The use of non-Euclidean space in van Gogh’s paintings
Our common sense teaches us that all that exists is a three-dimensional Euclidean space. But, no one can assert that this is actually all that exists around us. That is so only because we live in such a world, and the stretch of our imaginations is not enough for us to envision a more-dimensional world, or a world of some other, non-Euclidean geometry. To illustrate, imagine two people living on a piece of paper, in a two dimensional world, and looking at each other. Imagine now that a three-dimensional person comes, and lifts an arm of one of these people up, into the third dimension. What would the other person see? They would just see the arm disappearing! Or, we might always look at the projection of the 3-dimensional world into two dimensions. Then, if we lift the arm of one person, in the two-dimensional projection his arm would seem shrinked. Now, if we expand this for another dimension, that might as well apply to our, three-dimensional world. Another issue is the use of Euclidean or non-Euclidean space. The Euclidean geometry is based on a few ground axioms, which cannot be proved. This is why they are axioms. But, it is just our common sense that makes us take them for granted. For instance, one of the fundamental axioms is that given a line and a point not lying on it, through that point you can draw exactly one other line parallel to the given one. But what if this is not correct? What if you could actually draw an infinite number of parallel lines? Well, than you would get another, completely working and legitimate geometric system, called the geometry of Lobachevsky. Needles to say, the shapes would seem a bit different in the geometry of Euclid and of Lobachevsky.
There is reason to believe that van Gogh actually played with non-Euclidean spaces in his paintings, in particular, the painting of the Bedroom at Arles. Patrick Heelan’s article Toward a New Analysis of the Pictorial Space of Vincent Van Gogh states the possibility of van Gogh’s use of non-Euclidean space in the Amsterdam version of this painting. He says:
“When the Amsterdam painting is viewed at about arm’s length, or at the distance at which the artist would have been working at his easel, one receives an overwhelming impression of realism. One must, however, look at the full-sized painting, not a smaller reproduction. An analysis of the actual forms as represented in pictorial space by van Gogh reveals strange incongruities vis-a-vis Euclidean anticipations. A tracing of the perspective lines in the painting shows that he maintained neither the fixed viewpoint nor the fixed eye-level necessary for a conventional representation of Euclidean space; even single objects have multiple convergence points.” (Heelan, 484)