Why hexagons are the most extensively found shapes in nature?what is the mathematical reason?What makes them so efficient? Why heptagon,octagon etc are not preffered over hexagon?

Answer 1

Try this experiment:
cut out about 5 or 6 of each of
1. identical equilateral triangles
2. identical squares
3. identical pentagons
4. identical hexagons
5. identical heptagons (7 sided polygons)
...

Now for each set place them against each other, with sides matching up.
You should notice that you will be able to fit them exactly together for the triangle, the square and the hexagon
but you will have a "gap" for the pentagon and an "overlap" for the heptagon, (or for all regular polygons beyond that)

It all has to do with the interior angles, to complete 360° , (once around) of the adjoining vertices.

Triangle: interior angle = 60°, and 360/60 = 6
I can fit 6 equilateral triangle to make a flat surface

Square: interior angle = 90°, and 360 = 4
I can fit 4 squares to make a flat surface

Pentagon: interior angle = 108°, and 360/108 ≠ a whole number
I CANNOT fit pentagons together to make a flat surface

Hexagon: interior angle = 120° , and 360/120 = 3
I CAN fit 3 hexagons together to form a flat surface

Heptagon: interior angle = 360÷7 , which is not evenly divisible into 360
NO CAN DO!

Octogon: interior angle = 135° and 360/135 is not exact
etc.

So the only "nice-fitting" shapes are
the triangle, the square and the hexagon.

So why does mother nature use the hexagon more than the others ?
It has to do with circumference.
The shape that uses the smallest perimeter to have the largest area of course is the circle.
So of my three shapes which one would approximate the circle the best way ????
of course the hexagon !

One of our best examples of this is the structure of the honeycomb.
Not only has the bee figured out the above mathematics and uses the hexagon, but it also has calculated that by joining two adjacent chambers in a certain angle, it will minimize the wax needed in the construction of the honeycomb.
The amazing thing is that you would need advanced Calculus to actually find that angle, but the bee has it all figured out.

Answer 2

the interior angle of a heptagon = 900/7
and 360 ÷ (900/7) is not exact
(Same conclusion, don't know what I was typing above ! )