{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $p_1, p_2, p_3$ be the vertices of an equilateral triangle in $\\R^2$ and let $F_i(x) = (x + p_i)/2$ for $i = 1, 2, 3$.\nThe Sierpiński gasket $K$ is the unique nonempty compact set with $K = F_1(K) \\cup F_2(K) \\cup F_3(K)$, the attractor of this [[iterated function system|75B8VHDGSA]].\nLet $V_0 = \\set{p_1, p_2, p_3}$ and $V_{m+1} = F_1(V_m) \\cup F_2(V_m) \\cup F_3(V_m)$.\nLet $\\Gamma_m$ be the graph on $V_m$ in which $x \\sim_m y$ when $x$ and $y$ are two vertices of one triangle $F_{w_1} \\circ \\cdots \\circ F_{w_m}(\\text{triangle } p_1 p_2 p_3)$.\nFor $u \\colon V_m \\to \\R$, set\n$$\\mathcal{E}_m(u) = \\left(\\frac{5}{3}\\right)^m \\sum_{x \\sim_m y} (u(x) - u(y))^2,$$\nthe sum over edges of $\\Gamma_m$.\n\nThe factor $5/3$ is the one that makes these energies compatible.\nTake $u(p_1) = 1$ and $u(p_2) = u(p_3) = 0$, so $\\mathcal{E}_0(u) = 2$.\nAmong all extensions of $u$ to $V_1$, the one of least energy $\\mathcal{E}_1$ follows the \"$1/5$–$2/5$ rule\".\nAt the midpoint of $p_j p_k$ its value is $(2 u(p_j) + 2 u(p_k) + u(p_i))/5$, where $i$ is the third index.\nHere this gives $2/5$ at the midpoints of $p_1 p_2$ and $p_1 p_3$, and $1/5$ at the midpoint of $p_2 p_3$.\nThe three small triangles contribute $18/25$, $6/25$ and $6/25$ to the sum, in total $6/5$, and $(5/3) \\cdot (6/5) = 2 = \\mathcal{E}_0(u)$.\n\nBecause of this compatibility, $\\mathcal{E}_m(u|_{V_m})$ is nondecreasing in $m$ for every $u$, and its limit $\\mathcal{E}(u)$ is a [[resistance form|THCM7DWHNY]] on $K$.\nLet $\\mu$ be the self-similar probability measure that gives mass $3^{-m}$ to each level-$m$ cell.\nFor a suitable function $u$ and a point $x \\in V_m \\setminus V_0$, the Laplacian of $u$ with respect to $\\mu$ is\n$$\\Delta_\\mu u(x) = \\frac{3}{2} \\lim_{m \\to \\infty} 5^m \\sum_{y \\sim_m x} (u(y) - u(x)).$$\nThe factor $5^m = (5/3)^m \\cdot 3^m$ combines the energy scaling with the measure scaling.\nThe function on $V_1$ found above satisfies $\\sum_{y \\sim_1 x} (u(y) - u(x)) = 0$ at each of the three midpoints.\nFor example, at the midpoint of $p_2 p_3$ the neighbours have values $0, 0, 2/5, 2/5$, and $4/5 - 4 \\cdot (1/5) = 0$.\nIts continuation by the same rule at all levels is a harmonic function on $K$, that is, $\\Delta_\\mu u = 0$ off $V_0$.\n\n## Remarks\n\nThe eigenvalue counting function of $-\\Delta_\\mu$ with Dirichlet conditions on $V_0$ grows like $\\lambda^{d_s/2}$ up to bounded factors, with spectral dimension $d_s = 2 \\log 3 / \\log 5 \\approx 1.365$.\nThis differs from the [[Hausdorff dimension|SSNFTHZ6YJ]] $\\log 3 / \\log 2 \\approx 1.585$ of the gasket.\n\n## Explanations\n\n### L1 human\n\nThe Sierpiński gasket is a triangle with holes at every scale, so ordinary derivatives make no sense on it.\nInstead, one looks at finer and finer triangle networks inside the gasket and computes how the value at each vertex differs from the values at its neighbours.\nA small calculation on a single triangle shows that energies must be multiplied by $5/3$ at each step for the networks to agree.\nWith this scaling the network Laplacians converge, and their limit is the Laplacian of the gasket.\nHarmonic functions are then easy to compute: the value at each new midpoint is a fixed mix of $2/5$, $2/5$ and $1/5$ of the corner values.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "illustrates": [
      "VXB4G47Z3T"
    ],
    "kind": "example",
    "msc": [
      "31E05",
      "28A80"
    ],
    "name": "Laplacian on the Sierpiński gasket",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Analysis_on_fractals"
      }
    ],
    "tag": "W9V9JSAWYK"
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/W9V9JSAWYK/",
  "links": [
    {
      "field": "illustrates",
      "target": "VXB4G47Z3T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/W9V9JSAWYK.md",
  "msc": [
    "31E05",
    "28A80"
  ],
  "name": "Laplacian on the Sierpiński gasket",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W9V/W9V9JSAWYK.md",
  "status": null,
  "tag": "W9V9JSAWYK",
  "uri": "m2://W9V9JSAWYK",
  "url": "https://tamnd.github.io/math2/wiki/laplacian-on-the-sierpinski-gasket/",
  "writing": {
    "tag": "W9V9JSAWYK",
    "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
      }
    ]
  }
}