Why is a square? a triangle? a hexagon?

House is one of the most important fact for living. Not only mankind also others build their own houses. For example bees, spiders, ants and etc. Lots of scientist says hexagon shape house of bee is the powerful and scientific. Why the bees build their own houses as hexagon shapes? Nature is more efficient that our think.

If we want to use a space efficient and solid, we have to fill the spaces without spares. Lets assume that the shape of room is a regular polygon. Then answers will be a regular triangle, square and hexagon.

The one angle of regular n-polygon is

｛(n-2) ×180｝/n

and if number of m of regular n-polygon fill a space without spares the angle in one vertex has to be 360° so,

｛(n-2) ×180｝/n ×m = 360°

making this expression simply,

mn - 2m - 2n = 0

→ (n - 2)(m - 2) = 4

m and n are natural number, so (n,m) will be (3,6), (4,4) and (6,3).

That means six regular triangle, four regular square and three regular hexagon can make fill a space without spares.

And why did bee choose hexagon? Which shape can make bigger space with same amount of material?

On the assumption that the circumference of those shapes are 12 each,

A side of triangle is 4, so the extent is (√3)/4 ×4^2 ≒ 6.93.

A side of square is 3, so the extent is 3^2 = 9.

A side of hexagon is 2, so the extent is 6{(√3)/4 ×2^2}= 6 √ 3 ≒ 10.39

According to the extents, hexagon shape is efficient for making a bigger space with less materials.

Regular hexagon dose not make spares and very efficient for making a structure firmly. There are some more hexagon shape examples. The inside of big telescope has hexagon shape, and airplane has either. Inner part of hexagon shaped structure is empty, and both end points fixed with metal tubes. Due to that, this shaped structure weigh less and stronger than other shapes.