In the past, as part of my education, I have reproduced a few of Mondrian's images. Some of these appear in earlier posts on this blog. This has been my way of trying to find the systematic in his work. In this post, I hope to show how geometry played an increasing part in Mondrian's picture making. I hope to persuade you that the image shown above is a natural evolution of the methods that he deployed.
When a scientist writes about geometry (or any aspect of science) it is not often in a form that an artist would find accessible. Equally, when an artist writes about geometry, it is not in a way that a scientist might find accessible. It is often spiritual and metaphysical, rather than rational.Yet the scientific and (visual) artistic worlds collide precisely at this place we call geometry.
Geometry changed a lot in the 19th and early 20th centuries, when, among other things non-Euclidean geometries were investigated. There is evidence that modern artists of the day were aware of these ideas and some aspects of abstract art of the early 20th century were influenced by this awareness. For example, some people claim that cubism was influenced by contemporary writing on four-dimensional geometry [see for example http://uk.phaidon.com/agenda/art/articles/2012/july/19/picasso-einstein-and-the-fourth-dimension/].
Mondrian's use of geometry is a little less direct. His paintings evolved resolutely towards the geometric, just as surely as they did towards the abstract. Its seems that, for Mondrian, geometric abstraction was abstraction.
His early work is representational, but tends early in the 20th century towards the impressionistic and abstract. Think of his churches, mills, dunes, landscapes-at-night, particularly trees.
Slowly the representation in his work dissolves. First the landscape goes, leaving an interest in trees and water. Then the trees turn abstract and slowly become replaced by grids, until eventually we see the grids filled with coloured rectangles and and eventually his signature thick black lines.
At the very end of his life the lines begin to gain ascendancy. It makes me wonder what Mondrian would have done next, if he had lived a little longer. One possibility, with little evidence to support it,so this is pure surmise on my part, is that he may have begun to understand non-Euclidean geometry and to have found a new enthusiasm for curves.
First, the elementary objects displayed are all simple (random) variations on a cross. The angle at which the cross is shown and the relative positions of the two cross-pieces which form the cross have small variations, but apart from that each is displayed in a cell of regular grid.
The grid is not quite rectilinear (the sides are not quite straight and they are not quite at right angles. But it is close. Each cell has one cross, but that may not be located centrally in the cell. It will be sufficiently within the cell to avoid touching either neighbour.
Here is the grid (slightly smaller). In the published image each grid-cell does contain one cross, except not all have been painted. Some (a few) have just been omitted randomly [for artistic reasons]. But some are omitted because, to give the circular effect, I have chosen to paint them a shade of grey that gets lighter (and eventually white against a white background) as the crosses get further from the centre of the grid.
The final comment to make about the grid is that, while it is not obviously rectilinear, it actually is in a mathematical sense. It is a rectinlinear grid on a curved surface, chosen to give a false-perspective close to that in one of Mondrian's originals [according to my amateur forensics]..
It is interesting that this Mondrian image dates from 1915. It was not long after that that he moved more definitely to the Compositions with which he is most associated. By the mid 20s he was producing variations on a rectangular grid, not unlike the following.
And it seems that with this he persisted throughout the thirties and forties, refining as he went until (apparently) satisfied and moving on to the next in the series. It is only towards the end of his life that we see much departure from this when the lines become more prominent and carry more colours. It seems to me that, at the end of his life, he was looking for a way out of the straight-jacket that this grid-of-few-colours had become.
But suppose that Mondrian had lived on into the fifties and sixties, when non-Euclidean geometries (based on spheres and pseudo-spheres) were sufficiently understood that accessible presentations of the ideas were available. Would that have freed him from his straight-jacket?
Let's surmise that Mondrian had read a little about geometry on the surface of a hemisphere, or equivalently, geometry restricted to the interior of a circle. He might, like Escher did at the time, have begun to paint in circular, rather than straight grids.
To see how easy a transition that would have been for him, suppose someone had shown him a circular grid from a mathematics book (that's what happened to Escher in the fifties). Wouldn't he have been intrigued about the simple relationship between his beloved rectangles and their spherical equivalents?
To see how that works out, take a look at this simple transformation. We take Mondrian's composition from above, stretch it, and lay it on its side.
Assume now that it has been printed on a very pliable material that allows us to twist it around on itself. This allows us to deform it so that the rectangles in the original become curvilinear (so-called spherical) rectangles.
This image shows the Mondrian original being twisted back on itself. When we eventually close the gap, we end up with the image with which I began this piece. It's not far from Mondrian's original (in a mathematical sense) but getting close to something that Mondrian himself might have imagined if, as I imagine, he had encountered non-Euclidean geometry before he died.
editors: This piece is not complete. It will be some weeks now before I complete it.