Frame, J. Sutherland Formerly, Department of Mathematics, Michigan State University, East Lansing, Michigan.
A geometric figure bounded by three noncoplanar rays called edges that emanate from a common point called the vertex, and by the plane sectors called faces that are formed by each pair of edges (Fig. 1). A trihedron has three dihedrons formed by pairs of face planes, and three face angles formed by pairs of edges. A plane intersecting the edge of a dihedron cuts it into two trihedrons, whose trihedral angles are measures whose sum is the dihedral angle of the dihedron. Three planes having a common point but not a common line cut space into eight associated trihedrons, of which opposite ones are congruent but not necessarily superposable. (One is the mirror image of the other.) If one of these eight trihedrons has dihedral angles α, β, and γ and trihedral angle σ, its three neighbors that each share one of its faces will have trihedral angles α − σ, β − σ, and γ − σ. The sum of the four trihedral angles is 180°, so 2σ = α + β + γ − 180°.
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