System capabilities and traversal costs

For choosing a system, start with Choosing a grid. This page records system traits, boundary algorithms, and interoperability limits for implementors.

Registered global systems

The size column is cellsize in km at levels 0, 4 and 8 — the side of a square with the median cell's area, which is the only size a system whose cells differ in shape and area has. Ask levelfor for the level a given resolution wants rather than reading a level off this table.

system	cells at level l	≈ size (km) at l = 0 / 4 / 8	cell shape	equal-area
IGeo7System	10·7^l + 2	6520 / 146 / 3.0	hexagons + 12 pentagons	approximately; pentagons differ; see IGeo7.equal_area_steradians
H3System	120·7^l + 2	2026 / 42.6 / 0.87	hexagons + 12 pentagons	no (libh3's gnomonic faces)
HEALPixSystem	12·4^l	6520 / 408 / 25.5	curvilinear diamonds	yes, exactly 4π/(12·4^l)
A5System	12, 60, then 60·4^(l-1)	6524 / 365 / 22.8	pentagons (Cairo-style)	on its ellipsoid; unit-sphere cell_area varies by about 1%
S2System	6·4^l	9220 / 584 / 36.1	geodesic quadrilaterals	no; ~2.08× within-level spread
ISEA4RSystem	10·4^l	7142 / 446 / 27.9	rhombi on ten diamonds	yes, exactly 4π/(10·4^l)

Important cross-system traits:

• Neighbour degree varies, and Vertex/Edge need not coincide. maxneighbors is an upper bound. IGeo7/H3 have degree 6 except for 12 pentagons of degree 5, with identical connectivities. A5's 4-valent corners give maxneighbors(A5System(), Vertex()) == 11 against maxneighbors(A5System(), Edge()) == 5; at resolution 1, some cells have 11 vertex-neighbours and 3 edge-neighbours. Above level 1, ISEA4R has ten degree-9 cells at icosahedral vertices 0 and 11, thirty degree-7 cells, and degree 8 elsewhere; level 0 is 6-regular.

• node_extent. S2, ISEA4R, and HEALPix provide exact uninflated subtree caps. IGeo7, H3, and A5 use inflated caps; A5 sets cap_inflation to 1.75.

• has_sorted_subtrees. True except for A5, whose canonical order has not established the two-sided descendant_range contract.

• has_congruent_refinement. True for HEALPix, S2 and ISEA4R, whose four children tile their parent exactly; false for IGeo7, H3 and A5. A maxcells MultiOrderCellSet descends through meeting cells alone where it holds, and through cells that miss as well where it does not.

• has_direct_location. True for every system here: each names the cell containing a point from the point's coordinates, so a PartialGrid over any of them locates through its complete level rather than by searching its own cells.

• border on a subtree region. IGeo7, H3, HEALPix, ISEA4R, and S2 provide O(border) walkers; A5 uses the O(subtree) fallback. Each is a resumable EdgeCellIterator / InnerCellIterator in O(depth) memory, which is what border and interior read.

• halo on a subtree region. The same boundary from outside, as a resumable SubtreeHaloIterator in O(depth) memory. l == level(c) is the cell's own one-ring on every system, emitted ascending without a sort. Deeper, five of the six take a specialization and A5 does not:

  • HEALPix, S2 and ISEA4R walk the width-one band around their square block, one pruned quadtree descent per face the halo touches, taken in face order — which is canonical order, so the merge is concatenation. O(halo + depth) time. A block nowhere flush with its face edge has a closed-form count (4·side + 4 under Vertex, 4·side under Edge) and therefore a length; a block that crosses a seam has a conservative rectangle band that every candidate is filtered through the native one-ring before yielding, and declares SizeUnknown().

  • IGeo7 and H3 approach the halo from the root's own same-level neighbours. One level down the calibration is already the answer; deeper, each neighbour's subtree-border automaton is seeded with an exposed-direction arc calibrated by observation and walked to the target. Neighbouring subtrees occupy disjoint descendant_ranges, so walking them in range order is canonical without a heap or a seen-set, and every candidate goes through the native one-ring. Neither hexagonal engine declares a length: the observed counts (3^(d+1) + 3 around a hexagon, 5(3^d + 1)/2 around a pentagon) are validated by enumeration, not derived from the transition recurrence.

  • A5 has no descendant_range to prune a descent by and no validated boundary automaton, so it scans the target level in O(1) memory and O(ncells) time. Its aperture and Hilbert-like indexing are not evidence of a square fast path, and none is inferred from them.

  The generic outside-first hierarchy walk — canonical-order candidates with node_extent cap pruning — is still what every specialization's guards return to and what a newly registered system inherits.

• halo on any other region. The same question about an arbitrary subset, and always an iterator. A rooted grid holding a complete subtree delegates to SubtreeHaloIterator and keeps its system's specialization; everything else — a hole, a forgotten root, an arbitrary id list — takes an outside-first walk against membership, pruned by the subset's own index spans rather than by geometry: a block the subset holds entire is retired by one lookup, and a block no NEIGHBOUR of which it touches is retired by the coarse-containment law. The walk follows the subset's boundary, so its cost is the halo's and not the subset's. A cell punched out of the middle of a subset is outside it and touches it, so it joins the halo. A5 is again the exception to the delegation: without has_sorted_subtrees there is no way to recognise a held subtree, so even its rooted complete grid takes the subset walk — to the same answer.

• adjacency. Every one-ring of a region at once, CSR and counter-clockwise, with the out-of-region members dropped (halo = 0), addressed into a [region; halo] buffer (halo = 1), or marked with 0 in place (halo = :mark). The two complete-width shapes preserve slot indices against the canonical one_ring, so a direction code is a property of the cell; the clipped shape preserves order only. Rows exist for in-region indices alone, so halo above 1 throws and points at grow.

• Cross-level adjacency (member_neighbors). Boundary sharing in the geometric sense on HEALPix, S2 and ISEA4R, whose four children tile their parent exactly; the hierarchy's own relation on IGEO7, H3 and A5, where they do not and a member's footprint is not its descendants' union.

Interoperability caveats

• ISEA4R's face pairing, diamond numbering, and (x, y) orientations are package-defined and anchored on vertices (0, 11). Compatibility with external ISEA4R identifiers, including DGGAL, is not claimed; establish a fixture-derived permutation before interchange.

• S2's native 64-bit s2_cellid is not an available reindex scheme. The scaffold ordinal is the canonical identifier.

Use levels and levelgrid to construct queryable grids. Each system module documents its identifier codec and optimized operations.

Multi-order budget observations

Budget-mode coverage is exact on congruent systems. Tests also measure misses on noncongruent systems; those measurements are not guarantees for arbitrary targets. The initial implementation notes recorded these fractions of a state-sized target missed:

Measurement	IGeo7	Authalic IGeo7	H3	A5
Target outside emitted polygons	2%	3%	15%	30%
Intersecting reference-level cells not represented	1%	2%	2%	18%

Consult the multi-order budget tests for their actual targets and thresholds. Fixed-level queries can continue overhang descent to a fixed depth. Budget queries cannot rely on that same stopping rule, so their geometric and leaf coverage must be assessed separately. Expanding a budget set reports its represented descendants; it does not repair cells the budget traversal missed.