## A 2D algorithm for a 3D structure?

Whilst researching the techniques stone masons use when building walls out of irregular blocks of rock rubble I realised they select rocks so that the problem to be solved is 2-dimensional rather than the much harder 3-dimensional packing problem. To illustrate what I mean have a look at the diagrams below showing a retaining wall in cross-section and front-on.

Cross section and front view of a retaining wall

Each rock is irregular in 2-dimensions only. Its front face (the face which is visible after the wall is built) is an irregular polygon but the other faces give the rock the general shape of a prism so that when they are stacked together the joints between the rocks generally run perpendicular to the wall face. Solving a 2D packing problem is hard enough so it’s no wonder masons select rocks in this way. The prism-shaped rocks also lend themselves to making a stable dry stone structure.

It’s easy enough to see that single-skin structures built from prism-shaped rocks (retaining walls, paving and cladding) can be built by solving a 2D problem but what about something like a dry stone wall? Look at the diagram below and you’ll see that a dry stone wall can be thought of as two retaining walls leaning in on each other with the occasional “through stone” to help tie the two sides together. The middle of the wall consists of randomly placed smaller rocks and packed earth. So it’s still built by solving a 2D packing problem.

Dry stone wall in cross section and a mortared stone wall

Packing together rocks which are irregular in 3-dimensions is quite a difficult task but one which has several applications. For instance building a mortared wall were the aim is to minimise the amount of mortar and to use all available rocks rather than pre-selecting rocks for their ease of fit. Another example is packing the armour layer of a breakwater or seawall where each rock needs to key into its neighbours and into the irregular layer of rocks below.

The previous posts have shown how Rocksolver software is capable of solving the 2D packing problem and aid the building of real structures. The good news is that Rocksolver software already has the capability of handling rocks which are irregular in 3-dimensions. We are working on the 3D algorithm and the prototype 2D algorithm is giving us a few clues.