  In this paper, we introduce constructions of the high-dimensional
generalizations of the kagome and diamond crystals. The two-dimensional kagome
crystal and its three-dimensional counterpart, the pyrochlore crystal, have
been extensively studied in the context of geometric frustration in
antiferromagnetic materials. Similarly, the polymorphs of elemental carbon
include the diamond crystal and the corresponding two-dimensional honeycomb
structure, adopted by graphene. The kagome crystal in d Euclidean dimensions
consists of vertex-sharing d-dimensional simplices in which all of the points
are topologically equivalent. The d-dimensional generalization of the diamond
crystal can then be obtained from the centroids of each of the simplices, and
we show that this natural construction of the diamond crystal is distinct from
the D_d^+ family of crystals for all dimensions d\neq 3. We analyze the
structural properties of these high-dimensional crystals, including the packing
densities, coordination numbers, void exclusion probability functions, covering
radii, and quantizer errors. Our results demonstrate that the so-called
decorrelation principle, which formally states that unconstrained correlations
vanish in asymptotically high dimensions, remarkably applies to the case of
periodic point patterns with inherent long-range order. We show that the
decorrelation principle is already exhibited in periodic crystals in low
dimensions via a "smoothed" pair correlation function obtained by convolution
with a Gaussian kernel. These observations support the universality of the
decorrelation principle for any point pattern in high dimensions, whether
disordered or not. This universal property in turn suggests that the best
conjectural lower bound on the maximal sphere-packing density in high Euclidean
dimensions derived by Torquato and Stillinger [Expt. Math. 15, 307 (2006)] is
in fact optimal.
