Twice at least, "non-Euclidean" 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 three-dimensional 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 "non-Euclidean" 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 "non-Euclidean" 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 non-Euclidean 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 non-Euclidean 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 non-integer dimension with the non-Euclidean 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 non-integer (fractal) dimension is important in the claim that these shapes are non-Euclidean, 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 non-Euclidean 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 non-Euclidean 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 "non-Euclidian" 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 non-Euclidean would apply to

1. anything involving some extra dimension, beyond the three dimensions of our everyday life

2. 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 non-Euclidian surface. This does not help to clarify the discussion.

3. any shape with a non-integer "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 non-integer 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

Notes

Time as 4th dimension : in the Special Theory of Relativity, Einstein introduces a spatiotemporal continuum where time is linked to the three space dimensions. One of the main results is that clock measurements change when the clock is moving.

A 4th space dimension? In the General Theory of Relativity, Einstein concludes that our space-time continuum is "curved". In order to explain this concept of the curvature of a continuum, one speaks of the geometry of a spherical surface, which is a 2-dimension continuum curved in our usual 3-dimension space. A direct analogy leads to invoke an extra dimension in order to "understand" the curvature of the Einstein space-time continuum, that would be an extra 4th space dimension as the popular mind considers time as a special dimension.

In fact there is no need for any extra dimension. The whole geometry for a spherical surface can be carried out while remaining on the surface. One thus gets the results that follows, as simple as surprising.

The geometry of a spherical surface is not Euclidean. In any continuum where lengthes can be measured –as is the case along a sphere,– the "straight line" notion is replaced with the notion of geodesic line, i.e. the shortest way from a point to another one. The geodesic lines along a sphere are its great circles. Therefore, since two great circles are necessarily crossing, the spherical geometry does not satisfy the Euclid postulate, since from a point outside a "straight line", one cannot draw a second straight line "parallel" to the first one, i.e. uncrossing it. The spherical geometry is said Riemanian.

Euclid was a Greek mathematician who lived around the III-Vth century before J.C. and who undertook to write down the whole mathematics of his time. He and his followers thus established the classical geometry in the plane and in the space, demonstrating all the known properties of classical figures (straight lines, triangles, circles, ellipses, and so on) except one of them, of course, the Euclid postulate, which cannot be demonstrated: in a plane, from a point outside a straight line, one can draw one parallel to this line and a single one. The first historical meaning of the non-Euclidean word concerns the geometries that do not satisfy this postulate (Lobatchevski or Riemann), discovered by the end of the XIXth century. For mathematicians, the Euclidean spaces satisfy this property or its extension to spaces with more than 3 dimensions.

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