On Weiner's Proof of the Palásti Conjecture

Abstract

Letters to the editorinteresting to compare densities. To date, we have been unable to devise a non-trivial packing procedure for which these lemmas would both hold.Empirical evidence from computer simulations (Blaisdell and Solomon (1970), Akeda and Hori (1976), Jodrey and Tory (1979)) indicates that the conjecture itself is false and hence that x, y -coordinates are dependent, but it is not easy to see the effect of this dependence. If X is a random variable representing the x-coordinate of the centre of a rectangle and Y represents the corresponding y -coordinate (in the time-sequence mode of labelling), then symmetry implies that Cov (X, Y):= O. Though the simulated densities differ from the conjectured by eleven standard deviations, the actual difference in densities (0.0032) is small (Jodrey and Tory (1979)). Figure 1 suggests that filling B before A and C creates a slight tendency for the new rectangles to line up with those already in place. We speculate that this causes the small increase in packing density over that conjectured.References AKEDA, Y. AND HORI, M. (1976) On random sequential packing in two and three dimensions.