We analyze shapes of overdense regions (clusters and superclusters) in controlled N-body simulations of gravitational clustering with power-law initial spectra P(k) ∝ kn, n = -3, -2, -1, 0. At values of the density just above the percolation transition, the number of distinct (isolated) clusters peaks, and we use this "natural threshold" to study the shapes and multiplicity function of clusters and superclusters. We find that the extent of both filamentarity and pancakeness increases as the simulation evolves, the former being appreciably larger than the latter at virtually all epochs and for all spectra considered by us. Our results also show that high-density regions within very massive clusters/superclusters are likely to be noticeably filamentary or pancake/ribbon-like when compared to the less dense regions within these objects. We make a detailed study of two moment-based "shape statistics" proposed, respectively, by Babul and Starkman (BS) and Luo and Vishniac (LV) and find that both LV and BS have certain built-in limitations: LV is biased toward oblate structures and tends to overemphasize this property, and neither BS nor LV correctly describe the shape of strongly curved or topologically nontrivial objects. For instance, a thin filamentary torus and a ribbon are both described by BS and LV as being pancakes! By contrast, "Shapefinders," a new shape diagnostic not constructed from density moments but from Minskowski functionals, does not suffer from these limitations and appears to faithfully reproduce the shapes of both simple and topologically complex objects.
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