Mathematical properties of Hexagon.

feb 27 (3)Since this article is part of the Hexagon awareness series I am just restating what we all probably learnt at school. This is subjected more research and in depth understanding. Usually when we say hexagon, we mean regular hexagons, so is it here. but hexagonal does not necessarily refer to symmetric hexagons.

A hexagon is  a figure with 6 edges and and 6 vertices, a regular hexagon is convex figure with sides of the same leght, and the internal angles or 120 degrees. It has 6 rotational symmetries and 6 reflection symmetries.

Hexagons are the only regular polygons that can be subdivided into another regular polygon.

Hexagon-Tessellation-pic. courtsey internet
Hexagon-Tessellation-pic. courtsey internet

Hexagons are unique regular polygon such that the distance between the centre and each vertex is equal to the length of each side.

As for number six it is the first perfect number, the second smallest composite number.  It is the only number that is both the sum and the product of three consecutive numbers. That is

  • 1x2x3=6
  • 1+2+3=6

Hexagons can be tiled or tessellated, in a regular pattern. If like me you wondered what it was, then hexagon can bordered by six other hexagons which can themselves be bordered by six hexagons—including each other and so on ad infinitum, in any direction without leaving empty spaces. This kind of laying can be possible only with squares and triangles. This is probably why it is important in a variety of feilds.  Look at our Chinese chekkar boards or the beehives. They are all  hexagonal in pattern.  Somewhere the hexagonal tessellation is a dual of triangular tessellation these can be converted to one to the other with a little alteration of layout. The hexagonal tessellation, or lattices are they are called, provides an ideal blend of efficiency and strength, and economy of material as compared to the triangle or square.hexagon- triangle

When it comes to circles, hexagons helps to pack cicles on a flat plane very efficiently.  Though we are placing circles, their grid is hexagonal. Since cells tend to most efficient exposure of shapes, they tend to circularity, but lateral compression of the neighbours result in hexagonal tilting. This could be why hexagons are considered as natural shape.

The connect between triangles, and hexagons is a continuation of the relation of the hexagon with circle.  Like I shared before, triangles and hexagons are the only  two regular polygons. The hexagon can be divided into equilateral triangles while the triangle into subtriangles.

Hexagon-- Giant causeway  Ireland--image courtsey internet
Hexagon– Giant causeway Ireland–image courtsey internet

This is quite hectic, to go on, I am just teasing your thoughts as it is the month of Hexagon awareness. Hexagons pop up in unsual places like aromatic chemistry, healing, bio-sciences, mathematics, spirituality whatever. Like I shared before for me the benzene hexagon is the representative of vibrant life.

If you would like to know more – hexagon (Wikipedia.),  www.hexnet.org, www.wolframmathworld/hexagon

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