In the section of his Modern Painters entitled ‘The Truth of Vegetation’ Ruskin claims that in their painting of landscape none of the Old Masters (with the partial exception of Titian) knew how to depict a tree. It was plain that they had never really and properly looked at trees. They represented the trunk and the branches of trees as tapering, and this is simply not the case.

Neither the stems or the boughs of [European] trees taper, except where they fork. Wherever a stem sends off a branch, or a branch a lesser bough, or a lesser bough a bud, the stem of the branch is, on the instant, less in diameter by the exact quantity of the branch or the bough they have sent off, and they remain of the same diameter.

As a result, the trees in these Old Masters’ paintings look like anything but trees. ‘The stem of Gaspard Poussin’s tall tree, on the right of the La Riccia [Ariccia] in the National Gallery, is a painting of a carrot or a parsnip, not of the trunk of a tree’ and in Poussin’s View near Albano, opposite to it, the bough in the left-hand upper corner is ‘a representation of an ornamental group of elephants’ tusks, with feathers tied to the ends of them’. Salvator Rosa’s tree-branches ‘have the look of the long tentacula of some complicated marine monster, or of the waving endless threads of bunchy seaweed, instead of the firm, upholding, braced, and bending grace of natural boughs’. No painter before Turner really managed to render trees, to convey ‘the woody stiffness hinted through muscular line, and the inventive grace of the upper boughs’.

It was the same with foliage, wrote Ruskin. ‘Foliage will not be imitated, it must be reasoned out and suggested’; and he shows by a rational analysis – a brilliant and eloquent feat of prose which I shall come back to later – how the complete leafage of a tree is built up. Here again, according to Ruskin, only Turner has fully understood the problem, or has shown how every species of tree comes to have its own particular curve or profile.

But let me come back to tapering. To taper is to take the form of an (elongated) cone; and one soon realises that, as with other geometrical figures – the cylinder, the sphere etc. – you will never encounter a cone, with the straight lines of its converging sides or its rising to a peak or vanishing point, in the real world. For one thing tapering, in the real world, must itself taper. Let us think of water being emptied out of a bath. The quantity of water diminishes, which is a kind of tapering; and the rate of this diminution, one must suppose, will be changing from moment to moment because of the water’s lessening weight. Nor can we assume that this change of rate will itself be regular – or at least not after the water has ceased to cover the plug-hole completely, thus creating a vortex and allowing air to come up the pipe and cause jostling. What this teaches us (if we need to be taught it) is that in nature and the real world things only progress by discontinuous jumps, and these jumps may very likely not all be in the same direction.

The physicist Erwin Schrödinger gives a nice example of this in his What is Life? (1944). The same group of atoms can, he says, unite in more than one way to form a molecule, but a transition from one figuration to another will not happen in a simple and predictable way; it has to take place discontinuously, over intermediate configurations of considerably higher energy.

One can also learn quite a lot about discontinuity from James Gleick’s Chaos (1988), especially from what he tells us about the mathematician Benoît Mandelbrot, the discoverer and namer of the ‘fractal’. While an employee of the International Business Machines Corporation, Mandelbrot interested himself in the fact that telephone communication between one computer and another had a tendency to be interrupted, now and then, by ‘noise’. This was well-known and appeared to happen quite randomly; but he observed something strange. There was no obvious regularity to the bursts of interference; but insofar as they could be said to exhibit a pattern, this pattern, eccentric and discontinuous as it was, proved to be exactly the same over the space of twenty-four hours or over several months. There was, he decided, a hitherto unknown principle at work here, the essence of which lay in its total independence of scale, and he baptised it with the name ‘fractal’.

His investigations took him into economics, where it had generally been assumed that, on the supposed analogy of physics, prices passed smoothly through all the intervening levels on their way from one point to another. This, however, proved not to be the case. In actuality, prices changed in ‘instantaneous lumps’, as swiftly as an item of news could travel down a teletype wire. ‘A stock market was doomed to fail if it assumed that stock would have to have been sold at fifty dollars on its way down from sixty to ten dollars.’ One would look in vain here for tapering.

The fractal proved a key to wonders. Mandelbrot undertook a study of seashores, in all the complexity of their rough and jagged edges, and was led to ask ‘How do you decide the length of a shoreline?’ It turned out that you couldn’t. A shoreline was, in a sense, infinitely long; it all depended on what scale you were working to. If a surveyor takes as his standard unit a step of one foot, rather than of a yard, he will capture more of the detail of the shore, with its endless sub-bays and sub-peninsulas, and his estimate of length will, correspondingly, be longer. The estimate of a snail, who has to negotiate every pebble, would be longer still, and so on down to the atomic level.

The geometry of the fractal concerns itself, precisely, with the capture of detail, in ever-increasing quantity. Its characteristic operation is to divide the side of a triangle into three parts, removing the middle portion and using it as the basis of an additional, smaller triangle, a process possible, in theory, to repeat an infinite number of times. The new triangles are too small to get in one another’s way, and the overall area enclosed by the resulting curve is not substantially increased.

Mandelbrot hypothesised that the fractal, as a nearly endless repetition of the operation just described, could be thought of as the basis of some highly important physiological phenomena. Blood vessels, in tree-like manner, branch and branch again into ever smaller units, and their branching, he suggested, was fractal. It is the task of the circulatory system to squeeze a huge surface area into a limited volume, and (no doubt with the aid of natural selection) it has achieved this so efficiently that no cell is ever more than three or four cells away from a blood vessel. It owes its intricate compactness to the fractal.

This brings us back to Ruskin. For, as one cannot help feeling, the passage of his that I mentioned earlier, about the structure of the leafage of a tree, is intensely reminiscent of Mandelbrot and his fractals. I will quote it at length:

In every group of leaves some are seen sideways, forming merely long lines, some foreshortened, some crossing each other, every one differently turned and placed from all the others. The forms of the leaves, though themselves similar, give rise to a thousand strange and differing forms in the group; and the shadows of some, passing over the others, still farther disguise and confuse the mass, until the eye can distinguish nothing but graceful and flexible disorder of innumerable forms, with here and there a perfect leaf in the extremity.

Ruskin, of course, is considering an aesthetic question: how can a tree in full leaf best be painted? But what one notices, equally, is how close his analysis comes to science. The principle of photosynthesis demands that the pattern of growth of a plant or tree should be such as to give every leaf the fullest exposure possible to the sun’s rays; and this is clearly the law underlying Ruskin’s analysis. He has every right to the phrase ‘The Truth of Vegetation’.

*******

As for the ideal, clear-cut, symmetrical forms of Euclidean geometry, their straight lines and their tapering, it would be wrong to think of them as the enemy of painting. One just needs to be careful to assign to them their right role. The point is brought home to us rather satisfyingly by Cézanne, who urged his artist friend Émile Bernard to ‘see in nature the cylinder, the sphere, the cone’. They were – this is clearly Cézanne’s point – to be invoked as a help in defining the curves of the natural objects he was representing; and their hinted presence is a great originality in his own way of painting.

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