{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $p(z) = z^3 - 1$, with roots $1$, $\\omega = e^{2\\pi i/3}$ and $\\omega^2$.\n[[Newton's method|07EXVNF9VP]] for $p$ iterates the [[rational function|0XDK9CGYKJ]]\n$$N(z) = z - \\frac{p(z)}{p'(z)} = \\frac{2z^3 + 1}{3z^2}.$$\nThe *Newton fractal* of $p$ is the [[Julia set|5M4S7P04GR]] $J(N)$.\nIt is usually drawn with each point of the [[Fatou set|C0N0XGC6GK]] coloured by the root its orbit tends to.\n\nCheck of the main features:\n\n1. Each root $r$ is a superattracting fixed point: $N(r) = r$ since $p(r) = 0$, and $N'(z) = p(z) p''(z) / p'(z)^2$ vanishes at $r$.\n2. So each root has an open basin of attraction, contained in the Fatou set.\n3. The map $N$ commutes with $z \\mapsto \\omega z$, so the picture has threefold rotational symmetry.\n4. The Julia set is the common boundary of the three basins: every point of $J(N)$ lies in the closure of each basin.\n\n## Remarks\n\nNewton's method can therefore fail in a complicated way: arbitrarily close to the boundary of one basin there are starting points that converge to each of the other roots.\nFor a quadratic polynomial with two distinct roots, the Newton map is conjugate to $z \\mapsto z^2$.\nThen the Julia set is the perpendicular bisector of the two roots, a result going back to Cayley.\n\n## Explanations\n\n### L1 human\n\nNewton's method guesses a root of a polynomial and improves the guess again and again.\nColour each starting point in the plane by the root that the guesses settle on.\nFor $z^3 - 1$ there are three colours, and the borders between them form an endlessly detailed fractal.\nNear any border point all three colours appear, so a tiny change of the starting guess can change which root is found.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "5M4S7P04GR"
    ],
    "kind": "example",
    "msc": [
      "37F10",
      "65H05"
    ],
    "name": "Newton fractal",
    "same_as": {
      "wikidata": "Q1151001",
      "wikipedia": "Newton fractal"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Newton_fractal"
      }
    ],
    "tag": "CQE378ZVY5",
    "uses": [
      "07EXVNF9VP",
      "5M4S7P04GR",
      "C0N0XGC6GK",
      "0XDK9CGYKJ"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/CQE378ZVY5/",
  "links": [
    {
      "field": "instance_of",
      "target": "5M4S7P04GR"
    },
    {
      "field": "uses",
      "target": "07EXVNF9VP"
    },
    {
      "field": "uses",
      "target": "5M4S7P04GR"
    },
    {
      "field": "uses",
      "target": "C0N0XGC6GK"
    },
    {
      "field": "uses",
      "target": "0XDK9CGYKJ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CQE378ZVY5.md",
  "msc": [
    "37F10",
    "65H05"
  ],
  "name": "Newton fractal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CQE/CQE378ZVY5.md",
  "status": null,
  "tag": "CQE378ZVY5",
  "uri": "m2://CQE378ZVY5",
  "url": "https://tamnd.github.io/math2/wiki/newton-fractal/",
  "writing": {
    "tag": "CQE378ZVY5",
    "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
      }
    ]
  }
}