At t = 0 every tile is a point, sitting on the crossing that produced it, colored by mixing the two families whose lines met there. So what you are looking at is the pentagrid, with a dot on every intersection.
Push t up and the tiles grow. They cannot all grow in place — they are crowded in with their neighbors — so each slides to where the dual map sends it. The grid lines are never redrawn: each stays fastened to the midpoints of the two tile edges it passes between, so it is pulled along, bending as the tiles separate. At t = 1 the tiles are full size and edge to edge, every line has closed up seamlessly, and the five straight families have become the wandering ribbons of the tiling.
band at 0 leaves bare lines · 100% fills the tile
Drag to pan, scroll to zoom.
The page opens on the sun. Pick decagon from the reticulum's presets for Γ = 0 — every phase zero, the most singular pentagrid there is — and that turns out to be the best thing to watch. Where k lines meet at a point, all C(k, 2) rhombs that crossing makes start life at the same place: in view here, 54 crossings carry three rhombs each and one carries ten. At t = 0 they are superposed and you cannot tell a stack from a single tile. Push t up and they come apart, each to its own position — the hexagon opening into three, the decagon at the center into ten.
That is what a singularity is, shown rather than described: not a hole in the tiling but rhombs piled in a space that would hold them side by side. Move any phase off zero on the dials and nearly all the stacks vanish.
Tick P1 and each tile carries its share of the pentagon tiling. The pieces live in the tile's own coordinates, so at t = 0 every pentagon is gathered into the dots of the tiles it will occupy, and it assembles as they slide apart — the pentagons arrive with the tiling, not before it. What you are watching is one crossing becoming one face and its two gridlines becoming that face's edges; the pentagon is just what the face is wearing.
Tick 2k-gons to outline that space. It grows by the same law the tiles do — every corner starts on the crossing, scaled by the same 5/2, so the polygon opens out of the dot rather than appearing around it — and at t = 1 the rhombs exactly fill it. Three lines give a hexagon holding three rhombs, five give the decagon holding ten. Nothing picks which rhombic tiling of the polygon is the real one, because the construction does not: what is true is the space and what is stacked in it.
Nothing is faked in the middle. The attachment points start exactly on the straight grid line, and each tile's exit meets the next tile's entry exactly when t reaches 1 — the gap between them closes in a straight line at a constant rate. The wandering you end up with is not drift; it is the bounded wobble in f (x) = (5/2)x + const + wobble, and it never exceeds a rhomb edge or so.