Wiggle room

How much can γ move before the pattern changes?

The five offsets look like five degrees of freedom, and γ₀ + ··· + γ₄ = 0 cuts that to four. But two of those four do nothing at all. Shifting every γ_j by w · v_j moves each crossing to x₀ − w, and its K-tuple to

⌈ v_i · (x₀ − w) + γ_i + w · v_i ⌉  =  ⌈ v_i · x₀ + γ_i ⌉

— unchanged. Same tuple, same vertex. The pentagrid slides and the tiling does not move. So the pattern depends on just two numbers: γ's component in the perpendicular plane, spanned by the same five angles doubled. That plane is the panel on the right.

Fix a patch of the tiling and ask which γ keep it. Each vertex survives exactly when its five strips still share a point — a linear condition — so the answer is a convex region, and every vertex you add to the patch can only shave it down. That region is the wiggle room, and you can measure it.

drag the dot — double-click the right panel to re-anchor the patch here

Inside the region the marked patch is intact. Cross the boundary and a vertex fails — hollow circles are the ones that no longer exist. That crossing is a phason flip: the tiling rearranges locally while everything far away stays put. Growing the patch shrinks the room, but not smoothly — most vertices you add were already forced by the ones inside them, and the area only drops when a genuinely new constraint binds.