A wallpaper pattern is one motif repeated across the plane by rigid motions — moves that slide, flip, or turn the cloth without stretching it. The lattice already supplies the sideways slides; here you choose the flips and turns.
A mirror reflects across a fixed line. A glide reflects and then slides half a cell along that line — like left, right, left footprints. A rotation turns the motif about a centre; on a square cloth only half-turns and quarter-turns fit.
Switch on two perpendicular mirrors and watch closely: a rotation appears that you never asked for. Reflecting twice across crossing lines always equals one turn about their crossing — you get the half-turn for free.
Every combination lands on one of the plane symmetry groups. There are exactly seventeen; this square loom can weave ten of them, named p1, pm, pg, p2, pmm, pmg, pgg, p4, p4m, p4g.
A thread caught in the loom. The lesson continues without it — the pattern you were weaving is safe.