{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $\\Gamma \\subset \\R^n$ be a [[lattice|WN9V7WSJKF]], and let $T_\\Gamma = \\R^n / \\Gamma$ be the flat [[torus|03NM7DZXXC]] with the metric induced from $\\R^n$.\nLet $\\Gamma^* = \\set{v \\in \\R^n : \\ip{v}{\\gamma} \\in \\Z \\text{ for all } \\gamma \\in \\Gamma}$ be the dual lattice.\nFor each $\\gamma^* \\in \\Gamma^*$, the function $x \\mapsto e^{2\\pi i \\ip{\\gamma^*}{x}}$ is an eigenfunction of the Laplacian with eigenvalue $4\\pi^2 \\abs{\\gamma^*}^2$.\nThese functions form an orthogonal basis of $L^2(T_\\Gamma)$.\nSo the [[spectrum|B8TYKV7AA1]] of $T_\\Gamma$ is determined by the numbers $N_{\\Gamma^*}(r) = \\#\\set{\\gamma^* \\in \\Gamma^* : \\abs{\\gamma^*}^2 = r}$.\n\nTake $n = 16$, $\\Gamma_1 = E_8 \\oplus E_8$, with $E_8$ the [[E8 lattice|2CXH65EKY7]], and $\\Gamma_2 = D_{16}^+$.\nBoth are even unimodular lattices, so $\\Gamma_i^* = \\Gamma_i$.\nThe theta series $\\sum_{\\gamma \\in \\Gamma_i} q^{\\abs{\\gamma}^2/2}$ of an even unimodular lattice of rank $16$ is a [[modular form|GWKN5VG78A]] of weight $8$ for $SL_2(\\Z)$ with constant term $1$.\nThat space of modular forms has dimension $1$, so the two theta series are equal, and $N_{\\Gamma_1}(r) = N_{\\Gamma_2}(r)$ for every $r$.\nHence $T_{\\Gamma_1}$ and $T_{\\Gamma_2}$ are [[isospectral|18MQEGN13R]].\n\nThey are not isometric.\nAn isometry of flat tori lifts to an affine isometry of $\\R^{16}$ whose linear part maps $\\Gamma_1$ onto $\\Gamma_2$.\nBut the vectors of squared length $2$ generate a sublattice of index $1$ in $E_8 \\oplus E_8$, and a sublattice of index $2$ in $D_{16}^+$, namely $D_{16}$.\n\n## Explanations\n\n### L1 human\n\nA flat torus is a box whose opposite faces are glued, and its frequencies are given by the lengths of the vectors of a lattice.\nMilnor noticed in 1964 that two special lattices in $16$ dimensions have exactly the same number of vectors of each length, although they are shaped differently.\nSo the two tori sound the same but are not the same shape.\nThis was the first example showing that one cannot always hear the shape of a manifold.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "18MQEGN13R"
    ],
    "kind": "example",
    "msc": [
      "58J53",
      "11H06",
      "11F11"
    ],
    "name": "Milnor's isospectral flat tori",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Isospectral"
      }
    ],
    "tag": "JNR9YJTT54",
    "uses": [
      "18MQEGN13R",
      "B8TYKV7AA1",
      "03NM7DZXXC",
      "WN9V7WSJKF",
      "2CXH65EKY7",
      "GWKN5VG78A"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/JNR9YJTT54/",
  "links": [
    {
      "field": "instance_of",
      "target": "18MQEGN13R"
    },
    {
      "field": "uses",
      "target": "18MQEGN13R"
    },
    {
      "field": "uses",
      "target": "B8TYKV7AA1"
    },
    {
      "field": "uses",
      "target": "03NM7DZXXC"
    },
    {
      "field": "uses",
      "target": "WN9V7WSJKF"
    },
    {
      "field": "uses",
      "target": "2CXH65EKY7"
    },
    {
      "field": "uses",
      "target": "GWKN5VG78A"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/JNR9YJTT54.md",
  "msc": [
    "58J53",
    "11H06",
    "11F11"
  ],
  "name": "Milnor's isospectral flat tori",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JNR/JNR9YJTT54.md",
  "status": null,
  "tag": "JNR9YJTT54",
  "uri": "m2://JNR9YJTT54",
  "url": "https://tamnd.github.io/math2/wiki/milnors-isospectral-flat-tori/",
  "writing": {
    "tag": "JNR9YJTT54",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}