{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AH3G1HDB0D"
    },
    {
      "field": "uses",
      "target": "AH3G1HDB0D"
    }
  ],
  "body": "\n## Example\n\nLet $a \u003e 0$ and let $\\exp_a(x) = a^x$ be [[real exponentiation|091AV9YHZY]] to base $a$.\nFor integers $n \\ge 1$, the *tetration* ${}^{n}a$ is defined by\n$${}^{1}a = a, \\qquad {}^{n+1}a = a^{\\left({}^{n}a\\right)}.$$\nSo ${}^{n}a$ is a tower of $n$ copies of $a$, evaluated from the top down.\nWith ${}^{0}a = 1$, one has ${}^{n}a = \\exp_a^{\\,n}(1)$, the $n$-th [[iterate|AYVTR2ES5B]] of $\\exp_a$ applied to $1$.\n\nCheck for $a = 2$: ${}^{1}2 = 2$, ${}^{2}2 = 2^2 = 4$, ${}^{3}2 = 2^4 = 16$ and ${}^{4}2 = 2^{16} = 65536$.\nAnd $\\exp_2(1) = 2$, $\\exp_2(2) = 4$, $\\exp_2(4) = 16$, $\\exp_2(16) = 65536$, as claimed.\n\n## Remarks\n\nTetration is the next operation after addition, multiplication and exponentiation: repeated exponentiation, as exponentiation is repeated multiplication.\nIt grows very fast: ${}^{5}2 = 2^{65536}$ has $19729$ decimal digits.\nDefining ${}^{x}a$ for real heights $x$ is a problem of [[fractional iteration|5M3THA1S6E]] of $\\exp_a$.\nOne asks for a function $F$ with $F(0) = 1$ and $F(x + 1) = a^{F(x)}$; its inverse is a solution of the [[Abel equation|68SHB7Z721]] for $\\exp_a$.\nThese conditions alone do not determine $F$, so one adds regularity conditions such as real analyticity.\nThe infinite tower $a^{a^{a^{\\cdots}}}$ is the limit of ${}^{n}a$ as $n \\to \\infty$; see [[iterated exponentiation|RDC61V812J]].\n\n## Explanations\n\n### L1 human\n\nMultiplication repeats addition, and powers repeat multiplication.\nTetration repeats powers: ${}^{3}2$ means $2^{2^2} = 16$.\nThe values explode: ${}^{4}2 = 65536$, and ${}^{5}2$ is a number with almost twenty thousand digits.\nEach step applies the map $x \\mapsto 2^x$ once more, so tetration is a picture of iterating a function.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "illustrates": [
      "AYVTR2ES5B"
    ],
    "kind": "example",
    "msc": [
      "26A18",
      "39B12"
    ],
    "name": "Tetration",
    "same_as": {
      "wikidata": "Q194324",
      "wikipedia": "Tetration"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tetration"
      }
    ],
    "tag": "NT65WHM5VV",
    "uses": [
      "091AV9YHZY",
      "AYVTR2ES5B",
      "68SHB7Z721",
      "5M3THA1S6E"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/NT65WHM5VV/",
  "links": [
    {
      "field": "illustrates",
      "target": "AYVTR2ES5B"
    },
    {
      "field": "uses",
      "target": "091AV9YHZY"
    },
    {
      "field": "uses",
      "target": "AYVTR2ES5B"
    },
    {
      "field": "uses",
      "target": "68SHB7Z721"
    },
    {
      "field": "uses",
      "target": "5M3THA1S6E"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NT65WHM5VV.md",
  "msc": [
    "26A18",
    "39B12"
  ],
  "name": "Tetration",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NT6/NT65WHM5VV.md",
  "status": null,
  "tag": "NT65WHM5VV",
  "uri": "m2://NT65WHM5VV",
  "url": "https://tamnd.github.io/math2/wiki/tetration/",
  "writing": {
    "tag": "NT65WHM5VV",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}