(Geometry, especially Solid Geometry:)
"The" angle between two planes or other planar elements in three dimensional euclidian space R3, usually faces of a polyhedron which meet along an edge. Unlike a solid angle, only 2 planes are involved in a "dihedral" angle.

Suppose we have 2 planes α and β in R3, which meet along a line l=α∩β. What's the angle between them? Angles are inherently two dimensional objects, so we must reduce our problem to measuring some angle along some plane. Any plane γ not parallel to either of α,β will intersect each of them along a straight line. Thus, γ∩α and γ∩β are straight lines on the plane γ, which intersect at the point γ∩l. We can measure ∠(γ∩α,γ∩β) to get an angle between α and β.

Unfortunately, a little thought will show that this angle depends on the choice of γ! Which γ should we choose? A good idea in geometry is, other things being equal, to take something orthogonal.

Definition. The dihedral angle between α and β is the angle ∠(γ∩α,γ∩β), when γ is the plane perpendicular to l.

To give some taste of why this is a good idea, here are some equivalent definitions:

  • The minimum possible angle ∠(γ∩α,γ∩β) for all possible choices of γ.
  • The angle between the normals to the planes α,β through a point on l=α∩β.
  • The angle between the projections P(α) and P(β) when P is the projection onto a plane along l.
With so many equivalent formulations, it has got to be good for something. Since it's not clearly "THE" angle, we qualify by calling it the "dihedral angle".

Examples

  • The dihedral angle between faces of a cube is π/2=90°.
  • The dihedral angles of a prism formed by a polygon P are the angles of P, and π/2=90° between each side and the top.
  • The dihedral angle between 2 adjacent faces of an icosahedron are arccos(-sqrt(5)/3) [MathWorld].

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