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We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking arrangements include all five platonic solids. Criteria for interlocking based on transformations of the cross-sections of the elements in a 3D reconstruction of a layer from its middle cross-section are formulated. A generalization to higher dimensions is also given. In particular, an interlocking layer of four-dimensional cubes is described., Comment: 15 pages, 11 figures