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A periodic lattice in 
n is associated with an n-grid and its dual, and with a point symmetry group G. Given a subgroup H of G, a subspace m, m < n, of n, invariant under H, is chosen and a projection of the n-grid from n to m is defined. The translational and point symmetries of the projected n-grid are analyzed. A projection of the cubic n-grid from n to n-1 based on H = S(n) yields a periodic n-grid. A projection of the cubic 12-grid from 12 to 3 based on H = A(5) yields a non-periodic 12-grid. This 12- grid is characterized by three real numbers and from its projection has a well defined orientation. The dual to this 12-grid yields a generalization of the non-periodic Penrose patterns from two to three dimensions.
n is associated with an n-grid and its dual, and with a point symmetry group G. Given a subgroup H of G, a subspace m, m < n, of n, invariant under H, is chosen and a projection of the n-grid from n to m is defined. The translational and point symmetries of the projected n-grid are analyzed. A projection of the cubic n-grid from n to n-1 based on H = S(n) yields a periodic n-grid. A projection of the cubic 12-grid from 12 to 3 based on H = A(5) yields a non-periodic 12-grid. This 12- grid is characterized by three real numbers and from its projection has a well defined orientation. The dual to this 12-grid yields a generalization of the non-periodic Penrose patterns from two to three dimensions. 
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