Helical structure of space-filling mosaics based on 3D models of the 5D and 6D cubes

Our 3-dimensional framework (3-model) of any k -dimensional cube (kcube) can be produced either based on starting k edges arranged by rotational symmetry or as sequences of strut-chains originated from a separate one whose breakpoints join a single helix.Increasing the number of segments in the strut-chains to k = n (infinity) creates continuous helices, whose Minkowski sum can be called n-zo-notope.By combining 2 < j < k edges, we can build 3-models of j -cubes, as parts of a k -cube.The suitable combinations of these zonotope models can result in 3-dimensional space-filling mosaics.The investigated periodical tessellations keep the 3-model of the k-cube and necessary j-cubes derived from it and follow the helical structure of our models.Such a mosaic can have fractal structure as well, since we can replace it with a restructured one, built from multiplied solids.These are composed by the addition of 3-models of k-and j-cubes and are similar to the original ones.The intersections of a mosaic with planes allow limitless possibilities to produce periodical symmetric plane-tiling.Moving intersection planes results in a series of tessellations transforming into each other.Planar and spatial symmetry groups are the base of several works in different branches of art.Recent results can hopefully aid correlations between geometry, art and design.