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In this paper, we introduce constructions of the high-dimensionalgeneralizations of the kagome and diamond crystals. The two-dimensional kagomecrystal and its three-dimensional counterpart, the pyrochlore crystal, havebeen extensively studied in the context of geometric frustration inantiferromagnetic materials. Similarly, the polymorphs of elemental carboninclude the diamond crystal and the corresponding two-dimensional honeycombstructure, adopted by graphene. The kagome crystal in d Euclidean dimensionsconsists of vertex-sharing d-dimensional simplices in which all of the pointsare topologically equivalent. The d-dimensional generalization of the diamondcrystal can then be obtained from the centroids of each of the simplices, andwe show that this natural construction of the diamond crystal is distinct fromthe D_d^+ family of crystals for all dimensions d\neq 3. We analyze thestructural properties of these high-dimensional crystals, including the packingdensities, coordination numbers, void exclusion probability functions, coveringradii, and quantizer errors. Our results demonstrate that the so-calleddecorrelation principle, which formally states that unconstrained correlationsvanish in asymptotically high dimensions, remarkably applies to the case ofperiodic point patterns with inherent long-range order. We show that thedecorrelation principle is already exhibited in periodic crystals in lowdimensions via a "smoothed" pair correlation function obtained by convolutionwith a Gaussian kernel. These observations support the universality of thedecorrelation principle for any point pattern in high dimensions, whetherdisordered or not. This universal property in turn suggests that the bestconjectural lower bound on the maximal sphere-packing density in high Euclideandimensions derived by Torquato and Stillinger [Expt. Math. 15, 307 (2006)] isin fact optimal.

High-dimensional generalizations of the kagome and diamond crystals and the decorrelation principle for periodic sphere packings