9/42947 the triskelion

NON-TILER Does not tile space. Heesch number 1: copies can wrap it once, and provably never 2 times. The refutation is cube-and-conquer ledgers, every cube refuted.

shape corona 1

drag to orbit, scroll to zoom

cells: 9, bounding box 3×3×3
symmetry order: 3
chiral: yes, twin 9/42969
Heesch number: 1
record: data/9/42947/shape.json

Stages

stage	outcome	at
box	exhausted	96
torus	exhausted	72
corona	witnessed	1
corona	refuted	2

Downloads

model.stl the shape, printable
corona1.json the depth-1 witness, as placements
shape.json the record

The record, as stored

{
  "id": "9/42947",
  "n": 9,
  "cells": [[0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 1, 2], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 2, 0], [2, 0, 1]],
  "symmetry_order": 3,
  "chiral": true,
  "mirror_id": "9/42969",
  "verdict": "non_tiler",
  "tiles_rotations_only": false,
  "tiles_with_reflections": false,
  "certificate": {
    "adjacency": "26",
    "derived_from": null,
    "heesch": 1,
    "provenance": "computed",
    "refutation": "cnf/ledgers/c42947-{cubes,blade,blade2}.icnf.results.jsonl",
    "refutation_kind": "ledger",
    "refutation_sha256": null,
    "solved_with": "cube-and-conquer on kissat 4.0.4 (colo, 50 cores): 4096 sign-cubes + 475 clause-branch + 8225 clause-branch cubes, 12868 ledger entries, 12615 refutations, zero SATISFIABLE; cover verified by script/audit-cubes 2026-10-06; ledger-backed, per-cube DRAT campaign pending",
    "type": "heesch"
  },
  "heesch": 1,
  "budgets": { "box_max_volume": 96, "corona_depth": 2, "torus_max_index": 72 },
  "credits": { "prior_art": null, "solved_by": "polycube-census v0.1.0", "verified_by": null },
  "reached": { "corona_refuted": 2 },
  "stages": [
    { "outcome": "exhausted", "stage": "box", "volume": 96 },
    { "index": 72, "outcome": "exhausted", "stage": "torus" },
    { "depth": 1, "outcome": "witnessed", "stage": "corona" },
    { "depth": 2, "outcome": "refuted", "stage": "corona" }
  ]
}