Give all five families the same offset c and the pentagrid keeps its five-fold symmetry, so the tiling does too. Sliding c from 0 to 1 walks the whole family — and it passes through exactly two Suns and two Stars, paired by the mirror Σγ ↔ −Σγ.
Roll the wheel over a line to slide it along its own direction — hold shift for a tenth of the step. One line is free: it absorbs whatever the others do, holding Σγ. All offsets are read modulo 1, since only the fractional part moves the grid.
Drag to pan, scroll to zoom. Tiles are colored by the P1 pentagon that owns them — Pe5, Pe3, Pe1 — and the black ring marks the origin.
Every one of these caps shows five fat rhombs at the origin. That is why the vertex tells you nothing: a five-fat vertex is a star rhomb group wherever it appears, but it is a Pe5 center only when its Wieringa index is an extreme of the tiling's four levels. The origin's K-tuple is (1,1,1,1,1), so its index is always 5, while the levels run [Σγ+1, Σγ+4]:
A star's center belongs to no cluster at all, because it sits inside an St5 — a star-shaped gap. That is why hunting for a “star vertex” never finds one, and why this needed a cluster recognizer rather than a look at the rhombs.
The deca line is the other family here: phases (0, e, −e, −e, e), mirror-symmetric about the vertical axis with the total held at zero. At e = ±1/10, ±2/10, ±3/10 the origin carries the deca — a Pe3 on the axis with a Pe1 either side, and negating e flips it end for end, which is why 100 and 900 are a pair. At ±4/10 the origin sits in an St5 gap — a star; at 5/10 the line hits a singular couple; at 0 it is the 5-fold. The numbers are the thousandths the reticulum shows: 900 is −1/10.
At c = 0 all five lines pass through the origin. That is singular — no three lines may be concurrent, let alone five — and the dual of a five-fold crossing is not a rhomb but a decagon. It is the one cap that is not a tiling at all — and it is not the deca, which is the queen patch: one Pe3 with two Pe1, ten rhombs, what this singularity resolves into under a mirror-symmetric nudge (see the Caps row on the method and grow pages). The guard is switched off on this page so you can visit it.
Slide a single line off the others and the five-fold symmetry breaks: the grid is no longer the same seen from five directions, so the tiling inherits no rotation and the verdict stops applying. Everything here is a statement about equal offsets.