# Difference between revisions of "Trisection of an angle"

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− | The problem of dividing an angle $\phi$ into three equal parts | + | The problem of dividing an angle $\phi$ into three equal parts. |

+ | |||

+ | The special case of trisection using only ruler-and-compass construction was one of the classical problems of Antiquity. The solution of the problem of trisecting an angle reduces to finding rational roots of a cubic equation $4x^3-3x-\cos\phi=0$, where $x=\cos(\phi/3)$, which, in general, is not solvable by quadratic radicals: that is, the roots of the general cubic do not lie in the field of [[construcible number]]s. Thus, the problem of trisecting a general angle cannot be solved by means of ruler and compass, as was proved in 1837 by P. Wantzel. However, such a construction is possible for angles $m\cdot90^\circ/2^n$, where $n,m$ are integers. | ||

+ | |||

+ | The problem may be solved by using other means and instruments of construction (for example, the [[Dinostratus quadratrix]] or the [[conchoid]]). | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft. (1969) pp. 205–230 (Translated from Russian)</TD></TR></table> | + | <table> |

− | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft. (1969) pp. 205–230 (Translated from Russian)</TD></TR> | |

+ | </table> | ||

====Comments==== | ====Comments==== | ||

− | The problem of trisection of an angle, like [[ | + | The problem of trisection of an angle, like [[duplication of the cube]], is one of the problems dealt with in [[Galois theory]], cf. also [[#References|[a3]]]. |

A remarkable result on trisection of the angles of a triangle is F. Morley's theorem (1899), stating that the three points of intersection of the adjacent trisectors of the angles of an arbitrary triangle form an equilateral triangle (cf. [[#References|[a1]]]). | A remarkable result on trisection of the angles of a triangle is F. Morley's theorem (1899), stating that the three points of intersection of the adjacent trisectors of the angles of an arbitrary triangle form an equilateral triangle (cf. [[#References|[a1]]]). | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)</TD></TR> | ||

+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987)</TD></TR> | ||

+ | <TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5</TD></TR> | ||

+ | </table> |

## Revision as of 15:01, 7 December 2014

The problem of dividing an angle $\phi$ into three equal parts.

The special case of trisection using only ruler-and-compass construction was one of the classical problems of Antiquity. The solution of the problem of trisecting an angle reduces to finding rational roots of a cubic equation $4x^3-3x-\cos\phi=0$, where $x=\cos(\phi/3)$, which, in general, is not solvable by quadratic radicals: that is, the roots of the general cubic do not lie in the field of construcible numbers. Thus, the problem of trisecting a general angle cannot be solved by means of ruler and compass, as was proved in 1837 by P. Wantzel. However, such a construction is possible for angles $m\cdot90^\circ/2^n$, where $n,m$ are integers.

The problem may be solved by using other means and instruments of construction (for example, the Dinostratus quadratrix or the conchoid).

#### References

[1] | Yu.I. Manin, "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) pp. 205–230 (Translated from Russian) |

#### Comments

The problem of trisection of an angle, like duplication of the cube, is one of the problems dealt with in Galois theory, cf. also [a3].

A remarkable result on trisection of the angles of a triangle is F. Morley's theorem (1899), stating that the three points of intersection of the adjacent trisectors of the angles of an arbitrary triangle form an equilateral triangle (cf. [a1]).

#### References

[a1] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) |

[a2] | W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987) |

[a3] | I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5 |

**How to Cite This Entry:**

Trisection of an angle.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Trisection_of_an_angle&oldid=31664