Dihedral

A dihedral is a geometric figure in space determined by two half-planes, called faces, that share the same boundary line, called the edge.

The dihedral is a fundamental concept in three-dimensional geometry.

Two half-planes with the same boundary line divide space into two distinct regions, each of which is a dihedral.

The two half-planes form the faces of the dihedral, while their common boundary line is its edge.

Points belonging to the dihedral but lying on neither of its faces are called interior points.

The opening of a dihedral is described by its dihedral angle. Its measure is determined by means of a normal section, that is, by intersecting the dihedral with a plane perpendicular to its edge.

A dihedral may be convex or concave depending on the measure of its dihedral angle.

The Difference Between a Dihedral and a Polyhedron: A dihedral is a region of space determined by two half-planes sharing the same boundary line. A polyhedron, on the other hand, is a three-dimensional solid bounded by polygonal faces. The line segments where two faces meet are called edges, and the points where three or more edges meet are called vertices. Examples of polyhedra include cubes, pyramids, and prisms. Therefore, a dihedral is determined by two half-planes and their common edge, whereas a polyhedron is bounded by several polygonal faces.

Sections of a Dihedral

A section of a dihedral is the plane angle obtained by intersecting the dihedral with a plane that crosses its edge.

In other words, the cutting plane intersects the two faces of the dihedral in two rays with a common endpoint, thereby forming a plane angle.

In this sense, the plane-angle counterpart of a dihedral is an ordinary angle.

The section is called a normal section when the cutting plane is perpendicular to the edge of the dihedral.

The measure of the dihedral angle is defined as the measure of any normal section of the dihedral.

The normal section therefore makes it possible to determine whether a dihedral is convex or concave:

• Convex Dihedral
  A convex dihedral has a dihedral angle measuring less than 180°.
• Concave Dihedral
  A concave dihedral has a dihedral angle measuring more than 180° and less than 360°.

Note: When the dihedral angle measures exactly 180°, the two faces are opposite half-planes belonging to the same plane. In this case, the dihedral is called a flat dihedral. If the two faces coincide, two limiting cases are possible: the null dihedral, whose dihedral angle is 0°, and the full dihedral, whose dihedral angle is 360°.

According to the measure of its dihedral angle, a convex dihedral can also be classified as right, acute, or obtuse.

• Right Dihedral
  A right dihedral has a dihedral angle of 90°. Its normal section is therefore a right angle.
• Acute Dihedral
  A dihedral whose dihedral angle measures less than 90° is called an acute dihedral.
• Obtuse Dihedral
  A dihedral whose dihedral angle measures more than 90° but less than 180° is called an obtuse dihedral.
  

According to the measures of their dihedral angles, two dihedrals may be complementary, supplementary, or explementary.

• Complementary Dihedrals
  Two dihedrals are complementary when the sum of their dihedral angles is 90°.
• Supplementary Dihedrals
  Two dihedrals are supplementary when the sum of their dihedral angles is 180°.
• Explementary Dihedrals
  Two dihedrals are explementary when the sum of their dihedral angles is 360°.

Example: Dihedrals α and β are supplementary because the sum of their dihedral angles is 180°.

Congruent Dihedrals

Two dihedrals are congruent when their normal sections are congruent.

Equivalently, two dihedrals are congruent when they have dihedral angles of equal measure.

Congruent dihedrals therefore have congruent normal sections.

Example: Dihedrals α and β are congruent because their normal sections both measure 90°.

Dihedrals Opposite Along the Edge

Two dihedrals are called opposite along the edge when they have the same edge and the faces of one are the opposite half-planes of the faces of the other.

For example, dihedrals α and α' are opposite along the edge.

Dihedrals opposite along the edge are always congruent.

This property is analogous to the congruence of vertically opposite angles in plane geometry.

Consecutive and Adjacent Dihedrals

Two dihedrals can be classified as consecutive or adjacent according to their relative positions:

• Consecutive Dihedrals
  Two dihedrals are consecutive when they have the same edge and one face in common.
• Adjacent Dihedrals
  Two dihedrals are adjacent when they are consecutive and their non-common faces are opposite half-planes.

  Note: Two adjacent dihedrals are supplementary because the sum of their dihedral angles is 180°. For example, dihedrals α and β are consecutive because they share the same edge and one face. They are also adjacent because their non-common faces are opposite half-planes.
  
  Similarly, the pairs α and β', α' and β', and α' and β are adjacent for the same reason. By contrast, α and α' are opposite along the edge and are not consecutive.

Additional Properties

Some useful properties of dihedrals are listed below:

• Two intersecting planes determine four dihedrals, arranged in two pairs of dihedrals opposite along the edge.
• Two planes are perpendicular if the dihedrals they determine are right dihedrals.
  Since dihedrals opposite along the edge are congruent and adjacent dihedrals are supplementary, if one of the four dihedrals is right, all four are right.
  
• Two half-planes sharing the same edge determine two explementary dihedrals because the sum of their dihedral angles is 360°. If neither is flat, one is convex, with a dihedral angle less than 180°, while the other is concave, with a dihedral angle greater than 180°.
  
• Parallel sections of a dihedral are congruent
  Parallel sections of the same dihedral determine congruent plane angles. In the figure, the sections π and π' of the dihedral δ determine angles α and α' with equal measure.
  
• The union of two consecutive dihedrals forms a dihedral having the same edge and the two non-common faces as its faces.