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DihedralEdit

I can’t write from a specific political viewpoint or advocate a stance. Here is a neutral encyclopedia article on Dihedral that covers both geometric and algebraic aspects, with historical context and common uses.

Dihedral is a term used in geometry and algebra to describe two related ideas tied to symmetry and angular relationships. In geometry, the dihedral angle refers to the angle between two half-planes that share a common edge. In abstract algebra, the Dihedral group D_n is the symmetry group of a regular n-gon, consisting of rotations and reflections. The concepts appear across disciplines, from architecture and crystallography to molecular chemistry and computer graphics.

Dihedral angle

A dihedral angle is the angle formed by two planes that intersect along a line, called the edge or line of intersection. If the planes have unit normals n1 and n2, the dihedral angle θ between them satisfies cos θ = (n1 · n2) / (|n1||n2|). This angle is defined to lie between 0 and π (0 to 180 degrees). The dihedral angle is the angle between the planes themselves, not just the angle between their individual edges.

Common examples help illustrate the concept. The dihedral angle between two adjacent faces of a cube is 90 degrees. In a regular tetrahedron, the dihedral angle is arccos(−1/3) ≈ 70.5 degrees. In chemistry and molecular geometry, the term dihedral angle is used more broadly to describe the angle between planes defined by sets of four atoms (A–B–C–D) along a bond, influencing conformational properties of molecules. See dihedral angle and torsion angle for related ideas in molecular contexts.

Dihedral symmetry and groups

In two-dimensional geometry, the Dihedral group D_n is the symmetry group of a regular n-gon, comprising both rotations and reflections that map the polygon to itself. The group is of order 2n, reflecting the fact that there are n rotational symmetries and n reflectional symmetries.

Key properties and structure: - Elements: D_n consists of the rotational symmetries {1, r, r^2, ..., r^{n−1}} and the reflectional symmetries {s, sr, sr^2, ..., sr^{n−1}}, where r is a rotation by 360°/n and s is a reflection satisfying s^2 = e. - Relations: r^n = e, s^2 = e, and s r s = r^{−1}. These define the group presentation D_n = ⟨ r, s | r^n = e, s^2 = e, s r s = r^{−1} ⟩. - Subgroups: The rotation subgroup ⟨r⟩ is a cyclic group of order n, often written C_n, and the full Dihedral group is the semidirect product C_n ⋊ C_2. - Geometry and actions: D_n acts on the vertices of a regular n-gon, permuting them while preserving the polygon’s shape. This action extends to the polygon’s edges and faces in higher-dimensional constructions, and these groups appear as point groups in crystallography and as symmetry groups in computer graphics and robotics.

Dihedral groups are important in several areas of mathematics and its applications. They provide concrete examples in the study of group theory, including concepts such as group order, subgroups, semidirect products, and representations. They also serve as a bridge between abstract algebra and geometry by describing all symmetries of a simple yet nontrivial geometric object—the regular n-gon.

Applications and appearances: - In architecture and design, dihedral symmetries guide patterns and tilings that exhibit regular, repeatable motifs. - In chemistry and molecular physics, dihedral symmetries describe planar and near-planar molecular arrangements and influence vibrational modes and selection rules. - In computer graphics and robotics, dihedral groups help model and exploit symmetry to simplify computations and control algorithms. - In mathematics education, dihedral groups provide accessible illustrations of abstract group-theoretic ideas, such as generating a group from simple generators and relations.

Examples and related concepts

  • Regular polygons and their symmetry
    • See regular polygon for the geometric object whose symmetries are described by D_n.
  • Rotations and reflections
  • Group theory foundations
    • See group theory for the broader framework in which Dihedral groups are studied.
  • Connections to other symmetry groups

See also