Spaces:
Sleeping
Sleeping
| <html lang="en"> | |
| <head> | |
| <meta charset="UTF-8"> | |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |
| <title>rhombic — Keep Your Cube, Add Six Bridges</title> | |
| <link rel="preconnect" href="https://fonts.googleapis.com"> | |
| <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> | |
| <link href="https://fonts.googleapis.com/css2?family=Cormorant+Garamond:ital,wght@0,400;0,600;0,700;1,400&family=Inter:wght@300;400;500;600&display=swap" rel="stylesheet"> | |
| <link rel="stylesheet" href="style.css"> | |
| </head> | |
| <body> | |
| <!-- ═══ HERO ═══ --> | |
| <section id="hero"> | |
| <div class="hero-content"> | |
| <h1>Keep Your Cube,<br>Add Six Bridges.</h1> | |
| <p class="subtitle">Lattice topology research proving cubic networks leave | |
| <span class="stat-inline">6.1×</span> connectivity on the table.</p> | |
| <div class="hero-links"> | |
| <a href="https://pypi.org/project/rhombic/" class="btn btn-primary">pip install rhombic</a> | |
| <a href="https://github.com/promptcrafted/rhombic" class="btn btn-ghost">GitHub</a> | |
| <a href="https://huggingface.co/spaces/timotheospaul/rhombic" class="btn btn-ghost">Demo</a> | |
| </div> | |
| </div> | |
| <div class="hero-canvas" id="rd-canvas"></div> | |
| </section> | |
| <!-- ═══ THE PROBLEM ═══ --> | |
| <section id="problem"> | |
| <div class="section-inner"> | |
| <h2>The Unexamined Default</h2> | |
| <p>Every computation — neural networks, simulations, databases — defaults to | |
| <strong>cubic lattice topology</strong>: 6-connected grids. Each node talks to | |
| 6 neighbors along 3 axis-aligned directions. It's the geometry of spreadsheets, | |
| voxel grids, and attention heads. Nobody chose it. It was just… there.</p> | |
| <p>The <strong>FCC lattice</strong> gives each node 12 neighbors along 6 | |
| face-diagonal directions. Its Voronoi cell is the <em>rhombic dodecahedron</em> — | |
| a 14-vertex polyhedron whose 8 cube corners and 6 octahedral bridges tessellate | |
| all of 3D space. Kepler proved it packs spheres optimally in 1611. | |
| Hales confirmed it in 2005.</p> | |
| <div class="side-by-side"> | |
| <div class="card card-cubic"> | |
| <div class="card-label">Cubic</div> | |
| <div class="card-stat">6</div> | |
| <div class="card-desc">neighbors per node</div> | |
| <div class="card-desc">3 axis-aligned directions</div> | |
| </div> | |
| <div class="card card-fcc"> | |
| <div class="card-label">FCC</div> | |
| <div class="card-stat">12</div> | |
| <div class="card-desc">neighbors per node</div> | |
| <div class="card-desc">6 face-diagonal directions</div> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ THE NUMBERS ═══ --> | |
| <section id="numbers"> | |
| <div class="section-inner"> | |
| <h2>The Numbers</h2> | |
| <div class="stats-grid"> | |
| <div class="stat-block"> | |
| <div class="stat-value fcc-color">2.3×</div> | |
| <div class="stat-label">algebraic connectivity<br><span class="stat-context">uniform weights (Paper 1)</span></div> | |
| </div> | |
| <div class="stat-block"> | |
| <div class="stat-value fcc-color">6.1×</div> | |
| <div class="stat-label">algebraic connectivity<br><span class="stat-context">direction-weighted corpus (Paper 2)</span></div> | |
| </div> | |
| <div class="stat-block"> | |
| <div class="stat-value fcc-color">30%</div> | |
| <div class="stat-label">shorter average paths</div> | |
| </div> | |
| <div class="stat-block"> | |
| <div class="stat-value fcc-color">40%</div> | |
| <div class="stat-label">smaller graph diameter</div> | |
| </div> | |
| <div class="stat-block"> | |
