Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube?

A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.

Figure 1

Cube drilling. The upper row shows photos of the experimental setup; the lower panels are the corresponding numerical results. The faces of the cube are divided with a square-lattice mesh of linear size $L=6$, such that each face can be drilled $L^2=36$ times. From left to right, (a)–(d), the number of drilled holes per face are (a) 0, (b) 1, (c) 6, and (d) 8. From our numerical results for the position of the transition in the thermodynamic limit (see main text), one estimates that around 13 holes need to be drilled for the cube to disconnect.

Figure 2

(a) Double-logarithmic plot of the rescaled order parameter $P_{\infty }{L}^{\beta /\nu }$ as a function of the scaling variable $(p-p_c)L^{1/\nu }$ for different lattice sizes $L$. The linear part has a slope of $\beta =0.52\pm 0.04$, consistent with $\beta /\nu$ being the same as for three-dimensional percolation, but with a different exponent $1/\nu =0.92$. The present value of $\beta$ is different from the two-dimensional one $\beta =5/36\approx 0.139$ [5, 38] and the one in three dimensions, $\beta \approx 0.417$ [39]. (b) Double-logarithmic plot of the rescaled second moment $M_2^{\prime }{L}^{-\gamma /\nu }$, with $\gamma /\nu =2.0452$, as a function of the scaling variable $|p-p_c|L^{1/\nu }$, with $1/\nu =0.92$, for different lattice sizes $L$. The solid black line is a guide to the eye with a slope $-2.3$. The two sets of data correspond to the subcritical ($p<p_c$) and supercritical ($p>p_c$) regions.

Figure 3

Number of sites in the backbone of the spanning cluster, length of its shortest path (chemical distance), and red sites in the backbone, measured at $p=p_c=0.6339$, as a function of the lattice size $L$. Considering the local slopes of the data, we obtain the following fractal dimensions: $d_{\mathrm{BB}}=2.12\pm 0.08$, $d_{\mathrm{SP}}=1.30\pm 0.05$, and $d_{\mathrm{RS}}=0.92\pm 0.07$. The fractal dimension of the red sites is, within error bars, compatible with the value $1/\nu \approx 0.915$, found from the finite size scaling behavior of the $p_c$ estimators (see Fig. S1), and the relation $d_{\mathrm{RS}}=1/\nu$ [42]. Here, $d_{\mathrm{BB}}$ is larger than in classical three-dimensional percolation, where $d_{\mathrm{BB}}=1.875\pm 0.003$ [40, 43, 44], while $d_{\mathrm{SP}}$ is smaller than the classical value $d_{\mathrm{SP}}=1.3756\pm 0.0006$ [47]. The solid lines are guides to the eye. To extract the fractal dimensions, we analyzed the local slopes as proposed in Ref. [48]. Results are averages over at least $5\times {10}^3$ samples.

Figure 4

Main plot: Spanning probability $\mathrm{\Pi }$, at $p=0.63<p_c$, as a function of the lattice size $L_Z$, for different aspect ratios $r$. Solid black lines are guides to the eye with slopes, $-0.26$, $-0.58$, and $-1.22$, for $r=1$, 2, and 4. The inset shows the same probability as a function of the aspect ratio $r$, for different fixed lattice sizes $L_X$. Results are based on at least ${10}^7$ samples.