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arXiv · 2608.00022

Hex9: A Quasi-Authalic, Quasi-Continuous Hexagonal DGGS on the Reference Ellipsoid

Ben Griffin
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Abstract

Discrete global grid systems are conventionally designed over a prior coordinate reference system and inherit its compromises. Hex9 inverts the direction of design: we ask what requirements a hierarchical grid must satisfy to be geometrically coherent -- intrinsic orientability, flat mode transport, vertex closure, refinement commutativity -- and show that these requirements essentially determine the grid. The admissible cell primitive is the triangle; the admissible seed is the octahedral triangulation of S^2; admissible refinement is uniquely aperture 9, and uniquely up to chirality, with exactly one orientation of the hexagonal dual lattice per chirality -- fixed by a machine-verified exhaustive enumeration. The surviving structure is a shifted-aperture-9 hexagonal hierarchy in which every cell carries a unique address derived from the construction alone: truncated at level L, the address is a DGGS zonal identifier in the sense of OGC Topic 21; carried to the limit, a function recovers from it a point on the reference ellipsoid to arbitrary precision. The addressing is quasi-continuous -- position is recoverable everywhere except on a measure-zero set of seams -- rather than continuous in the strict ISO 19111 sense; the same object serves as both zonal identifier and position-recovery coordinate, with no prior coordinate reference system as input. A separable geometric realisation -- an analytical octahedral base projection composed with an optimal-transport-derived area-correcting warp -- places the grid on WGS84 with quasi-uniform cell areas: at level 5, 99% of the 708,588 cells lie within 0.005% of ideal area, with residual deviation confined to the six octahedral vertices required by the topology. The combinatorial grid is projection- and ellipsoid-independent; only the warp is specific to the reference body, and is recomputable for any ellipsoid, terrestrial or planetary.

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Ben Griffin. 2026-07-06. Hex9: A Quasi-Authalic, Quasi-Continuous Hexagonal DGGS on the Reference Ellipsoid. https://arxiv.org/abs/2608.00022

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