hexagonen

Hexagonen (Dutch for hexagons) are polygons with six sides and six interior angles. A regular hexagon has equal sides and equal interior angles of 120 degrees. In a regular hexagon, the diagonals form three pairs of equal lengths; the longest diagonals connect opposite vertices and equal twice the side length. It can be partitioned into six equilateral triangles, illustrating its high degree of symmetry.

The symmetry group of a regular hexagon is the dihedral group D6, comprising six rotations (including the identity) by 60-degree increments and six mirror reflections. This gives the hexagon a 6-fold rotational symmetry and multiple reflection axes.

Area and perimeter follow standard formulas for regular polygons. For a regular hexagon with side length s, the area is (3√3/2)s² and the perimeter is 6s. The circumradius (distance from the center to a vertex) equals the side length s.

Tilings and applications: Regular hexagons tile the plane in a honeycomb arrangement, with three hexagons meeting at each vertex. This tessellation is common in nature and design, appearing in honeycombs, crystalline lattices such as graphene, benzene rings, and in various tiling-based patterns used in games and architecture.

Variants and terminology: Hexagons that are not regular are called irregular hexagons. Convex hexagons have all interior angles less than 180 degrees, while concave hexagons contain an indentation. The term derives from Greek: hex, six, and gonia, angle.