# Difference between revisions of "Dihedral angle"

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− | A figure in space formed by two half- | + | {{TEX|done}} |

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+ | A figure in space formed by two [[half-plane]]s emanating from the same straight line and the part of space bounded by these half-planes. The half-planes are said to be the [[face]]s of the dihedral angle, while their common straight line is known as the edge. The dihedral angle is measured by the [[linear angle]], i.e. by the angle between two [[normal]]s to the edge emanating from the same point and lying in different faces, or, in other words, by the angle formed by the intersection of the dihedral angle and a plane normal to the edge. | ||

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

− | Dihedral angles play a role in elementary geometry as well as in the metrical theory of [[ | + | |

+ | Dihedral angles play a role in elementary geometry as well as in the metrical theory of [[regular polyhedra]] (in which invariants of such polyhedra are computed in terms of, e.g., the dihedral angles, cf., e.g., [[#References|[1]]], [[#References|[4]]]) and, in a more general context, in the theory of convex polytopes (see [[Convex polyhedron]]). For their relation to general valuations on [[Convex body|convex bodies]] cf. [[#References|[3]]] (see also [[Geometry of numbers]]). | ||

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

− | <table><TR><TD valign="top">[ | + | |

+ | <table> | ||

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973)</TD></TR> | ||

+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> P. McMullen, "Non-linear angle-sum relations for polyhedral cones and polytopes" ''Math. Proc. Cambr. Phil. Soc.'' , '''78''' (1975) pp. 247–261</TD></TR> | ||

+ | <TR><TD valign="top">[3]</TD> <TD valign="top"> R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 170–247</TD></TR> | ||

+ | <TR><TD valign="top">[4]</TD> <TD valign="top"> M. Berger, "Geometry" , '''II''' , Springer (1987)</TD></TR> | ||

+ | <TR><TD valign="top">[5]</TD> <TD valign="top"> J. Cheeger, W. Müller, R. Schrader, "On the curvature of piecewise flat spaces" ''Comm. Math. Physics'' , '''92''' (1984) pp. 405–454</TD></TR> | ||

+ | <TR><TD valign="top">[6]</TD> <TD valign="top"> J.L. Coolidge, "A treatise on the circle and the sphere" , Clarendon Press (1916)</TD></TR> | ||

+ | </table> |

## Latest revision as of 09:48, 26 May 2016

A figure in space formed by two half-planes emanating from the same straight line and the part of space bounded by these half-planes. The half-planes are said to be the faces of the dihedral angle, while their common straight line is known as the edge. The dihedral angle is measured by the linear angle, i.e. by the angle between two normals to the edge emanating from the same point and lying in different faces, or, in other words, by the angle formed by the intersection of the dihedral angle and a plane normal to the edge.

#### Comments

Dihedral angles play a role in elementary geometry as well as in the metrical theory of regular polyhedra (in which invariants of such polyhedra are computed in terms of, e.g., the dihedral angles, cf., e.g., [1], [4]) and, in a more general context, in the theory of convex polytopes (see Convex polyhedron). For their relation to general valuations on convex bodies cf. [3] (see also Geometry of numbers).

#### References

[1] | H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973) |

[2] | P. McMullen, "Non-linear angle-sum relations for polyhedral cones and polytopes" Math. Proc. Cambr. Phil. Soc. , 78 (1975) pp. 247–261 |

[3] | R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 170–247 |

[4] | M. Berger, "Geometry" , II , Springer (1987) |

[5] | J. Cheeger, W. Müller, R. Schrader, "On the curvature of piecewise flat spaces" Comm. Math. Physics , 92 (1984) pp. 405–454 |

[6] | J.L. Coolidge, "A treatise on the circle and the sphere" , Clarendon Press (1916) |

**How to Cite This Entry:**

Dihedral angle.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dihedral_angle&oldid=38843