Hexagon

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Regular hexagon 1

Hexagon r12 symmetry

Hexagon i4 symmetry

Hexagon d6 symmetry

Hexagon g6 symmetry

Hexagon is a polygon with six edges and six vertices. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Hexagons are seen extensively in nature, science, and engineering due to their interesting properties and aesthetic symmetry.

Properties and Types

A hexagon can be classified as regular or irregular. A regular hexagon has all sides of the same length and all internal angles equal to 120°. In contrast, an irregular hexagon lacks equal side lengths and angles. Regular hexagons are of particular interest because they can fill a plane with no gaps, which is known as tessellation. This property is not only mathematically significant but also widely utilized in various fields such as architecture and design.

Tiling and Tessellation

Hexagons are the most efficient shape for tiling a plane without any gaps or overlaps, a property that bees exploit in constructing their honeycombs. This efficiency arises because hexagons have the highest number of sides among polygons that can tessellate. As a result, hexagonal patterns are prevalent in both natural and human-made structures.

Applications

1. 1. 1. In Nature###

The hexagonal shapes of honeycomb cells are one of the most well-known examples of hexagons in nature. This structure provides a high strength-to-weight ratio and efficient use of space and materials. The hexagonal pattern is also observed in the basalt columns of the Giant's Causeway in Northern Ireland and in the eyes of insects, where thousands of hexagonally shaped ommatidia make up the compound eyes.

1. 1. 1. In Science and Technology###

In science, the hexagon appears in the structure of graphene, a single layer of carbon atoms arranged in a hexagonal lattice. This arrangement gives graphene its extraordinary strength and electrical conductivity. In technology, hexagonal designs are used in creating efficient cooling networks, particularly in computing where heat distribution can be optimized using a hexagonal grid.

1. 1. 1. In Culture and Design###

Hexagons are also prevalent in design and culture. The board game Settlers of Catan uses hexagonal tiles to create a dynamic playing field. In architecture, hexagonal patterns are used for tiling floors and decorating surfaces, showcasing both aesthetic appeal and structural efficiency.

Mathematical Properties

The area of a regular hexagon with side length a is given by the formula: \[ \text{Area} = \frac{3\sqrt{3}}{2}a^2 \] This formula derives from the hexagon being composed of six equilateral triangles.

The perimeter of a regular hexagon is simply six times the length of one side: \[ \text{Perimeter} = 6a \]

See Also

• Polygon
• Tessellation
• Graphene
• Settlers of Catan