On periodic and non-periodic space fillings ofEmobtained by projection

A periodic lattice in En is associated with an n-grid and its dual, and with a point symmetry group G. Given a subgroup H of G, a subspace Em, m < n, of En, invariant under H, is chosen and a projection of the n-grid from En to Em is defined. The translational and point symmetries of the projected n-grid are analyzed. A projection of the cubic n-grid from En to En-1 based on H = S(n) yields a periodic n-grid. A projection of the cubic 12-grid from E12 to E3 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.