{
  "id": "translational-tile-with-no-periodic-tiling",
  "title": "A translational tile with no periodic tiling in dimension three",
  "description": "A finite set T in Z^3 tiles Z^3 by translations, but no tiling of Z^3 by T is invariant under a subgroup of finite index, and the union of the unit cubes at the points of T has the same property in R^3, even with arbitrary real translations. An arithmetic rule for arrays along lines of integer slope has a solution with no constant column but none with a vertical period; tiling equations in Z^2 x V with V cyclic encode the rule through the dependence of outputs on input digits; one explicit common solution, a stacking with a fresh prime and a rigid passage to three dimensions complete the proof. The course is an exposition of OpenAI's proof (2026) that the periodic tiling conjecture fails in dimension three.",
  "units": [
    {
      "id": "tilings-of-abelian-groups-and-periodicity",
      "title": "Tilings of abelian groups and periodicity",
      "reader": "tilings-of-abelian-groups-and-periodicity.html",
      "source": "src/tilings-of-abelian-groups-and-periodicity.md",
      "source_sha256": "b829b6a1dbbb7549b81fe941135bf6e54824a526ed4ebcdbef7c293c3d601eec",
      "status": "author_self_checked"
    },
    {
      "id": "an-arithmetic-word-rule-with-no-vertical-period",
      "title": "An arithmetic word rule with no vertical period",
      "reader": "an-arithmetic-word-rule-with-no-vertical-period.html",
      "source": "src/an-arithmetic-word-rule-with-no-vertical-period.md",
      "source_sha256": "953de41a371af9d9c88e03b7f6c8755f2c0a22edf953786317057619f188e754",
      "status": "author_self_checked"
    },
    {
      "id": "encoding-symbols-by-tiling-equations",
      "title": "Encoding symbols by tiling equations",
      "reader": "encoding-symbols-by-tiling-equations.html",
      "source": "src/encoding-symbols-by-tiling-equations.md",
      "source_sha256": "3692f4c21ce1b6a800729cafd16b44fa8b83486f2caa28ae06ee5fc0b24c298c",
      "status": "author_self_checked"
    },
    {
      "id": "forcing-activity-and-excluding-periodicity",
      "title": "Forcing activity and excluding periodicity",
      "reader": "forcing-activity-and-excluding-periodicity.html",
      "source": "src/forcing-activity-and-excluding-periodicity.md",
      "source_sha256": "e8a07e3d49de7c812c18a8ffe67d09bc5a6cf51411329a74f026ccd82cb01d1e",
      "status": "author_self_checked"
    },
    {
      "id": "a-common-solution-of-the-tiling-equations",
      "title": "A common solution of the tiling equations",
      "reader": "a-common-solution-of-the-tiling-equations.html",
      "source": "src/a-common-solution-of-the-tiling-equations.md",
      "source_sha256": "7b50a78a02e5b0a6d7f452c37f3fd34aa78dad9ca5fe6ddc631204ff77cd1b48",
      "status": "author_self_checked"
    },
    {
      "id": "one-tile-with-a-cyclic-finite-coordinate",
      "title": "One tile with a cyclic finite coordinate",
      "reader": "one-tile-with-a-cyclic-finite-coordinate.html",
      "source": "src/one-tile-with-a-cyclic-finite-coordinate.md",
      "source_sha256": "20b52b4403cec43c95e01a98442bd5bce9c4ba6aea4ff96a7c37482dc5129896",
      "status": "author_self_checked"
    },
    {
      "id": "passing-to-three-dimensional-space",
      "title": "Passing to three-dimensional space",
      "reader": "passing-to-three-dimensional-space.html",
      "source": "src/passing-to-three-dimensional-space.md",
      "source_sha256": "c5c4095bc0160acb96bcd5f8f1c17820db99b6b6a58de2e232ffe26d15db001e",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Abstract Algebra I (Judson, Abstract Algebra: Theory and Applications): Bezout's identity (Corollary 2.2.3), infinitely many primes (Theorem 2.2.6), orders of elements divide the order of a finite group (Corollary 6.2.3), products of cyclic groups of coprime orders (Theorem 9.2.9, Corollary 9.2.10), the Chinese remainder theorem (Theorem 16.5.3)",
    "core course Linear Algebra (Hefferon, Linear Algebra, fourth edition): in an n-dimensional space n linearly independent vectors span (Chapter Two, Section III, Corollary 2.14)",
    "core course Measure and Integration (Fremlin, Measure Theory, Volume 1): countable additivity (112C), Borel sets and the measure of boxes (115G), translation invariance (134A)",
    "finite probability spaces: independence and the union bound"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}