{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nStart from two points at distance $1$ in the plane, and use only straightedge and compass.\n\n1. It is impossible to construct a segment of length $\\sqrt[3]{2}$, that is, to double the volume of a cube with edge $1$.\n2. It is impossible to trisect every angle; in particular the angle $\\pi/3$ cannot be trisected.\n3. It is impossible to construct a square with the same area as a circle of radius $1$, that is, a segment of length $\\sqrt{\\pi}$.\n\n## Remarks\n\nA length is constructible exactly when it is a [[constructible number|0FMN2MN7TM]], and [[the degree of a constructible number over $\\Q$ is a power of $2$|0SSSM1D475]].\nFor clause 1, $\\sqrt[3]{2}$ has minimal polynomial $x^3 - 2$, of degree $3$; see [[impossibility of doubling the cube|0CQ357M0QS]].\nFor clause 2, $2\\cos(\\pi/9)$ has minimal polynomial $x^3 - 3x - 1$, of degree $3$; see [[impossibility of trisecting the angle|0EWXPZCSZ0]].\nFor clause 3, constructible numbers are algebraic, while $\\pi$, and hence $\\sqrt{\\pi}$, is transcendental by the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]]; see [[impossibility of squaring the circle|0X7CVDE3BG]].\nSome angles can be trisected, such as $\\pi/2$, since $\\pi/6$ is constructible; clause 2 says that no method works for all angles.\n\n## Explanations\n\n### L1 human\n\nThe ancient Greeks posed three famous drawing problems: double a cube, divide any angle into three equal parts, and turn a circle into a square of the same area.\nThey allowed only an unmarked ruler and a compass.\nTwo thousand years later, algebra showed that all three are impossible.\nThe reason is that these tools can only produce numbers reached by repeated square roots.\n\n### L2 human\n\nEach straightedge-and-compass step solves linear or quadratic equations, so each new point lies in an extension of degree $1$ or $2$ of the field of the previous points.\nA constructible number therefore lies at the top of a tower of quadratic extensions, and by the tower law its degree is a power of $2$.\nClauses 1 and 2 then follow from degree $3$ minimal polynomials, while clause 3 needs the much deeper transcendence of $\\pi$.\nThe same method, with [[cyclotomic extensions|VD6VC3ZMMY]], decides which regular polygons are constructible.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "0FMN2MN7TM",
      "0DAECMZVRT"
    ],
    "kind": "statement",
    "msc": [
      "12F05",
      "51M15"
    ],
    "name": "Impossibility of the classical construction problems",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Straightedge_and_compass_construction"
      }
    ],
    "tag": "W8J42M1G28",
    "uses": [
      "0FMN2MN7TM",
      "0SSSM1D475",
      "0CQ357M0QS",
      "0EWXPZCSZ0",
      "0X7CVDE3BG",
      "0JNCBFMPXR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/W8J42M1G28/",
  "links": [
    {
      "field": "about",
      "target": "0FMN2MN7TM"
    },
    {
      "field": "about",
      "target": "0DAECMZVRT"
    },
    {
      "field": "uses",
      "target": "0FMN2MN7TM"
    },
    {
      "field": "uses",
      "target": "0SSSM1D475"
    },
    {
      "field": "uses",
      "target": "0CQ357M0QS"
    },
    {
      "field": "uses",
      "target": "0EWXPZCSZ0"
    },
    {
      "field": "uses",
      "target": "0X7CVDE3BG"
    },
    {
      "field": "uses",
      "target": "0JNCBFMPXR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W8J42M1G28.md",
  "msc": [
    "12F05",
    "51M15"
  ],
  "name": "Impossibility of the classical construction problems",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W8J/W8J42M1G28.md",
  "status": null,
  "tag": "W8J42M1G28",
  "uri": "m2://W8J42M1G28",
  "url": "https://math2.tamnd.com/wiki/impossibility-of-the-classical-construction-problems/",
  "writing": {
    "tag": "W8J42M1G28",
    "kind": "statement",
    "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
      }
    ]
  }
}