Why some patterns repeat invisibly and others give the join away.
A tiling covers the plane with shapes, no gaps and no overlaps. Periodic ones repeat on a lattice. Aperiodic ones, of which Roger Penrose's 1974 tilings are the famous case, never repeat exactly and still cover the plane. They look organised and restless at the same time, which is a hard effect to get any other way.
Every repeating two-dimensional pattern belongs to one of exactly seventeen symmetry groups. Fedorov proved it in 1891. Each group is a different combination of translation, rotation, reflection and glide reflection, and knowing which one you are working in tells you what will happen at the edges before you try it.
Two approaches work reliably. Mirroring reflects the tile so every edge meets its own reflection. It is guaranteed for any artwork at all, and you pay in four-fold symmetry that the eye notices sooner or later.
Wrapping is subtler. Any mark crossing an edge gets drawn again on the opposite side, offset by exactly one tile. The repeat stays asymmetric, which usually looks better, but it needs the artwork to be measurable geometry rather than a flat image.
Why can I see a grid in my repeat? Nearly always one dominant element landing in the same spot in every tile. Vary its scale and rotation across the tile, or move to a half-drop repeat.
Mirror or wrap? Mirror when the artwork is dense and you want a guarantee. Wrap when the design has a direction to it and the symmetry would be obvious.