Studies of DLA are carried out through numerical simulations of the growth algorithm, such that the clusters usually 'exist' only in the memory of a computer. On occasion of the 50th birthday of Paul Meakin - a distinguished member of our research group who has contributed much to the present understanding of DLA - a macroscopic wood model of a three-dimensional DLA cluster was built. To our knowledge, this is the first time such an enterprise has been undertaken, and in the following we describe the construction process in detail.
Although the growth of DLA clusters is concentrated on their outmost tips, they have deep 'fjords' that can be penetrated by the random walkers during their diffusive motion . As a result, there is always a chance that a new particle is added deep inside the existing cluster, at a position which is hard to reach from the outside. Consequently, it would be difficult to build the model of a three-dimensional DLA cluster in the way the cluster is grown, i.e. by adding one particle after the other. Instead, we first grew a cluster of a given mass on a computer, and then assembled the corresponding model from appropriate construction plans.
Figure 3: Perspective view of a three-dimensional, cubic lattice DLA cluster of mass . The individual particles are identical cubes of side length , but the cluster is displayed as if consisting of vertical columns of different length. The 'upper' and the 'lower' part are indicated, cf. text.
Very 'light' clusters (small ) do not display the characteristic fractal structure of DLA. On the other hand, the assembly of say a model with particles would require an enormous amount of time. Additionally, even the largest cluster branches are usually attached to the cluster seed through only one single particle, such that very 'heavy' (large ) DLA clusters would be mechanically unstable. For our model the quite moderate mass was chosen.
The cluster was created using a lattice model in which the random walkers were confined to the sites of a cubic lattice with lattice constant . The particles were identical cubes of size , such that all particle-particle junctions were full side-side contacts. Figure 3 displays a perspective view picture of the cluster, in which each vertical row of cluster particles is represented by a column of length . Using such columns instead of elementary cubes as the construction elements increases the stability of the model and makes its assembly easier, faster, and more precise. None of the individual columns crosses the midplane that cuts the cluster into two parts of approximately the same size. The 'upper' and the 'lower' part are indicated in Fig. 3 by using different colors for the columns that belong to different parts. When the cluster model was constructed, these two parts were assembled separately.
With a particle size of cm the model has a diameter of about cm, and fits nicely on a book shelf.
Figure 4: The blueprint of the cross section through the upper part of the DLA cluster in Fig. 3 at height to the midplane. The cluster model was assembled with the help of such blueprints.
Construction plans for each of the two parts of the model were obtained from the positions of all the cluster particles. The complete plan for the upper part consisted of a number of blueprints of horizontal cross-sections through the cluster at different heights, , , to the midplane. In each of the blueprints, all columns that passed through the corresponding section plane were represented as an outlined square, placed at the appropriate position. The length of the corresponding column was printed inside this square. Additionally, columns that cross , but not , were represented as shaded squares in order to improve the readability of the blueprints. As an example, the blueprint of the upper half of the cluster is reproduced in Fig. 4. The plan to construct the lower part of the cluster was obtained in a similar way.
On the basis of these blueprints, the entire hardware model of the DLA cluster was assembled in a five-hour session involving a team of six students. The building material was balsa wood. This type of wood is delivered in rods of various shape and size. It combines extreme lightness with sufficient stiffness, is readily available, and cheap. The rods were cut into rectangular columns of the necessary lengths with a wood saw and a mitra box. Rough edges were smoothed using sandpaper.
The upper part of the cluster was assembled in the following way: All balsa columns emerging from the section plane were positioned on a sheet of paper according to the first blueprint. Neighbouring columns were glued together with superfast glue. Then, subsequent layers of the model were completed by putting all columns emerging from the corresponding section plane into place. Occasionally it was convenient to assemble small cluster branches separately before adding them to the model.
When the upper part of the model was finished, it was turned upside down and the lower half assembled on top of the upper one. At this stage, the model had to be supported from below by hand. Figure 5 shows the cluster in an intermediate and in its final state, when it was given to Paul Meakin.
Figure 5: The DLA cluster model at a) some intermediate state, and b) the final state of the assembly, when given to Paul Meakin.