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?
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.
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 ! )
Hexagons are indeed one of the most extensively found shapes in nature, and there are several mathematical reasons behind this observation.
One of the key factors that make hexagons efficient in nature is their ability to tessellate, which means they can fit together without leaving any gaps. If you closely examine a honeycomb or a beehive, you'll notice that the cells are hexagonal and neatly arranged side by side. This hexagonal pattern allows bees to maximize the use of space and store the greatest amount of honey with the least amount of materials.
So why are hexagons able to tessellate so well? The answer lies in the sum of angles around a point. In Euclidean geometry, the angles around a point add up to 360 degrees. Since a hexagon has six sides, each interior angle of a regular hexagon is 120 degrees. This allows hexagons to fit together perfectly, forming a seamless pattern with no wasted space.
In comparison, other polygons like heptagons (seven sides) or octagons (eight sides) have interior angles that are larger than 120 degrees. This means that when these shapes are tiled together, there will be gaps left between them, resulting in inefficient use of space. Hence, nature seems to favor hexagons because they can pack together in a more compact and efficient manner.
Additionally, hexagons have other desirable properties in nature. They have a greater ratio of perimeter to area compared to other polygons. This means that for a given area, a hexagon has the smallest perimeter among all regular polygons, providing a more economical use of resources. This property is advantageous for biological systems where energy conservation is important.
To summarize, the mathematical reason why hexagons are extensively found in nature is their ability to efficiently tessellate and maximize the use of space, thanks to their interior angle of 120 degrees. Their properties of compactness and optimized perimeter-to-area ratio make them advantageous for many natural processes, including honeycomb construction, cellular structures, and more.