hhg9 0.1.0a6

Hierarchical hexagonal grid on an octahedral earth projection

hhg9 / Hex9

This is Hex9 is an ongoing project exploring (and developing) a novel hierarchical hexagonal grid (HHG) system for global projections. It is well-suited to population mapping, environmental modeling, heat-mapping, hex-binning, and other geospatial analyses.

Examples and further documentation may be found on the git repository.

Changelog

0.1.0a5 More bug-fixes - especially regarding grids.
        A new example, showing multiple hexagons over Chiginagak volcano, AK.
        A little more documentation.
0.1.0a4 Couple of small bug-fixes,
        A couple of new features
0.1.0a3 Rewrite of the hex address and hex polygon code.
        Hex addresses are not backward compatible, so several files
        (documentation) will need to be revised.
        Examples have all been refactored, and tested accordingly.
0.1.0a2 Improved some documentation; added more licence declr.
0.1.0a1 Modifying Numpy minimum - bitwise_count dependency.
        Removed redundant methods in neighbours.
        hex_layer is still unreliable
0.1.0a0 Initial Package

Why hexagonal tiling

• Hexagonal tiling is unique among regular tilings: all neighbors share an edge.
• Hexagonal tiling corresponds to optimal circle packing.
• However...
  • Hexagons cannot be tiled with hexagons.
  • The sphere cannot be covered by hexagonal tilings
• So, although ideal HHG have a long history in geospatial modeling...
  • Most global HHG are either flat or approximate, or not entirely hexagonal.
  • Existing approaches involve trade-offs, such as
    • Approximating the sphere with slight distortions.
    • Limiting the number of hierarchical layers.
    • Supporting only partial support for transitions between layers.
    • Deriving the hexagonal grid from spherical geometry
    • Needing precomputed databases of fixed points.
    • Requiring additional polygon types (commonly pentagons) for closure
• Hex9 presents a new approach to the “holy grail” of HHG; It aims to reduce some of these constraints while remaining early-stage and not yet production-ready (Autumn 2025). Where does it fail? The main concern is that the current Hex9 octahedral projection is not amazingly equal-area, especially across a global level. However, it's hexagons are (for the main part) pretty round, and the area is pretty stable (±7% or so, globally), however local variation is far more stable. For example, when examining nations, the variation is not so strong (ok maybe not so much Russia, or Canada). The authalicity example reveals global variation very clearly.

Why Hex9

Grid-Projection Decoupling

The Hex9 grid is fully decoupled from the underlying global projection. The logical hex grid exists independently of any coordinate reference system and can be applied within any octahedral global projection. While Hex9 comes with a derived projection for this project, the grid itself is projection-agnostic, while the derived projection relies on the Hex9 grid for root-finding operations.

This separation ensures:

• The grid can be reused across projections without loss of structure.
• Analyses and visualizations remain consistent, regardless of map distortions in the underlying projection.

Accuracy

Hex9 supports near-lossless forward and inverse mappings between grid addresses and geodetic coordinates. Accuracy is maintained even at extreme resolution: At layer 30, hexagons are accurate to about 1µm. Example round-trip accuracy for several landmarks:

The grid addresses below might be non-canonical (still prone to change). Current Grid address semantics are as follows. The Great Pyramid at Giza is identified at 29°58'45.817792004858"N, 31°8'3.457294813097"E This can be projected to a (reversible) hex-grid address at any given layer. Here it is at layer 36. 0070143470686461861005464283175018506 Broken down.. 0-0701434706864618610054642831750185-06 The first digit is the root hexagon at layer 0. This is one of 12 hexagons that cover the earth's surface, so the range of values are 0...B (where A,B = hexagons 10,11 respectively). 07...185 These are the sub-hexagons at each respective layer; so as there are 35 of those, along with the root hexagon, we know this address is at layer 36. 06 - these act as 'metadata tail', and provide the minimum amount of metadata to convert the address back to it's longitude/latitude. (The calculations for this are found in h9/addressing.py).

There is also a 'key' version of the address. This is used for identifying which points are within a specific hexagon when hex-binning (a main purpose for this sort of grid). In this case, the 'metadata tail' is reduced. In this case, the great pyramid is: 0070143470686461861005464283175018500 Importantly, and despite best efforts, the hexagon key of a higher layer cannot be solely derived by chopping the string. The c2 identity is an important aspect for decoding ids 6,7,8 Likewise the octant face mode is very useful. For the first 11 layers, the pyramid address key is as follows. 0:00 1:002 2:0072 3:00704 4:007012 5:0070140 6:00701430 7:007014344 8:0070143474 9:00701434700 10:007014347064 *

Great Pyramid

29°58'45.817792004858"N, 31°8'3.457294813097"E (Reference Coordinates)
29°58'45.817792004871"N, 31°8'3.457294813071"E (Roundtrip Coordinates)
0070143470686461861005464283175018506 (L35 Grid Address)
∂0.984436nm (roundtrip via GCD<->Hex9 Label) in Geodesic distance (nanometres)

Stonehenge

51°10'43.672800075871"N, 1°49'34.283450385600"W (Reference Coordinates)
51°10'43.672800075845"N, 1°49'34.283450385640"W (Roundtrip via Grid Address)
4352164061084274326815104253457062812 (L35 Grid Address)
∂1.314516nm delta (Geodesic.DISTANCE)

Moai on Rapa Nui

27°7'32.827199567155"S, 109°16'36.740870832014"W (Reference Coordinates)
27°7'32.827199567142"S, 109°16'36.740870832014"W (Roundtrip via Grid Address)
b501888670665528318830382731266380195 (L35 Grid Address)  
0.352774nm  delta (Geodesic.DISTANCE)

North Pole (edge case)

90°0'0.000000000000"N, 0°0'0.000000000000"E (Reference Coordinates)
89°59'59.999999999898"N, 12°16'35.957736079345"E (Roundtrip via Grid Address)
422222222222222222222222222222224832a (Grid Address)
∂3.174534nm  delta (Geodesic.DISTANCE)

Intuitive uint64 Addresses

Hex9 supports various uint64 addresses in a directly intuitive manner.

For example, one of the Nazca Spirals - at 14.679806S, 75.101925W has the uint64 address 0xb404124850835306 (in hexadecimal). 'b' here represents hexagon 11 This is (by default) a 'Layer 13' Address; The hexagons are in bytes 0..13 The final 2 bytes are the meta, used to convert the address back to another location. To convert the address to an identifying hex key, this can be done by masking the final byte with 0x70 (or using a far more reliable internal method).

0x40412...
  ↑↑↑↑↑
  ||||└─── Layer 4, hexagon 1
  |||└──── Layer 3, hexagon 4
  ||└───── Layer 2, hexagon 0
  |└────── Layer 1, hexagon 4
  └─────── Hexagon 'B'

When the uint64 address is depicted in hexadecimal, the global address is revealed and may be readily eye-balled with a crib - see the following image that traces the first 5 regions 8,2,5,4,A - each one covering 1/9th the area of the preceding layer.

Performance

Hex9 efficiently handles large datasets. For example, 25 million sparse GCD points can be mapped in far less than 20 minutes on standard desktop hardware.

Summary

Thanks to its decoupled, fractal-based structure, Hex9 allows direct projection of spatial data onto hexagonal grids. It has strong roundtrip integrity. It is reasonably authalic, with a global RSME of about 7%, but smooth authalic change - meaning that at local levels, data sampling is representative.