| <div class="stat-value accent-color">p = 0.000025</div> | |
| <div class="stat-label">prime-vertex mapping<br><span class="stat-context">non-accidental geometry affinity</span></div> | |
| </div> | |
| <div class="stat-block"> | |
| <div class="stat-value cubic-color">255</div> | |
| <div class="stat-label">passing tests<br><span class="stat-context">Python 3.10-3.12, MPL-2.0</span></div> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ THE MECHANISM ═══ --> | |
| <section id="mechanism"> | |
| <div class="section-inner"> | |
| <h2>Bottleneck Resilience</h2> | |
| <p>Heterogeneous weights create near-disconnection cuts — bottlenecks where | |
| information barely flows. The <strong>Fiedler value</strong> (algebraic connectivity) | |
| measures the worst such bottleneck.</p> | |
| <p>In a cubic lattice, each node has <strong>5 alternative routes</strong> around | |
| a blocked connection. In the FCC lattice, it has <strong>11</strong>. The more | |
| heterogeneous the weights, the more the FCC advantage grows.</p> | |
| <div class="amplification-chart"> | |
| <h3>Fiedler Ratio Amplification at Scale 1,000</h3> | |
| <div class="chart-bars"> | |
| <div class="chart-row"> | |
| <span class="chart-label">Uniform</span> | |
| <div class="bar-pair"> | |
| <div class="bar bar-ec" style="width: 41.7%"><span>2.55×</span></div> | |
| <div class="bar bar-dw" style="width: 41.7%"><span>2.55×</span></div> | |
| </div> | |
| </div> | |
| <div class="chart-row"> | |
| <span class="chart-label">Random</span> | |
| <div class="bar-pair"> | |
| <div class="bar bar-ec" style="width: 43.3%"><span>2.64×</span></div> | |
| <div class="bar bar-dw" style="width: 59.7%"><span>3.65×</span></div> | |
| </div> | |
| </div> | |
| <div class="chart-row"> | |
| <span class="chart-label">Power-law</span> | |
| <div class="bar-pair"> | |
| <div class="bar bar-ec" style="width: 50.1%"><span>3.06×</span></div> | |
| <div class="bar bar-dw" style="width: 55.2%"><span>3.37×</span></div> | |
| </div> | |
| </div> | |
| <div class="chart-row"> | |
| <span class="chart-label">Corpus</span> | |
| <div class="bar-pair"> | |
| <div class="bar bar-ec" style="width: 50.9%"><span>3.11×</span></div> | |
| <div class="bar bar-dw" style="width: 100%"><span>6.11×</span></div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="chart-legend"> | |
| <span class="legend-ec">Edge-cycled</span> | |
| <span class="legend-dw">Direction-weighted</span> | |
| <span class="legend-base">Paper 1 baseline: 2.31×</span> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ THE AGENT ═══ --> | |
| <section id="agent"> | |
| <div class="section-inner"> | |
| <h2>The Agent</h2> | |
| <p>We built a <a href="https://github.com/NousResearch/hermes-agent">Hermes Agent</a> | |
| equipped with the <code>rhombic</code> library as live tools. It runs experiments, | |
| generates visualizations, and explains the geometry — all in natural conversation.</p> | |
| <div class="agent-demo"> | |
| <div class="demo-placeholder" id="video-embed"> | |
| <p class="demo-text">Demo video coming soon</p> | |
| </div> | |
| </div> | |
| <div class="tool-grid"> | |
| <div class="tool-card"> | |
| <code>lattice_compare</code> | |
| <p>Build matched SC/FCC lattices, compute stats</p> | |
| </div> | |
| <div class="tool-card"> | |
| <code>fiedler_ratio</code> | |
| <p>Weighted algebraic connectivity comparison</p> | |
| </div> | |
| <div class="tool-card"> | |
| <code>direction_weights</code> | |
| <p>Direction-pair weighting (6.1× finding)</p> | |
| </div> | |
| <div class="tool-card"> | |
