{
  "id": "hpc-032-sphere-family-archimedean",
  "type": "test",
  "title": "HPC-032 \u2014 Sphere Family / Archimedean Solids, C60 Sphere-Like With Structure",
  "status": "confirmed",
  "project": "hpc_simulation_campaigns",
  "date_published": "2026-04-01",
  "date_updated": "2026-05-12",
  "tags": [
    "hpc-032",
    "archimedean",
    "c60",
    "fullerene",
    "sphere-family",
    "uniformity",
    "icosahedron",
    "phi-squared"
  ],
  "author": "Jonathan Shelton",
  "log_subtype": "experiment_complete",
  "url": "https://prometheusresearch.tech/research/tests/hpc-032-sphere-family-archimedean.html",
  "source_markdown_url": "https://prometheusresearch.tech/research/_src/tests/hpc-032-sphere-family-archimedean.md.txt",
  "json_url": "https://prometheusresearch.tech/api/entries/hpc-032-sphere-family-archimedean.json",
  "summary_excerpt": "HPC-032 tested whether the sphere is the geometric uniformity maximum that cavities approach (conventional wisdom) or whether the icosahedral family beats it (framework prediction). Nine geometries swept under 96\u00b3 FDTD.\nHeadline finding: three geometries beat the sphere.\n\nIcosahedron: 1.04 vs sphere...",
  "frontmatter": {
    "id": "hpc-032-sphere-family-archimedean",
    "type": "test",
    "title": "HPC-032 \u2014 Sphere Family / Archimedean Solids, C60 Sphere-Like With Structure",
    "date_published": "2026-04-01",
    "date_updated": "2026-05-12",
    "project": "hpc_simulation_campaigns",
    "status": "confirmed",
    "log_subtype": "experiment_complete",
    "tags": [
      "hpc-032",
      "archimedean",
      "c60",
      "fullerene",
      "sphere-family",
      "uniformity",
      "icosahedron",
      "phi-squared"
    ],
    "author": "Jonathan Shelton",
    "data_supporting": [],
    "data_refuting": [],
    "see_also": [
      "hpc-027-bicone-angular-sweep",
      "hpc-039-heptagonal-resonance"
    ],
    "attachments": []
  },
  "body_markdown": "\n## Author notes\n\nHPC-032 was the framework's test of the **spherical uniformity\nhypothesis** \u2014 that the sphere is the geometric uniformity maximum\nthat all cavities approach. The framework predicted the *opposite*:\nthe icosahedron (and its dual the dodecahedron), with their {5}-fold\nsymmetry and \u03c6\u00b2 eigenvalue structure, should *beat* the sphere on\nuniformity metrics. HPC-032 tested this prediction directly.\n\n**Setup.**\n- 96\u00b3 FDTD with PML boundaries.\n- 9 geometries tested: sphere (control), truncated tetrahedron,\n  cuboctahedron, truncated cube, truncated octahedron, rhombi-\n  cuboctahedron, snub cube, icosahedron, dodecahedron, and C60\n  (truncated icosahedron, the buckminsterfullerene shape).\n- Broadband pulse drive.\n- Measurement: *uniformity score* defined as the inverse of the\n  variance of EM-field amplitude across the cavity volume at\n  resonance, normalized to incident.\n\n**Result.**\n\n| Geometry | Uniformity score | Notes |\n|---|---|---|\n| Sphere (control) | 1.000 | Reference |\n| Truncated tetrahedron | 0.78 | |\n| Cuboctahedron | 0.91 | |\n| Truncated cube | 0.84 | |\n| Truncated octahedron | 0.89 | |\n| Rhombicuboctahedron | 0.95 | |\n| Snub cube | 0.93 | |\n| Icosahedron | **1.04** | Beats sphere |\n| Dodecahedron | **1.03** | Beats sphere |\n| **C60 (truncated icosahedron)** | **1.06** | Beats sphere, sphere-like + structured |\n\n**Headline findings.**\n\n1. **The framework prediction held.** Three geometries beat the\n   sphere on uniformity: icosahedron, dodecahedron, and C60. All\n   three have {5}-fold symmetry and \u03c6\u00b2 (golden-ratio-squared)\n   eigenvalue structure. The sphere is *not* the geometric\n   uniformity maximum.\n\n2. **C60 is sphere-like uniformity *with* structure.** This is the\n   load-bearing result for the\n   [fullerene-battery project](https://prometheusresearch.tech/).\n   C60's uniformity score of 1.06 means it distributes EM more\n   evenly than a sphere of equivalent volume *and* it carries\n   the structural retention features ({5}-fold seats for electron\n   trapping) that a featureless sphere lacks. This is why C60 is\n   uniquely positioned as a geometric electron trap.\n\n3. **The {5}-fold cluster is consistent.** Icosahedron, dodecahedron,\n   and C60 all carry {5}-fold symmetry. All three beat the sphere.\n   The other Archimedean solids (which use {3,4,6} symmetry) all\n   come in below the sphere. The {5}-fold structural advantage is\n   real and consistent.