8/1309 the square ring

NON-TILER Does not tile space. Heesch number 1: copies can wrap it once, and provably never 2 times. The refutation is a DRAT proof checked by drat-trim.

shape corona 1

drag to orbit, scroll to zoom

cells: 8, bounding box 1×3×3
symmetry order: 8
chiral: no, its own mirror image
Heesch number: 1
record: data/8/1309/shape.json

Stages

stage	outcome	at
box	exhausted	128
torus	exhausted	152
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": "8/1309",
  "n": 8,
  "cells": [[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 2], [0, 2, 0], [0, 2, 1], [0, 2, 2]],
  "symmetry_order": 8,
  "chiral": false,
  "mirror_id": "8/1309",
  "verdict": "non_tiler",
  "tiles_rotations_only": false,
  "tiles_with_reflections": false,
  "certificate": {
    "adjacency": "26",
    "checked": {
      "checker": "drat-trim 2e3b2dc",
      "date": "2026-10-06",
      "seconds": 24.0,
      "verdict": "VERIFIED"
    },
    "derived_from": null,
    "heesch": 1,
    "provenance": "computed",
    "refutation": "cnf/corona2.drat",
    "refutation_kind": "proof",
    "refutation_sha256": "1c8a3d6a1992d85268fb758b9effe2dd2ef007439f55cface220c06d117b9bd0",
    "solved_with": "kissat 4.0.4, re-solved 2026-10-06 on the byte-identical formula; replaces the 2026-07 proof (kissat version unrecorded, sha256 621091ca…, drat-trim VERIFIED 2026-07-07 and 2026-10-06); corona-2 UNSAT also reproduced by CaDiCaL in 2026-07",
    "type": "heesch"
  },
  "heesch": 1,
  "budgets": { "box_max_volume": 128, "corona_depth": 2, "torus_max_index": 152 },
  "credits": {
    "prior_art": null,
    "solved_by": "polycube-census v0.1.0",
    "verified_by": "Shane Becker via drat-trim (s VERIFIED) and CaDiCaL, 2026-07-07"
  },
  "reached": { "corona_refuted": 2 },
  "stages": [
    { "outcome": "exhausted", "stage": "box", "volume": 128 },
    { "index": 152, "outcome": "exhausted", "stage": "torus" },
    { "depth": 1, "outcome": "witnessed", "stage": "corona" },
    { "depth": 2, "outcome": "refuted", "stage": "corona" }
  ]
}