This 2D companion applies the same rotation and mirror math as the 3D version directly on a flat canvas: every drawn segment is rotated by 360°/K about the center to build K copies (cyclic group Ck), and in Mirror mode each of those copies is additionally reflected to build the dihedral group Dk with twice as many segments. Rotational symmetry of order k means a pattern looks unchanged after turning it by 360°/k — the same principle that governs snowflakes, wallpaper patterns, and the crystallographic restriction that forbids ordinary crystals from having 5-fold symmetry.