SAME MOVES

With Trace on, focus this field and use Left or Right to select one of all 720 orderings; Shift changes by ten. Use the Step slider to inspect individual moves. Global keys: 1–3 ordering, R trace, C commute, T theme, Space play, brackets step.

One frozen catalogue of six rigid motions of the plane. Move k turns in place by its angle, then walks its radius straight ahead along the new heading. Composition runs left to right in the body frame. The six turn angles are 120, 30, 20, 40, 60 and 90 degrees and sum to 360 exactly; the six radii are 0.26, 0.41, 0.26, 0.28, 0.59 and 0.30 field units and sum to 2.10 exactly. Every one of the 720 orderings at the original λ = 1 therefore turns through the same total angle, ends on the starting heading, and walks the same total length; only the endpoint differs. Net turn is printed to twelve decimals as a check and is order-independent to machine precision, with a maximum deviation of 8.9e-16 radians across all 720 orderings. Trace is a complete enumeration of 6 factorial orderings, not a sample. At λ = 1, all 720 endpoints are distinct: the closest pair is separated by 0.00829 field units against a trace diameter of 2.15599, a ratio of about 1 to 260, so the closest pair can render within a few pixels of each other. Spread is the exact diameter of that endpoint set, computed by brute force over all 258,840 pairs. Shuffled is one permutation fixed in the source, not a draw made on load. Commute rescales all six turn angles by a factor λ and is a control rather than a fourth ordering: it changes the moves. At λ = 0 the moves are pure translations, the group is abelian, and every ordering ends at (2.100000, 0.000000); the 720 computed endpoints agree there to within 5e-16 units, the residue of floating-point summation order rather than a property of the geometry. At intermediate λ the net turn is 360λ degrees, so all orderings still share a final heading, which differs from the starting heading except at integer λ. During a reordering the drawn path is a straight-line interpolation between two vertex lists and is not itself a realisable ordering; readout statistics are withheld until it settles. The field is drawn with equal x and y units in a single fixed window that accommodates every path at every λ, so no comparison is rescaled.