How do you make sure that tall buildings, like the Shard in London, don't buckle under their own weight? Ahmer Wadee, Professor of Nonlinear Mechanics at Imperial College, London, explains how a little geometry goes a long way.
If the tower had 4 columns of uniform cross-section it would be most critically loaded near its ground level. This is because the greatest weight is taken here but the ground itself provides some lateral support. So to design such a structure to be most efficient in resisting buckling there should be cross connecting bars spread all the way up and you can see them in the picture. If we are limited in the number of these cross linkages and we know how the load due to gravity grows from zero at the top to the maximum at the ground, it provides us with a need for finding the optimal way the material should be spread. I suspect that the distribution of cross-section as one goes down, varies with the cube of the distance down, with slight increases at points between the cross-braces. This is for when the distribution of the lateral supports is uniform and their weight and strength are not taken into account here. It might also show how the shape of the Eiffel Tower was derived.