We introduce TriHex-Cluster, a hierarchical overlapping clustering framework built on the self-similar geometry of the triangular lattice (6-regular planar graph). The primary algorithm is regime C (greedy 2-packing followed by Voronoi completion), a practical hierarchical clustering method producing disjoint clusters with the Voronoi-contact graph
as the next-level graph and aggregation complexity
(embedding cost excluded). On regular triangular domains with near-perfect packings, regime C achieves
per level; the measured depth on finite data is
. Two variants complete the framework: regime A (full-overlap edge-induced,
) adds native overlap semantics by preserving the EI meta-graph 6-regularity without reducing the vertex count; regime B (deterministic index-7 sublattice,
with
) is a theoretical construction establishing an exact sublattice density ratio of 7 per level on the infinite lattice
, and exact termination in
levels on finite periodic domains with
. Unconditional results: EI 6-regularity in regime A; perfect star-tiling and exact index-7 structure in regime B; strict hierarchy via Voronoi-completed clusters in regime C; tile-shape alternation proven at levels 1–2 (hexagonal, then triangular-like) and conjectured, with numerical verification, beyond;
(as an unnormalised SSE) strictly admissible and
weakly admissible. Aggregation complexity, embedding excluded:
in regime C,
in regime B,
in regime A. We provide a fully reproducible reference implementation (trihex2, MIT-licensed) with extensive parameter sweeps on UCI benchmarks, synthetic Gaussians, non-convex shapes, and overlapping distributions. The genuine contributions of the framework are the multi-scale hierarchical structure with provable geometric guarantees and, in regime A, native overlap semantics that no hard-clustering baseline can provide. A central empirical finding concerns the embedding: an ablation isolating the 2D-lattice projection shows it to be the main bottleneck, and a lattice-free variant that runs the same combinatorial core directly on a
k-nearest-neighbour graph in the original feature space—with no embedding and no quantisation—removes the projection entirely and improves accuracy on six of seven pilot datasets. With a frozen, fully unsupervised meta-selection rule (graph-geodesic arbitration between a convex-consensus and a graph-min-cut candidate, no per-dataset tuning), this variant reaches ARI
on
moons and
on
circles, where
k-means, HAC, and GMM all collapse to
and
, respectively. On a 73-dataset benchmark (23 real UCI, 50 synthetic, all loaded with validated class labels), TriHex is the
most frequently best method on the synthetic panel (
win rate) and close behind GMM overall (
versus
), while having the
lowest mean ARI—the signature of a specialist: it dominates on non-convex structure (rings, spirals, manifolds) and is outperformed on convex tabular data, where we make no claim of superiority. We also report a genuine robustness limitation: with the
default configuration, TriHex fails on Cancer (ARI
, essentially uncorrelated with the ground truth) because the default lattice over-fragments a two-class problem; competitive performance requires a dataset-appropriate configuration, and we report this explicitly rather than only the best configurations. On overlapping Gaussians, regime A detects the boundary points that the data-generating process itself classifies as ambiguous with precision
at heavy overlap (
); the detector
over-flags as the clusters separate (precision falls to
at
), so its usefulness is confined to the strong-overlap regime. Within that regime, it provides a measurable capability unavailable to hard-clustering baselines.
Full article