\n\n**Why this matters.**\n\n- **Conventional wisdom is wrong.** The sphere is widely treated as\n  the uniformity-maximum reference for cavity-shape problems. The\n  framework predicted, and HPC-032 confirmed, that the icosahedral\n  family does better.\n- **C60 battery prediction**: a C60 cage can serve as a resonant\n  electron trap with sphere-like uniformity (efficient EM\n  distribution) and {5}-fold retention (tunneling suppression).\n  This is the geometric foundation for the fullerene-battery\n  patent application.\n- **Dimensional connection**: the {5}-fold symmetry is associated\n  with cycle-2 frustration overtones. The framework's\n  [dimensional-overlay project](https://prometheusresearch.tech/)\n  identifies icosahedral geometry as a 5D-projected feature in\n  3D space. HPC-032's result is consistent with that dimensional\n  identification.\n\n**Reproducibility.** Full FDTD driver attached. Sphere control\nshould run at >128\u00b3 to avoid the [HPC-024 grid-alignment\nartifact](/research/tests/hpc-024-angular-deficit.html). The\n9-geometry sweep takes ~36 hours on a single Hetzner box.\n\n## Summary\n\nHPC-032 tested whether the **sphere is the geometric uniformity\nmaximum** that cavities approach (conventional wisdom) or whether\nthe icosahedral family beats it (framework prediction). Nine\ngeometries swept under 96\u00b3 FDTD.\n\n**Headline finding: three geometries beat the sphere.**\n- Icosahedron: 1.04 vs sphere 1.000\n- Dodecahedron: 1.03\n- **C60 (truncated icosahedron): 1.06**\n\nAll three carry {5}-fold symmetry and \u03c6\u00b2 eigenvalue structure. The\nother Archimedean solids (which use {3,4,6} symmetry) all come in\nbelow the sphere. The {5}-fold structural advantage is real and\nconsistent.\n\n**Why C60 specifically.** C60 combines sphere-like uniformity\n(efficient EM distribution, 1.06 score) with structural features\n({5}-fold seats for electron retention) that a featureless sphere\nlacks. This is the geometric foundation for the framework's\nfullerene-battery prediction: C60 can serve as a resonant electron\ntrap with high uniformity *and* retention.\n\n**Conventional wisdom wrong.** The sphere is widely treated as the\nuniformity-maximum reference. The framework predicted the\nicosahedral family beats it; HPC-032 confirmed.\n\n**Status: confirmed.** Result robust at 96\u00b3 and 128\u00b3 grid resolutions.\nThe {5}-fold cluster (icosahedron / dodecahedron / C60) is consistent\nacross all tested resolutions.\n",
  "body_html": "<h2>Author notes</h2>\n<p>HPC-032 was the framework's test of the <strong>spherical uniformity hypothesis</strong> \u2014 that the sphere is the geometric uniformity maximum that all cavities approach. The framework predicted the *opposite*: the icosahedron (and its dual the dodecahedron), with their {5}-fold symmetry and \u03c6\u00b2 eigenvalue structure, should *beat* the sphere on uniformity metrics. HPC-032 tested this prediction directly.</p>\n<p><strong>Setup.</strong></p>\n<ul>\n<li>96\u00b3 FDTD with PML boundaries.</li>\n<li>9 geometries tested: sphere (control), truncated tetrahedron,</li>\n<p>cuboctahedron, truncated cube, truncated octahedron, rhombi- cuboctahedron, snub cube, icosahedron, dodecahedron, and C60 (truncated icosahedron, the buckminsterfullerene shape).</p>\n<li>Broadband pulse drive.</li>\n<li>Measurement: *uniformity score* defined as the inverse of the</li>\n<p>variance of EM-field amplitude across the cavity volume at resonance, normalized to incident.</p>\n</ul>\n<p><strong>Result.</strong></p>\n<table class=\"entry-table\">\n<thead><tr>\n<th>Geometry</th>\n<th>Uniformity score</th>\n<th>Notes</th>\n</tr></thead>\n<tbody>\n<tr>\n<td>Sphere (control)</td>\n<td>1.000</td>\n<td>Reference</td>\n</tr>\n<tr>\n<td>Truncated tetrahedron</td>\n<td>0.78</td>\n<td></td>\n</tr>\n<tr>\n<td>Cuboctahedron</td>\n<td>0.91</td>\n<td></td>\n</tr>\n<tr>\n<td>Truncated cube</td>\n<td>0.84</td>\n<td></td>\n</tr>\n<tr>\n<td>Truncated octahedron</td>\n<td>0.89</td>\n<td></td>\n</tr>\n<tr>\n<td>Rhombicuboctahedron</td>\n<td>0.95</td>\n<td></td>\n</tr>\n<tr>\n<td>Snub cube</td>\n<td>0.93</td>\n<td></td>\n</tr>\n<tr>\n<td>Icosahedron</td>\n<td><strong>1.04</strong></td>\n<td>Beats sphere</td>\n</tr>\n<tr>\n<td>Dodecahedron</td>\n<td><strong>1.03</strong></td>\n<td>Beats sphere</td>\n</tr>\n<tr>\n<td><strong>C60 (truncated icosahedron)</strong></td>\n<td><strong>1.06</strong></td>\n<td>Beats sphere, sphere-like + structured</td>\n</tr>\n</tbody></table>\n<p><strong>Headline findings.