| <code>spectral_analysis</code> | |
| <p>RD Laplacian spectrum under 4 distributions</p> | |
| </div> | |
| <div class="tool-card"> | |
| <code>prime_vertex_map</code> | |
| <p>Exhaustive prime-vertex optimization</p> | |
| </div> | |
| <div class="tool-card"> | |
| <code>visualize_rd</code> | |
| <p>3D rhombic dodecahedron visualization</p> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ FORWARD VISION ═══ --> | |
| <section id="vision"> | |
| <div class="section-inner"> | |
| <h2>RhombiLoRA</h2> | |
| <p>The cube isn't the enemy — it's the skeleton. The rhombic dodecahedron | |
| <em>contains</em> the cube: its 8 trivalent vertices ARE the cube's corners. | |
| The 6 tetravalent vertices are the bridges that convert cubic topology into | |
| space-filling FCC.</p> | |
| <p><strong>RhombiLoRA</strong> is a geometric LoRA adapter that adds 6 diagonal | |
| bridge connections to transformer attention heads. Keep your existing | |
| infrastructure. Add six bridges. Gain up to 6.1× bottleneck resilience.</p> | |
| <blockquote> | |
| "Data symmetry analytically determines representation geometry." | |
| <cite>— Karkada et al., 2026</cite> | |
| </blockquote> | |
| <p>If data symmetry determines embedding geometry, then the adapter's topology | |
| should match the data's symmetry structure. RhombiLoRA is that match.</p> | |
| <div class="paper-arc"> | |
| <div class="arc-step"> | |
| <div class="arc-num">Paper 1</div> | |
| <div class="arc-q">"What happens when you replace the cube?"</div> | |
| <div class="arc-a">The comparison.</div> | |
| </div> | |
| <div class="arc-arrow">→</div> | |
| <div class="arc-step"> | |
| <div class="arc-num">Paper 2</div> | |
| <div class="arc-q">"Does structure amplify or attenuate?"</div> | |
| <div class="arc-a">The mechanism.</div> | |
| </div> | |
| <div class="arc-arrow">→</div> | |
| <div class="arc-step"> | |
| <div class="arc-num">Paper 3</div> | |
| <div class="arc-q">"What happens when you embrace the cube?"</div> | |
| <div class="arc-a">The product.</div> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ INSTALL ═══ --> | |
| <section id="install"> | |
| <div class="section-inner"> | |
| <h2>Try It</h2> | |
| <div class="install-block"> | |
| <code class="install-cmd">pip install rhombic</code> | |
| </div> | |
| <div class="install-links"> | |
| <a href="https://github.com/promptcrafted/rhombic">GitHub</a> | |
| <a href="https://pypi.org/project/rhombic/">PyPI</a> | |
| <a href="https://huggingface.co/spaces/timotheospaul/rhombic">HF Space</a> | |
| </div> | |
| <div class="citation"> | |
| <h3>Citation</h3> | |
| <pre><code>@unpublished{bielec2026shape, | |
| author = {Timothy Paul Bielec}, | |
| title = {The Shape of the Cell: Empirical Comparison of | |
| Cubic and {FCC} Lattice Topologies}, | |
| note = {Promptcrafted LLC}, | |
| year = {2026} | |
| } | |
| @unpublished{bielec2026weighted, | |
| author = {Timothy Paul Bielec}, | |
| title = {Structured Edge Weights Amplify {FCC} Lattice | |
| Topology Advantages via Bottleneck Resilience}, | |
| note = {Promptcrafted LLC}, | |
| year = {2026} | |
| }</code></pre> | |
| </div> | |
| </div> | |
| </section> | |
| <!-- ═══ FOOTER ═══ --> | |
| <footer> | |
| <div class="section-inner"> | |
| <p class="footer-brand">rhombic — <a href="https://promptcrafted.com">Promptcrafted</a></p> | |
| <p class="footer-meta">v0.3.0 · 255 tests · MPL-2.0 · Python 3.10-3.12</p> | |
| </div> | |
| </footer> | |
| <script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/r128/three.min.js"></script> | |
| <script src="rd.js"></script> | |
| </body> | |
| </html> | |