Hex9

Hex9 is an ongoing project developing a novel Discrete Global Grid System (DGGS) built on a purpose-designed octahedral coordinate reference system (CRS). These are two distinct, layered components:

• The CRS pipeline maps any point on the WGS84 ellipsoid through a chain of coordinate domains — ending in a pair of authalic barycentric coordinates (b_oct) on the surface of a unit octahedron. This mapping is continuous, bijective, and achieves sub-nanometre round-trip fidelity.

• The DGGS lives entirely in b_oct space. The octahedral face is subdivided hierarchically into half-hexagons, producing a hexagonal grid whose geometry, hierarchy, and addresses are all deterministically derivable from the coordinates — no look-up tables, no precomputed databases.

Hex9 supports population mapping, environmental modelling, heat-mapping, hex-binning, and other geospatial analyses.

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The CRS Pipeline

The pipeline from geodetic coordinates to the grid's native space is:

g_gcd  →  r_gcd  →  c_ell  →  c_oct  →  b_raw  →  b_oct

Step	Domain	Description
g_gcd	Geodetic (degrees)	Standard lon/lat on WGS84
r_gcd	Geodetic (radians)	Same, converted to radians for internal use
c_ell	ECEF Cartesian	3D Cartesian on the WGS84 ellipsoid
c_oct	Octahedral Cartesian	Projected onto the unit octahedron via the AK formula
b_raw	Geometric barycentric	Per-face barycentric coordinates (unwarped)
b_oct	Authalic-ish barycentric	Near Equal-area–warped barycentric — the grid's home space

The non-trivial step is c_ell ↔ c_oct: the forward direction uses the AK tangent–normalisation formula; the inverse has no reliable closed form and is solved numerically with a domain-specific hierarchical root-finder. A PROJ-compatible plugin (h9_boct) exposes the full g_gcd ↔ b_oct pipeline to any PROJ-aware application.

For display, b_oct coordinates are projected into a 2D net (n_oct) that is fully independent of any cartographic map projection chosen for the final output.

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The DGGS

Why hexagonal tiling?

• Hexagonal tiling is unique among regular tilings: every neighbour shares a full edge.
• Hexagonal tiling corresponds to optimal circle packing.
• However:
  • Hexagons cannot be tiled with hexagons.
  • The sphere cannot be covered by a pure hexagonal tiling.
• So, although ideal HHGs have a long history in geospatial modelling, most global HHGs involve trade-offs:
  • Slight distortion to approximate the sphere.
  • A finite number of hierarchical layers.
  • Partial or approximate inter-layer transitions.
  • Additional polygon types (commonly pentagons) for closure.
  • Precomputed databases of fixed reference points.

Hex9 takes a new approach: the grid is defined in the warped barycentric space b_oct, where the tiling is exact and the hierarchy is unlimited in depth. The display is decoupled from the grid — the net projection (b_oct → n_oct) handles rendering, not addressing.

Addressing

A location encodes to a 128-bit UUID — the canonical Hex9 address. The Great Pyramid at Giza (29°58′45.82″N, 31°8′3.46″E) encodes to:

00701523-8841-1584-7742-040444242844

The same address as a Hex9 label (body.key):

0070152388411584774204044424284 . 4
│╰──────────── 31 body digits (layers 0–30) ────╯   ╰─ single-nibble key
╰─ root hexagon (0–B, 12 global roots)

The first digit is one of 12 root hexagons that cover the Earth; each subsequent digit subdivides the enclosing hexagon by 9, down to layer 30. The single key nibble carries the minimum metadata — the c2 identity and octant face mode — needed to reverse the address back to geodetic coordinates, so the UUID alone is self-inverting.

The same point truncated to a bin at the first 11 layers:

 0:  0.0
 1:  00.2
 2:  007.2
 3:  0070.4
 4:  00701.2
 5:  007015.4
 6:  0070152.4
 7:  00701523.4
 8:  007015238.0
 9:  0070152388.2
10:  00701523884.0

The key nibble (after the dot) is recomputed at each layer: truncating a deeper address does not directly yield the parent-layer bin — the c2 identity and octant face mode must be carried explicitly.

Accuracy

Hex9 addresses are binned — a coordinate resolves to the cell that contains it, and every point in that cell shares one address. The cell area at layer L is E / (12 × 9^L) for total Earth surface area E, so:

• UUID (layer 30) resolves to ≈ 1,000 nm² (≈ 30 nm across).
• uint64 (layer 14) resolves to ≈ 1.9 m² (≈ 1.4 m across).

For maximum positional fidelity, a raw representative string at layer 35 round-trips to within 7 nm globally — projection-limited, since the layer-35 cell itself is far smaller (≈ 0.02 nm²). ('nm' here is SI nanometres, not nautical miles.)

Great Pyramid — layer-35 raw

reference : 29°58'45.817792004858"N, 31°8'3.457294813097"E
address   : 00701523884115847742040444242847137308
∂ 1.029 nm  geodesic round-trip error

Stonehenge — layer-35 raw

reference : 51°10'43.672800075871"N, 1°49'34.283450385600"W
address   : 4352166438362084635124244718646587130d
∂ 3.162 nm  geodesic round-trip error

Compact uint64 addresses

A Nazca Spiral at 14.680°S, 75.102°W has the uint64 address 0xB404155374658884. Reading the hex nibbles from the top:

0xB404155374658884
  ↑↑↑↑↑
  ||||└─── Layer 4, hexagon 1
  |||└──── Layer 3, hexagon 4
  ||└───── Layer 2, hexagon 0
  |└────── Layer 1, hexagon 4
  └─────── Layer 0, hexagon B (root)

When written in hex, the leading digits spell out the root-to-leaf path through the hierarchy — the address is directly eye-readable with a crib: the first regions B → 4 → 0 → 4 → 1 each cover 1/9 the area of the preceding layer.

The layer area formula: for total Earth surface area E, a hexagon at layer L covers E / (12 × 9^L).

Hex9 natively supports:

• UUID / uint128 (128-bit, 32 hex nibbles) — canonical address to layer 30.
• uint64 (64-bit, 16 hex nibbles) — compact address to layer 14.

At layer 30, a single hexagon covers roughly 1,000 nm² (one thousand square nanometres).

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What's included

• A working Python implementation of the full CRS pipeline and DGGS, precise to sub-micrometre accuracy using geodesics.
• A PROJ-compatible plugin (h9_boct) for integration with PROJ-aware tools.
• Unit tests for the H9 grid engine.
• Examples and tutorials for geospatial tasks.

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Documentation

• Introduction — foundational insight and grid geometry
• Precision — domain table, projection table, root-finder detail, accuracy benchmarks
• Enumeration — mathematical foundations of the half-hexagon hierarchy
• Examples — step-by-step example guide (updated for 0.1.1a1)
• Early thoughts — historical notes

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How can I help?

• Explore the grid and experiment with it.
• Suggest improvements to enhance understanding or usability.
• Contribute bug fixes or enhancements via pull requests.

Why NOT H9?

Hex9 is a self-funded solo project written as a proof-of-concept and demonstrator, to show that new approaches to the HHG question remain open. It is not trying to compete with other HHG systems — it only aims to offer another approach, one that works particularly well for grid-layer transitions.