Continuum percolation threshold for interpenetrating squares and cubes

Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold ${\ensuremath{\varphi}}_{c}=0.6666\ifmmode\pm\else\textpm\fi{}0.0004,$ while for randomly oriented squares ${\ensuremath{\varphi}}_{c}=0.6254\ifmmode\pm\else\textpm\fi{}0.0002,$ 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold ${\ensuremath{\varphi}}_{c}=0.2773\ifmmode\pm\else\textpm\fi{}0.0002,$ while for randomly oriented cubes ${\ensuremath{\varphi}}_{c}=0.2168\ifmmode\pm\else\textpm\fi{}0.0002,$ 22% smaller.