Hexagon

A hexagon or hexagon [ hɛksago ː n] (Greek ἑξα, hexa, "six" and γωνία, Gonia, " angle, corner " ) is a polygon (polygon ), composed of six corners and six sides. Are all six sides of equal length, one speaks of an equilateral hexagon. Additionally, all angles at the six corners of the same size, then the hexagon is called regular or regular basis.

Mathematical relationships

The underlying relationships of the regular hexagon first described Euclid in his 15 mathematical proposition of the 4th book The Elements. If the opposite corners of the hexagon interconnected, there are six equilateral triangles. If, however, all non- opposite corners together so you get a hexagram.

Construction

A regular hexagon is very easy to construct a circle by the radius of the circle is six times removed on the circle edge ( construction only with a compass ). The points obtained are the corners of the hexagon. Alternatively, the two-time ablation on the circle edge is sufficient. The missing corners can then use the straight line through the center of the circumcircle and the already known corners are constructed (see animated graphic ).

Construction of a hexagon with a ruler and compass

Approximate construction

A ( in the absence of circles ) for practical purposes often sufficient approximation to a regular hexagon is obtained when constructing the hexagon of approximately equilateral triangles, which have a ratio of height to base side of 7:8. This approach is used in applications where an error of less than 1 % can be tolerated. Thus, for example, cards or tiles are divided into easy to draw and easy to be calculated sectors.

The points P (x | y) are numbered in the following according to the digits of a clock, the center is ( 0 | 0):

• Coordinates of a horizontal hexagon

P1 ( 4 | 7), P3 (8 | 0), P5 ( 4 | -7), P7 ( -4 | -7), P9 (-8 | 0), P11 ( -4 | 7)

• Coordinates of a set at a point hexagon

P2 ( 7 | 4), P4 ( 7 | -4), P6 ( 0 | -8), P8 ( -7 | -4), P10 ( -7 | 4), P12 ( 0 | 8)

A better approximation is obtained by replacing, in the above coordinates, the figures 4, 7 and 8 by 15, 26 and 30 respectively. Other possible triplets (56, 97, 112) or (209, 362, 418). In all these triples the triangles formed from the corresponding side lengths differ only very little of a right triangle with acute angles 30 ° and 60 °. Another such triangles can be found iteratively by from " cathetus " a and b ( a < b) of an already known such a new triangle catheti 2a b, 3a 2 b is formed. The " hypotenuse " is because of the 30 ° - property always twice the short cathetus.

Hexagonal tilings

The regular hexagon is adjacent to the equilateral triangle and the rhombus (including square ) is the only equilateral polygon with a plane without any gaps filled ( cradled ) can be. Unlike the tiling of triangles or rhombuses, the individual polygon in a hexagonal tiling only neighbors are connected via full edges, but none that are connected only via corners or edge parts. The distance of the center points of two directly adjacent hexagons

The ability to form hexagonal tessellation is a cause that hexagons are significantly more likely than regular pentagons, heptagons and higher than the polygons.

Occurrence of hexagons and hexagonal structures

Regular hexagons are often found in art, culture, nature and technology. The following list shows just a few examples.

Nature

• Crystal structures: Many crystals form from hexagonal shapes. Well-known examples are natural (water) ice and six-pointed snowflakes.
• Honeycomb: The hexagonal shape of the walls results in an optimum ratio of wall material to volume and provides high stability.
• One of the basic structures of organic molecules is the benzene ring which is a regular hexagon of six carbon atoms. In the crystal structure of the graphite carbon forms planar surfaces of hexagons.
• The superfluidity of liquids: array of quantized vortices mechanical
• The north pole of the planet Saturn is the center of a stable atmospheric structure in the form of an almost regular hexagon. It has a diameter of almost 25,000 kilometers.
• The Wintersechseck is a constellation that forms an irregular hexagon.
• Basalt forms when it freezes slowly, often meter long hexagonal columns.
• The convection in the Bénard experiment can train as regular hexagons.

Arts and Culture

In architecture, painting and graphics of Judaism and Christianity is the use of the hexagon, the symbolism of the number 6 is based, the meaning of the sum of the first three numbers (1 2 3 ) and their number symbolism results. You and therefore the equilateral hexagon symbolize the omnipotence of God in both religions. However, they are are also available for balance and harmony of the divine and the mundane, which also imaginary in the equilateral and the hexagon in its composition from six or geometrically visible equilateral triangles, including the symbolism of the number 3 included. The number 6 and the hexagon, depending on the context, also the symbol of the six -day work of creation ( first book of Moses ) be.

An example from the Christian painting of the late Middle Ages is the hexagonal table in the " Garden of Eden " of the Upper Rhenish Master ( 1410; Frankfurt, Städel ): He has mastered color bright in form and symbolic power of the hexagon the image and is crucial to the understanding of the image, which is emphasized by its role in the composition.

Other examples of the use of the hexagon, but only limited symbolic importance:

• Architecture: In many cultures, the Hexagon is a fundamental element for the design of windows, tiles and mosaics. Examples are frescoes at the cathedral in Pisa or wall elements of some buildings in Pompeii; well there is the Glastonbury Abbey within one ( imaginary) hexagon. The private villa Kentuck Knob by architect Frank Lloyd Wright, the hexagon is the continuous design principle.
• MC Escher: Many of his mosaic variations based on hexagons.
• France: Due to its approximately hexagonal shape which is located on the European mainland territory of France is also referred to as l' hexagons. Therefore, is located on the back of the French 1 - and 2- euro coins, a stylized tree in a hexagon, and the Marshal of France carrying its seven- star located on the shoulder pieces in the hexagon.
• Music Electronics: The first commercial electronic drums of the British firm Simmons were hexagonally shaped.

Technology

• The heads of bolts are often hexagonal, either as external hex or Allen (Allen). Also hex nuts are common.
• A sandwich panel with a honeycomb core, which is a composite structure having a honeycomb-shaped support core, is used in constructions which require a high strength with low weight and material consumption.

Games

In many games, especially in conflict simulation games, is the game plan of a hexagonal grid. This allows, among other distances between two fields is determined as the simple case of a square grid (for example a chessboard ).