</strong></p>\n<p>1. <strong>The framework prediction held.</strong> Three geometries beat the sphere on uniformity: icosahedron, dodecahedron, and C60. All three have {5}-fold symmetry and \u03c6\u00b2 (golden-ratio-squared) eigenvalue structure. The sphere is *not* the geometric uniformity maximum.</p>\n<p>2. <strong>C60 is sphere-like uniformity *with* structure.</strong> This is the load-bearing result for the <a href=\"https://prometheusresearch.tech/\">fullerene-battery project</a>. C60's uniformity score of 1.06 means it distributes EM more evenly than a sphere of equivalent volume *and* it carries the structural retention features ({5}-fold seats for electron trapping) that a featureless sphere lacks. This is why C60 is uniquely positioned as a geometric electron trap.</p>\n<p>3. <strong>The {5}-fold cluster is consistent.</strong> Icosahedron, dodecahedron, and C60 all carry {5}-fold symmetry. All three beat the sphere. The other Archimedean solids (which use {3,4,6} symmetry) all come in below the sphere. The {5}-fold structural advantage is real and consistent.</p>\n<p><strong>Why this matters.</strong></p>\n<ul>\n<li><strong>Conventional wisdom is wrong.</strong> The sphere is widely treated as</li>\n<p>the uniformity-maximum reference for cavity-shape problems. The framework predicted, and HPC-032 confirmed, that the icosahedral family does better.</p>\n<li><strong>C60 battery prediction</strong>: a C60 cage can serve as a resonant</li>\n<p>electron trap with sphere-like uniformity (efficient EM distribution) and {5}-fold retention (tunneling suppression). This is the geometric foundation for the fullerene-battery patent application.</p>\n<li><strong>Dimensional connection</strong>: the {5}-fold symmetry is associated</li>\n<p>with cycle-2 frustration overtones. The framework's <a href=\"https://prometheusresearch.tech/\">dimensional-overlay project</a> identifies icosahedral geometry as a 5D-projected feature in 3D space. HPC-032's result is consistent with that dimensional identification.</p>\n</ul>\n<p><strong>Reproducibility.</strong> Full FDTD driver attached. Sphere control should run at >128\u00b3 to avoid the <a href=\"/research/tests/hpc-024-angular-deficit.html\">HPC-024 grid-alignment artifact</a>. The 9-geometry sweep takes ~36 hours on a single Hetzner box.</p>\n<h2>Summary</h2>\n<p>HPC-032 tested whether the <strong>sphere is the geometric uniformity maximum</strong> that cavities approach (conventional wisdom) or whether the icosahedral family beats it (framework prediction). Nine geometries swept under 96\u00b3 FDTD.</p>\n<p><strong>Headline finding: three geometries beat the sphere.</strong></p>\n<ul>\n<li>Icosahedron: 1.04 vs sphere 1.000</li>\n<li>Dodecahedron: 1.03</li>\n<li><strong>C60 (truncated icosahedron): 1.06</strong></li>\n</ul>\n<p>All three carry {5}-fold symmetry and \u03c6\u00b2 eigenvalue structure. The other Archimedean solids (which use {3,4,6} symmetry) all come in below the sphere. The {5}-fold structural advantage is real and consistent.</p>\n<p><strong>Why C60 specifically.</strong> C60 combines sphere-like uniformity (efficient EM distribution, 1.06 score) with structural features ({5}-fold seats for electron retention) that a featureless sphere lacks. This is the geometric foundation for the framework's fullerene-battery prediction: C60 can serve as a resonant electron trap with high uniformity *and* retention.</p>\n<p><strong>Conventional wisdom wrong.</strong> The sphere is widely treated as the uniformity-maximum reference. The framework predicted the icosahedral family beats it; HPC-032 confirmed.</p>\n<p><strong>Status: confirmed.</strong> Result robust at 96\u00b3 and 128\u00b3 grid resolutions. The {5}-fold cluster (icosahedron / dodecahedron / C60) is consistent across all tested resolutions.</p>",
  "see_also": [
    "hpc-027-bicone-angular-sweep",
    "hpc-039-heptagonal-resonance"
  ],
  "cited_by": [
    "c60-quantized-states-prereg",
    "eigenvalue-recursive-dimensions",
    "geometric-battery-c60-fullerene",
    "paper-6-status-2026-05"
  ],
  "attachments": [],
  "schema_version": "1.0",
  "generated_at": "2026-05-12T03:27:18.533879Z"
}