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_octspace. 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.
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 barycentric | 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.
- 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.
The Great Pyramid at Giza (29°58′45.82″N, 31°8′3.46″E) at layer 36:
0070143470686461861005464283175018506
Breaking it down:
0 - 0701434706864618610054642831750185 - 06
│ ╰──────────── 35 layer digits ───────────╯ ╰─ 2-digit metadata tail
╰─ root hexagon (0–B, 12 global roots)
The root hexagon is one of 12 that cover the Earth. Each subsequent digit subdivides the enclosing hexagon by 9. The two-digit tail carries the minimum metadata needed to reverse the address back to geodetic coordinates.
There is also a key form — used for hex-binning — where the tail encodes only containment, not full reversal. The Great Pyramid key at layer 36 is:
0070143470686461861005464283175018500
Truncating a key to a shorter length does not directly yield the parent-layer
key; the c2 identity and octant face mode must be carried explicitly.
First 11 layers of the pyramid address key:
0: 00
1: 002
2: 0072
3: 00704
4: 007012
5: 0070140
6: 00701430
7: 007014344
8: 0070143474
9: 00701434700
10: 007014347064
(here, 'nm' is being used in its SI context- so it means nanometres and not nautical miles!)
Great Pyramid
29°58'45.817792004858"N, 31°8'3.457294813097"E (reference)
29°58'45.817792004871"N, 31°8'3.457294813071"E (round-trip)
0070143470686461861005464283175018506 (L35 address)
∂ 0.984 nm geodesic error
Stonehenge
51°10'43.672800075871"N, 1°49'34.283450385600"W (reference)
51°10'43.672800075845"N, 1°49'34.283450385640"W (round-trip)
4352164061084274326815104253457062812 (L35 address)
∂ 1.315 nm geodesic error
A Nazca Spiral at 14.680°S, 75.102°W has the uint64 address
0x8515044362475050. Reading the hex nibbles from the top:
0x8515044362475050
↑↑↑↑↑
||||└─── Layer 4, hexagon 0
|||└──── Layer 3, hexagon 5
||└───── Layer 2, hexagon 1
|└────── Layer 1, hexagon 5
└─────── Layer 0, hexagon 8
When written in hex, the first digits of the address spell out the root-to-leaf path through the hierarchy — the address is directly eye-readable with a crib. See the image below for the first five regions 8 → 5 → 1 → 5 → 0, each covering 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:
- uint128 string addresses (32 hex nibbles) — indexes to layer 28 and beyond.
- uint64 integer addresses (16 hex nibbles) — layer 13 by default.
At layer 30, a single hexagon covers roughly 1,000 nm² (one thousand square nanometres).
- 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.
- 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
- Explore the grid and experiment with it.
- Suggest improvements to enhance understanding or usability.
- Contribute bug fixes or enhancements via pull requests.
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.
