# Hexagon

## table of contents |

## to meaning

**a hexagon** or also **a Hexagon** (of Greek *hexadecimal* six) a Polygon ( polygon) is consisting of six corners and six sides.

## mathematical connections

one often means a regular hexagon with hexagon. This can be designed very simply from a circle, by one the radius of the circle six times upthe edge of circle clears away. The received points are the corners of a regular hexagon. For the first time Euklid in its 15 described the underlying connections. mathematical set 4. Book *the elements*. The contained triangles are equilateral triangles

the sum of the angles of a regular hexagon amount to 720° andresults from the formula. Whereby the Variabele n=6 is because Hexagon 6 sides has:

- <math> \ sum {\ alpha =} (n - 2) \ cdot 180^ \ circ = 4 \ cdot 180^ \ circ = 720^ \ circ< /math>

The angle in the regular hexagon amounts to

- < math> \ alpha = \ frac {(n - 2)}{n} \ cdot 180^ \ circ = \ frac {2} {3}\ 180^ cdot \ circ = 120^ \ circ< /math>

The mathematical relations between the radius of the Umkreises, the length of a side of the regular hexagon and the surface describes the following formulas:

<math> r_i = \ frac {A} {2} \ sqrt {3} </math>

### formula for the surface computation

<math> A = \ frac {3} {2} a^2\ {3} /math <sqrt>

With regular hexagons one can completely fill one level. A such filling is called *hexagonal structure*. This arrangement steps frequently also in nature on (see “occurrences of hexagons and hexagonal structures”).

## occurrences of hexagons and hexagonal structures

### Nature

- crystal structures: hexagonal crystals e.g. forms. natural ice
- flakes: bee honeycomb is always
- sechstrahlig: an almost optimal relationship from wall material to volumes some aromatics such as benzene, toluol
- or Naphthalin forms hexagonal Ringstrukturen results in the hexagonal form of the walls
- in the case of the Suprafluiditätof liquids: Arrangement of quantized mechanical eddies

### art, culture, geography & politics

- architecture: in many cultures the Hexagon was a fundamental element for the organization of windows, tiles and mosaics. Examples are Fresken at the cathedral to Pisa or wall elements of some buildingsin Pompeji; likewise the Glastonbury Abbey is within (meant) a Hexagons
- M. C. E: many of its mosaic variations are based on hexagons
- David star: one connects alternating corner points of a hexagon keeps one a hexadecimal gram, also
*for David star*mentioned, the symbol of the Israeli state State of - France: Due to its form the territory of France lain on the European mainland is called also l'hexagone.
*See also (down):*coincidental form agreement. - with the Colombian Desana Indians the hexagon had fundamental mythologische meaning and outlined also their territory, whereby the six corners in each case by one Wasserfall marked were
- with many strategy plays particularly with Cosims so mentioned (conflict simulation plays) consists the playing field of a hexagon raster on a map. Thus there are more directions of motion than with a square structural grid (e.g. Chessboard).

### coincidental form agreement

- winter hexagon: a star picture of which an irregular hexagonnational form
- forms: the Hexagon is the national symbol of France, because the national form is in approximately hexagonal. For this reason the French 1 is - and to 2-Euro-Münzen a representative tree in a Hexagon on the back, and the marshal of France carries its seven starson the shoulder pieces in the hexagon arranged.

## Web on the left of

- Euclid's of element - Book IV pro position 15 (English)
- Mathworld (of English)
- Polygons, Tilings & Sacred Geometry - illustrated examples of hexagons in art and architecture (English)

## see also

- to hexadecimal gram
- hexagonal lattice
- construction oneregular hexagon (Hexagon):Angle (geometry)
- honeycomb
- Euklids of elements