MODEL FRONTIER 2026 · MATH 2025 · under KEEPER ERIS

The Noperthedron · Not Rupert

The first convex polyhedron proved not to have Rupert's property: no copy can pass through a straight tunnel in another copy. It has 90 vertices, 240 edges and 152 faces, rebuilt here from the authors' coordinates.

Dave (t242u): "perfect, continue on building out pages please, anything you want browse the internet for guidance"

LIT Steininger and Yurkevich (arXiv 2508.18475, Aug 2025) disproved the 2017 conjecture that every convex polyhedron is Rupert. The Noperthedron is the orbit of three points C1, C2, C3 under the 30-element group C30 = {±R_z(2πk/15)}. It has 150 triangles and 2 regular 15-gons. The proof checked about 18 million orientation blocks with certified computation.

LIT I rebuilt the hull from the published C1–C3. It has 90 vertices and 176 hull triangles, which merge into exactly 152 planar faces (150 + 2 fifteen-gons). Euler: 90 − 240 + 152 = 2.

MODEL Tie to Dave's world: A shape that cannot pass through itself, like a seal: write-if-missing.

MODEL Art: flat-shaded crystal, slowly turning, with its 90 vertices lit.

Sources (browsed Oct 2026)

  1. Steininger & Yurkevich, A convex polyhedron without Rupert's property (arXiv)
  2. Quanta: first shape found that can't pass through itself (24 Oct 2025)
  3. Scientific American: Noperthedron
  4. Univ. Vienna colloquium slides

MODEL Cooperative agents: BUILDER (points 1–5) and WEAVER (points 6–10) take turns generating moves toward one shared goal, the target form (faint ghost points). The CRITIC scores the whole form, and both builders receive that same shared reward; neither can win alone. Every page also posts its best shared reward to a cross-page board in localStorage, so agents on all the Frontier pages share one team score. Rendered with Three.js r160 from a pinned CDN, devicePixelRatio capped at 2, antialiased. Drag to orbit, scroll to zoom.

LIT on load: the facts on this page are re-checked as arithmetic (it throws on mismatch), the target scores 1, 200 rounds improve, the workspace loop logs 5 entries, and the OBJ and SVG exports are consistent.

Gallery of kept attempts (WebGL snapshots):

Workspace improvement loop: instructions for agents MODEL

  1. Measure: every 40 rounds, record the acceptance rate, the current score and how many rounds since the last improvement.
  2. Try next: if the score has been flat for 40 rounds, widen the mutation step. If acceptance is under 15%, shrink it; over 45%, widen it. Otherwise keep it.
  3. Log back: append {entry, round, measured, tried, step} to the workspace log (it's in the export JSON and the list below), and propose one change to this world on the shared board.
  4. Repeat, forever: never stop improving. Re-check against the real data before you claim anything.

Live workspace log:

    Cooperative board, shared across all Frontier pages MODEL

    Shared localStorage key mayhem-frontier-board: each page writes its best shared reward. The team score is the sum. Open several pages and they all add to the same board.

      Shared board MODEL

        Background art: AMBER MODEL a generative field seeded from this page's seal.

        proving…

        Seal sha256(name|slug|blurb): f268f2a0392ca54793aa64ab10e425500cb93f06d29b9e0e8311231f849ddd62

        ↑ Frontier 2026 index · ← ERIS, keeper of Mayhem