Twice at least, "nonEuclidean"
has been a label of modernism for artistic works in the XXth century.
It first occurred with cubist painters. There was already a firm
tradition for the advanced Art of this time to go beyond the limits
imposed by traditional conventions, so as to handle the unknown and
the hidden. Among the constraints to be shoved out, there was the fact that we
are living in a threedimensional world, and that our vision of it obeys
the rules of perspective. Cubists were not the first to attack
perspective, but they went very far. Various comments accompanied them and
one of them refered to a fourth dimension that artists could have
attempted to put on canvas
(see Luc Ferry, Le sens du beau, Editions Cercle d'Art, 1998).
This fourth dimension would have become fashionable with the first
popularizations of Einstein theories, though it is not clear whether
one was mentioning time
(in this case, cubist representations would have been related with
stroboscopies, but this looks unlikely) or some extra
spatial dimension.
By itself, a fourth spatial dimension does not justify the "nonEuclidean"
attribute since mathematicians easily consider Euclidean spaces with 4,
5, 6... dimensions, but, most probably, this simply meant that the artist
was going out from the everyday space analyzed by
Euclid.
This "nonEuclidean" attribute appeared again in recent comments
(in French only, sorry)
around the fractalist painters... together with
meanings, not always very clear.
 The journalist Henri François Debailleux connects this term with what
I called the "vertigo of the
infinite", but undoubtly this dive into the infinite needs some
time, so that one recovers the interpretation of the 4th dimension in
terms of time. The nonEuclidean feature comes from this extra dimension:
"the Euclidean principle has been given up so as to find the
perspective which lies in time".
 Seemingly, the philosopher Christine Buci Glucksmann uses this term
with two meanings.
First she says that fractals do not belong to the tools which are
described in the Euclid geometry:
"Fractals fascinate artists because they answer to a definite
question, namely, how to grasp and create a nonEuclidean and non
formalist shape, whatever it may be, a seashore, a cloud, a heap of leaves,
the roots of a plant or a cauliflower...", which looks banal
enough, insofar as classical artists did not wait for Mandelbrot to
paint a cloud or a cauliflower in a realistic manner.
Then she introduces the notion of dimension
with no real explanation:
"Fractals are curves with an infinite complexity in a finite space.
Hence the key notion of dimension, which no longer is an integer, in
the sense of spaces with 2 or 3 dimensions" (what can be understood
by the layman?) and seemingly she connects this noninteger dimension
with the nonEuclidean character:
"on the opposite of the Euclidean handling of shapes, with its
constants (right/left, top/bottom, heavy/light...), here, the notion
of dimension cannot be separated from a viewpoint." I must confess
that I am not sure of the deep meaning of this sentence, but I am
afraid that the invocation of the noninteger (fractal)
dimension is important in the claim that
these shapes are nonEuclidean, i.e. they do not belong to our everyday
life.

More cautiously, the writer Susan Condé sticks to the fact that fractals
do not belong to the set of the Euclid classical figures:
"First, let us state simply that a fractal is a kind of shape
designed within a nonEuclidean geometry, in the sense that a square
comes from the Euclidean geometry",
and she thinks that these new shapes will be very important because
"every shape embodies a way of thinking... the shape represents a
system of thinking, a philosophy of behaviour, a way for
modeling", and she expects that new attitudes will emerge from
these new shapes.
Incidentally, this text from Susan Condé comes from a book which
should have been published in 1998. It should be interesting to know
whether
she is aware of
the works carried out with the
help of Fractint, Tierazon,
Flarium and so on, closer to my idea of a popular fractal art, and
if so, how she reacts to them.
 In its 4th point, the manifesto of the Art and Complexity Group
(where fractalist painters are gathered) claims:
"We give up the Euclidean rationality in favor of unexpected and
unplanned processes." Well...
 Lastly, one of these painters (Cesar Henao) states "a new
existential fact: the geometries of the nonEuclidean illusion put
forward the fractal and protofractal multimomenta that are crossing the
reflexive nature and its projection in the art..."
Humph!
As a conclusion, the word "nonEuclidian" is ambiguous, because
on the one hand it has a very
accurate mathematical meaning, while, on the other hand, it was given
several popular acceptations which poorly agree. For instance, the label
nonEuclidean would apply to
 anything involving some extra dimension, beyond the three dimensions of
our everyday life
 any shape outside the family described by Euclid and his
followers, i.e. straight lines, triangles, squares, circles, ellipses...
(and the analytical curves, for scientists).
Notice that a spherical surface would be Euclidean in these two senses
while mathematicians regard it as a
nonEuclidian surface.
This does not help to clarify the discussion.
 any shape with a noninteger "dimension".
In my opinion, this definition belongs to either an intellectual
terrorism or a trick. How could the layman access to the subtle notion of
fractal dimension?
Initiated people know that this implies an infinite length, hence the
"infinite complexity in a finite space", thus sending us back to to the
second definition. But also, this mysterious noninteger dimension suggests
that these shapes do not really belong to our everyday space with its neat 2 or 3
dimensions, i.e. it unfairly suggests a return to the first definition
Preceding Page
[ Return to home page ]
[ Art and Fractals Contents ]
 Discovering the fractal world:

Introduction 
Mandelbrot Exploration 
Lyapounov Exploration 
Von Koch Curves 
IFS Fractals 
Fractal Dimension 
Mandelbrot Relatives 
Finest Fractal Pictures 
Software 
Biblio and Links
 Fractals and mysticism:

Introduction 
The Mysticism of Infinite 
NonEuclidean Art?
 A new Art?

Introduction 
Fascination of Fractals 
Fractals and Photography 
Definitions of Art 
The Colour Choice 
Other Colour Choices 
Fractalists Painters 
Compositions with Mandelbrot 
Put a pretty girl 
Algorithmic Art 
Beyond the "Fractal" Art
Charles.Rosemarie.Vassallo@wanadoo.fr
