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.
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
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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.