Namespaces
Variants
Actions

Difference between revisions of "Trisection of an angle"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (ce)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
The problem of dividing an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094310/t0943101.png" /> into three equal parts; one of the classical problems of Antiquity on ruler-and-compass construction. The solution of the problem of trisecting an angle reduces to finding rational roots of a cubic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094310/t0943102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094310/t0943103.png" />, which, in general, is not solvable by quadratic radicals. Thus, the problem of trisecting an 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094310/t0943104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094310/t0943105.png" /> are integers, as well as by using other means and instruments of construction (for example, the [[Dinostratus quadratrix|Dinostratus quadratrix]] or the [[Conchoid|conchoid]]).
+
{{TEX|done}}{{MSC|51M04|01A}}
 +
 
 +
The problem of dividing an angle 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 $\phi$ 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 [[constructible 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 [[Duplication of the cube|duplication of the cube]], is one of the problems dealt with in [[Galois theory|Galois theory]], cf. also [[#References|[a3]]].
+
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 &amp; 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 &amp; Hall  (1973)  pp. Chapt. 5</TD></TR>
 +
</table>

Latest revision as of 15:05, 7 December 2014

2010 Mathematics Subject Classification: Primary: 51M04 Secondary: 01A [MSN][ZBL]

The problem of dividing an angle 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 $\phi$ 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 constructible 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=16472
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article