Dualizing the discrete directions

The grid on the left, its dual on the right — one γ, one view

De Bruijn's construction needs only a set of directions, a phase for each, and the rule that every crossing becomes a tile. None of that asks the directions to be evenly spread. So: hand it the discrete five and look at what comes out.

Two things the pentagrid never has to separate come apart here, and they are the two switches.

Spacing — how far apart family j's lines are. All equal isolates the effect of direction alone; the wheel's own lengths (1.0462, 1, 1.0288, 1.0288, 1) are what the lattice actually supplies, and no two families then have the same spacing.
Edges — the vector the dual builds each tile edge from. Unit vectors are de Bruijn's, and give rhombs. The wheel vectors themselves give parallelograms with unequal sides, which is the reading that could reproduce the quadrille tiles, whose edges are 4, √13 and √17.

geometry wheel spacing wheel edges ghost the real grid generation limit

tile shapes
gains of the dual map
spacing λ
edge lengths
∑v_j
directions

the multigrid — lines, crossings

its dual — a tile per crossing

Drag to pan, scroll to zoom, on either side.

What to watch for

Six shapes, not two. The ten family pairs give six distinct angles — 34.6438°, 36.4939°, 37.7245°, 69.2876°, 71.1377°, 74.2184° — where the pentagrid gives two, 72 and 144. The mirror symmetry is what keeps it to six rather than ten.

The dual map shears. Its gain comes from ∑vvᵀ, which for evenly spread directions is (n/2)I — a similarity, one number. Here it is diagonal but not isotropic, so there are two gains and the dual is stretched along one axis relative to the other. The readout gives both and the gap between them.

The index almost survives. ∑v_j over the five directions is (0, 0.0012) — exactly zero across the mirror, and only just off it the other way, where the pentagrid's is zero outright. So ∑K is not a height function but drifts from one slowly, which is a more interesting failure than a flat one: there is no Wieringa lift and no four levels, but a patch may look as though there were. The rhomb groups, the arrows and the exact regularity criterion are five's, and are not offered.

They look regular until you have something to compare them with. Five families of evenly spaced lines read as a regular mesh whatever their angles, so ghost the real grid draws the pentagrid behind in dashed gray: same γ, same spacing, 36° apart. Where the red lines leave the gray ones is the whole of the difference. At the limit the gaps are 34.64, 36.49, 37.72, 36.49, 34.64; at generation 1 they are 43.60, 31.33, 36.87, 36.87, 31.33, which is a long way off five 36s.

Periodic at every finite generation — wind the slider back. The wheel's points are lattice vectors until the limit, so the slopes are rational and the grid, with it the tiling, is periodic. Generation 1 is (0, 6), (5, 2), (3, −4) and their mirrors — 3-4-5 slopes — and its rhombs are 31.33°, 36.87°, 43.60°, nothing like the limit's 34.64°, 36.49°, 37.72°. Wind forward and the angles crawl in: 33.69/34.21/39.21 at 2, 34.99/35.57/38.88 at 3, 34.51/36.86/37.26 at 4. The quasiperiodicity is the limit of a sequence of periodic tilings, and you can watch the period grow out of sight.