequent visualization of the corresponding curve can be obtained with standard routines
implemented in Mathematica 6.
Acknowledgement
Partially supported by MIUR project Metriche riemanniane e varietà differenziali, by GNSAGA of the Istituto di Alta Matematica F.Severi and by RIA project Varietá differenziabili,
sistemi integrabili e dinamiche caotiche of the University of L’Aquila.
References
[1] CALABI, E., HAKER,S., OLVER, P., SHAKIBAN, C., TANNENBAUM, A., Differential
and numerically invariant signature applied to object recognizon. In Int. J. Comput. Vision,
26 (1998), pp. 107–135.
[2] GRIFFITS, P., Exterior differential systems and the calculus of variations . Birkäuser,
Boston, 1983.
[3] HWANG, C.C., A differential-geometric criterion for a space curve to be closed. In Proc.
AMS, 83 (1981), pp. 357–361.
[4] KANDINSKI, W., Punto Linea Superficie. Edizioni Adelphi, Milano, 1968.
[5] KEMP, M., Immagine e veritá. Il Saggiatore, Milano, 2006.
[6] KLINE, M., Mathematical thought from ancient to modern times. Oxford University Press,
New York, Oxford 1990.
[7] NIKIFOROV, A., OUVAROV, V., Fonctions spéciales de la physique mathématique. Mir,
Moscou, 1983.
[8] OLVER, P., Moving Frames - in Geometry, Algebra, Computer Vision, and Numerical
Analysis. In Foundations of Computational Mathematics , R. De Vore, A.Iserles, E.Suli
eds., London Math. Soc. Lect. Notes Series, vol. 284, Cambridge University Press, Cambridge (2001), 267–297.
[9] THOM, R., Structural Stability and Morphogenesis. Addison Wesley Publishing Company,
1989.
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[10] D’ARCY THOMSON, W., On Growth and Form. Cambridge University Press 2000.
[11] TRESIDDER, J., Complete dictionary of symbols. Duncan Baird Publishers, 2005.
Current address
Emilio Musso, Professor
Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio,
I-67010 Coppito (L’Aquila), Italy, [email protected], +39-0862-433128
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