rhombic 0.3.0

Lattice topology benchmarks: what happens when you replace the cube

rhombic

The bottleneck is not the processor. It is the shape of the cell.

A benchmarking library that compares cubic (6-connected) and FCC/rhombic dodecahedral (12-connected) lattice topologies across graph theory, spatial operations, and signal processing.

The Numbers

Metric	FCC vs Cubic	Scale
Average shortest path	30% shorter	125 – 8,000 nodes
Graph diameter	40% smaller	125 – 8,000 nodes
Algebraic connectivity	2.4× higher	125 – 8,000 nodes
Flood fill reach	55% more nodes	125 – 8,000 nodes
NN query speed	17% faster	125 – 8,000 nodes
Signal reconstruction	5-10× lower MSE	216 – 1,000 samples
Reconstruction isotropy	5-20× more uniform	216 – 1,000 samples
Embedding neighbor recall	+15-26pp at 1-hop	125 – 1,000 nodes
Information diffusion	1.4-2× faster	125 – 1,000 nodes
Edge cost	~2× more edges	(the price)

These ratios are stable across all tested scales. They hold at every size tested, consistent with derivation from Voronoi cell geometry rather than sample size.

Under Structured Weights (Paper 2)

Metric	Corpus vs Uniform	Scale
Fiedler ratio (direction-weighted)	2.3x → 6.1x	125 – 8,000 nodes
Path advantage	30% → 60% shorter	125 – 1,000 nodes
Consensus speedup	1.0x → 6.7x	125 nodes
Prime-vertex coherence	p = 0.000025	Single cell (40,320 permutations)

Heterogeneous edge weights amplify the FCC advantage. Direction-based weighting — mapping structured values to the 6 direction pairs of the FCC lattice — nearly triples the Fiedler ratio. The mechanism is bottleneck resilience: FCC routes around suppressed edges that strangle cubic lattices.

• Raw data and tables
• What the numbers mean

The Question

Computation is built on the cube. Memory is linear. Pixels are square. Voxels are cubic. Nobody chose this — it accumulated. Descartes gave us orthogonal coordinates. Von Neumann gave us linear memory. The cubic lattice is the spatial expression of Cartesian geometry.

Is the cube optimal? This library measures the alternative: the face-centered cubic lattice, whose Voronoi cells are rhombic dodecahedra. 12 faces instead of 6. The densest sphere packing in three dimensions (Kepler, proved by Hales 2005, formally verified 2017). The lattice that nature uses for copper, aluminum, and gold.

Read the full thesis →

Quick Start

pip install rhombic            # minimal (numpy + networkx)
pip install "rhombic[viz]"     # add matplotlib for plots
pip install "rhombic[all]"     # everything including dev tools

Reproduce all results:

python -m rhombic.benchmark

Use in code:

from rhombic.lattice import CubicLattice, FCCLattice

cubic = CubicLattice(n=10)     # 1000 nodes, 6-connected
fcc = FCCLattice(n=6)          # ~864 nodes, 12-connected

# Convert to networkx for any graph analysis
G_cubic = cubic.to_networkx()
G_fcc = fcc.to_networkx()

Results

Rung 1: Graph Theory (complete)

Four metrics, three scales, consistent ratios. The FCC lattice outperforms the cubic lattice on every measure of routing efficiency and structural robustness. The cost is bounded: ~2× edges for ~30% shorter paths and ~2.4× robustness.

• Raw data and tables
• What the numbers mean

Rung 2: Spatial Operations (complete)

The routing advantage translates. FCC flood fill reaches 55% more nodes per hop. Nearest-neighbor queries are 17% faster. Range queries return 24% more nodes per volume (denser packing). The cost: range query time scales with density — 3-5× slower for sphere/box queries at 8,000 nodes.

• Raw data and tables
• What the numbers mean

Rung 3: Signal Processing (complete)

Direct empirical measurements confirm the FCC advantage. FCC spatial sampling produces 5-10× lower MSE and 5-20× more isotropic reconstruction than cubic sampling at matched sample counts. The advantage peaks in the mid-frequency range (10-60% of Nyquist) and grows with scale — from +6 dB at 216 samples to +10 dB at 1,000. Above Nyquist, both lattices alias and cubic's axis alignment accidentally helps.

• Raw data and tables
• What the numbers mean

Rung 4: Context Architecture (complete)

Does the FCC advantage survive when the lattice organizes high-dimensional embedding data? FCC captures 15-26 more percentage points of an embedding's true nearest neighbors at 1-hop. Information diffuses 1.4-2× faster. Consensus converges 1.58× faster at moderate scale (500 nodes), though per-neighbor weight dilution reduces the advantage at 1,000 nodes.

• Raw data and tables
• What the numbers mean

Full experimental ladder →

FCC Embedding Index (complete)

A proof-of-concept ANN index that organizes high-dimensional embeddings on lattice topology. At matched node counts, the FCC index captures +7 to +20 percentage points more true nearest neighbors at 1-hop than the cubic index. The only variable is the connectivity pattern.

from rhombic.index import FCCIndex, CubicIndex, brute_force_knn

fcc = FCCIndex.from_target_nodes(dim=384, target_nodes=500).build(embeddings)
results = fcc.query(query_vector, k=10, hops=1)
recall = fcc.recall_at_k(queries, ground_truth, k=10, hops=1)

• Raw data and tables
• What the numbers mean

Paper 2: Weighted Extensions (complete)

What happens when edges carry heterogeneous weights? Seven experiments across two scales (lattice and single-cell). The FCC advantage amplifies under structured weights — direction-based corpus weighting pushes the Fiedler ratio from 2.3x to 6.1x. Prime-vertex coherence is significant at the optimal mapping (p = 0.000025 vs 40,320 alternatives). Spectral bottleneck creation is universal across 24-edge polytopes, not RD-specific.

• Raw data and tables
• What the numbers mean

Synthesis

The complete argument across all four rungs — cultural genealogy, empirical evidence, cybernetic interpretation, and practical recommendations.

• The Shape of the Cell: Cubic vs FCC Lattice Topology Across Four Domains

Philosophy

• Reproducible by default. Every result has code that generates it.
• The geometry is the argument. The numbers are the evidence.
• Cost is always reported alongside benefit.
• Sparse results are data, not failure.

Ecosystem

• Interactive demo — try it in your browser
• The essay — the thesis for humans
• PyPI — pip install rhombic
• Full synthesis — the complete argument across all four rungs
• Weighted extensions — what happens under heterogeneous weights

Papers

• Paper 1: The Shape of the Cell — four-domain topology comparison (arXiv cs.DS)
• Paper 2: Pure Number Architecture — weighted amplification and spectral structure

Contributing

See CONTRIBUTING.md. We're looking for new topologies, new metrics, and new rungs on the experimental ladder.

License

MPL-2.0 — Use freely. Modifications to library files shared back to the commons.

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