Here is a picture of a piece of wood.

The wood comes from a place where the trunk was beginning diverge into two unequal branches.  The diagram below is a simulation of the electric field around two unequal charges, which represents the stress in the space around them.

The resemblance to the field of cracks generated by the stresses in the wood is quite strong.  One reason that it isn't perfect is that it doesn't include the boundary effect.  Where the cracks are near the edge, they turn so that they reach the surface at about a right angle.  The reason is that the cracks are generated by tension, which can only be parallel to the surface in the boundary layer.  At the top left of the wood picture, you can see the cracks curving towards the surface.  Does this resemblance tell us anything about the stresses?

The clumps always have smaller trees near the edges.  This is probably because those trees are not competing for light with as many neighbours as  those in the middle.

To test this a simple calculation was made using the assumption that each tree in a row grows at a standard rate plus a contribution proportional to the sum of the heights of its two neighbours.  Random numbers were used to select one tree at a time for a spurt of growth.  The sixth picture used a hundred times as many random numbers as the others, to reduce statistical fluctuations.

This is only a one-dimensional model, but it shows the way that modelling can be done. 

In a real case the results of modelling must be compared with reality.  Even if there is agreement, it could be fortuitous, and it would be advisable to choose examples with different values of the variables, to see whether the model and the reality vary in the same way.

The next set of pictures was made using such a model.  The assumption was that a taller tree had ten times the effect than a shorter one.  These pictures look more like the shapes of the clumps in Wiltshire.

The model is still very crude.  You might think that the effect of one tree on another would be proportional to the difference in height, rather than an all or nothing effect.  One can go on changing the model, but soon one is limited by the accuracy of making observations.

One obvious fault is that the model makes the tress at the edge much too small.  The rate of growth should include a standard minimum value.  This is a trivial model, but it illustrates a few points about modelling.  Better to go and the wonderful views and the wind in the trees.

On metsät Pohlolassa sankat, tummat,

ne ikisalat, haaveet hurjat loi.

Asunnot Tapion on siellä kummat,

haltiat väikkyy, hämyn äänet soi.

Wide-spread they stand,

the Northlands dusky forests,

ancient, mysterious, brooding savage dreams;

within them dwells the Forest's mighty god,

and wood-sprites in the gloom weave magic secrets.

_______________________

You don't have to have read these words, you don't have to have seen a forest, you don't have to have been to Finland, you don't have to know Tapio, to comprehend the great construction that is Tapiola.

Nobody knows why a set of sounds, engineered by a master builder, can excite such depths of feeling.  Nobody knows why these sounds can do this almost more precisely than words.  Nobody knows why the engineering is necessary at all.  Why can't the sounds have no pattern at all?

But if you have seen the immense forests of Finland, seen the beautiful lakes, seen the sombre conifers, and seen the lovely birches that complement them so wonderfully, you will have had an experience of a totally different kind.  There is no structure here at all.  Chance has decided which tree grew where.  Chance has decided where a glacier gouged out a depression which would one day hold a lake.  Only the roads were built with purpose.  Only from these roads and from the lakes can you see the forest at all.

Then you can have some idea of what it would be like to be lost as the dusk deepens, and strange animals emerge.  Then you can imagine the depths of winter, and the impossibility of escape from hundreds of miles of trees and snow.

Driving through a plantation of conifers is a completely different experience.  The rows of trees provide avenues of vision in many directions, through which you can often see the edge of the forest.  A crystallographer would know all about these lines.

But what if you stand inside the primitive forest.  Can you hope to see outside?  Because the trees are randomly placed, a ray of light has a certain probability of interception by a tree.  From this we could calculate the mean free path of a light ray.

Suppose it were 100 metres.  Then after 100 metres, it would have a 36.7 % chance of not being caught.  After 200 metres, 13.5 %, after 300 metres, 5 %, after 400 metres, 1.8 %, and so on.  After 800 metres the result is 0.033 %.  It doesn't need much of a forest to seem dark and impenetrable.

The diagram below shows how some rays of light are intercepted, so that they do not reach an observer.

The next picture represents the probability that an observer will see any light, for different distances to the edge of a forest.  The observer is at the top left.  The average brightness of the diagram at any point represents the probability that the observer will see light in that direction, if the forest ends at that point.

If you are lost in a big forest, with no map, no compass, and no sun, the best hope of escape is to keep looking for the tiniest glimmer of light, and then to go towards it.  You also need to avoid going round in a circle.  Following a river might help, but if it meanders you will walk a long way without making much progress.

Let's reverse the situation.  Suppose the trees are fluorescent lamp tubes.  Can we look around and see any dark strips among the strips of light?  By the same reasoning we can see that a large enough forest of lights will produce an almost solid wall of light.

The same idea works in three dimensions, with spherical lamps.  So when we look out at the stars, from high in the mountains, or on a wild moor, far from the lights, why is there so much black between the stars?

In 1826, the German astronomer Heinrich Olbers asked this question.  He thought that if the universe has no edge it must be infinite, and that it would therefore be bright in every direction.  Interstellar dust does not answer the question, because it would be heated to the same average luminosity as the stars.

With the idea of universal expansion, and finite age, the paradox disappears.  The further galaxies, receding faster, look dimmer than the nearer ones, and so the light fades faster with distance than in a static world.  We can see only a finite region of space and time.

We can't see the universe for the stars.  We can't get outside it and look in at it to see what it is like.  By definition it includes everything, and all the light for seeing.

This spirit from Finland is more benign than Tapio - a good note to end on.

One reason is that many problems are "inverse problems", which are harder to solve than the same thing reversed.  Anyone who has tried to integrate a complicated function will have wished that they only had to differentiate it.  Sir Arthur Conan Doyle found it much harder to solve real crimes than to write Sherlock Holmes stories - though few detectives have found it easy to write stories either.  In the same way, doing science or musical analysis are activities which are very different from creating a world or writing music.