Hexagon in nature: why is the honeycomb hexagonal?

2년 전

The honeycomb is a perfect engineering marvel that is made up of hexagonal cells.

The bees hexagonal honeycomb cells may look pretty and easy to the eyes, but it has been a problem that had puzzled mathematicians for over a millennium. The problem is known as the hexagonal conjecture.


Image Credits Pixabay: A Honeycomb

Around 2000 years ago (36 BC), a Roman citizen, Marcus Terentius Varro, cited an answer that the most efficient way a planar surface could be in unit area size is through division into hexagons. This hypothesis is known as the honeybee conjecture.

That was one of the theory of the hexagonal structure. The other theory was of the opinion that the hexagon is the best suited shape that could support the bee's six feet.

The former theory was one that was more supported by the mathematicians of the day.


The old conjecture is now a theorem as it was proved to be right in 1999 by the American mathematician, Thomas Callister Hales.

The hexagonal shape above takes the least perimeter (most efficient) method of division of a plane into unit areas.

Take a look at the circular packing; you could see that rigid circles can only cover at most 90% of the area of the plane, which leaves 10% of the plane empty. Take note of the open area marked with a red rhombus.

To tile a plane with no area wasted, we have to choose between these regular polygons: the triangle (3 sides), the square (4 sides) and the hexagon (4sides).

Since we want to achieve the least total perimeter per given area, the hexagon is the shape which satisfies this requirement.

It gives us a greater degree of filling with fewer edges.

Let us do a little geometry here to check if indeed the hexagonal shape takes the least perimeter.

We would be checking the plane division for an equilateral triangle (triangle with equal sides), the square and of course the hexagons.

Let us assume the area of the hexagon that made up each bee cell is 23.4 square millimetre, mm2.

Let us take an area of 100 x 100 mm or 10,000 square millimter, mm2.

The number of cells that could be formed in that area = 10,000/23.4 = 427.350 or approximately= 428

  • Equilateral Triangle:
Area of an equilaterial Triangle A= (√3/4) a2 a= side of the equilateral triangle.

Since we already know A= 23.4mm2

size of each side of the equilateral triangle, a= 7.35mm2

Since there are 428 cells of equilateral triangles, each side of the triangle is shared by two. Therefore to get the total length, we will have to divide it by 2

Total length of the equalteral triangles = 3 (triangle has 3 sides) x 7.35mm * (428 cells ÷ 2) =4718.7mm or 471.87cm

Repeating the same for the squares we have:

Area of a square A= a2
where
the a= side of the equilateral triangle.

Since we already know A= 23.4mm2

a= √23.4= 4.837mm

Total lenght of squares while making allownace for two squares that share same side = 4 (sides of a square) x 4.837 x (428 ÷ 2) = 4140.472mm = 414.047cm

Area of a regular hexagon A= (3√3÷2) a2

where the a= length of the side

since A= 23.4mm, a= 3mm

Total lenght of the hexagon= 6 (sides of a hexagon) x 3mm x (428 ÷ 2) = 3852mm= 385.2cm

You can see that hexagon offers the most economic division of a planar surface area into units with respect to the length of the material required for division of the cells.


Why is this important to the bees?


Making beeswax is an "expensive" affair for the bees. For every one ounce of wax the bees produced, it will have to consume eight ounces of honey! The housing of nectar and honey is an efficient work. The cells made of beeswax is made by young bees using a high-energy substance which involves loads of nectar. Efficiency is critical as it ensures there is no waste of resources that they need to produce the structure (honeycomb).


More Hexagons in Nature


The dragonfly has no identified sense of smell or hearing. What it does depend majorly on its eyesight.

But they have extremely incredible compound eyes of all the insects. Their visual field span almost 360 degrees which makes it possible to see predator and prey coming from behind them. At least that explains why it is hard to catch one!


Each eye contain a whooping 30,000 ommatidia , singular ommatidium, hexagonal facets (lens) that collects lights from different directions.

The hexagonal packing is the most efficient as it covers the eye with the highest number of facets.

[Image Credits]Pixio


The more the facet the insect has, the more the resolution (detail) the insect can see. Unlike the human vision where only one "facet" is needed, and retinal sensory cells make the final image resolution, the insects need a larger number of facets to create a final image since all that facets "compound" to produce a final image.

In the case of the bees honeycombs, some are of the opinion that the hexagonal shape was created by the bees but not on purpose. The proponent of this school of thoughts say the bees started off the cells as a circle which due to heat melts. The wax then flattens, the surface tension then helps create a perfect junction that is 120 degrees apart forming the hexagon.

What do you think?


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Nature is actually a perfect engineer. Technological advancements have mainly come from studying nature. Well done !

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It gets things done. That is why most engineering marvels copied nature.

It amazes me how intelligent nature is! Almost all of our modern technology was perfected by nature long before humans figured it out.

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Nature is almost like the perfect architect. We just steal ideas from. Her.

I love the mathematics of nature. Perhaps humans are not the smartest in creation if it took us all that time to figure out what the bees had always known. Beautiful post.

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The humans really has a lot of catching up to do when it come to beating or even getting close to nature.

"You can see that hexagon offers the most economic division of a planar surface area into units with respect to the length of the material required for division of the cells." Another fact of life that doesn't stop making me wonder.

Were there also populations of bees making squared or round honeycombs and they died out because of the lack of adaptation?

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That is a perspective which I'd love to know about too.

I think nature does not do any thing by mistake. All of nature is perfect and so why can't the bees build in a hexagons?
This is most revealing especially the dragonfly's eyesight. I have definitely learnt some thing new. Thanks @greenrun

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I'd agree with you that nature is the real genius here, we can only learn from it. Thank you.

I'm just thinking to myself whats the inspiration for your post. How do u choose such amazing areas like this. Its really uncommon thanks for sharing @greenrun

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I just see it :)

I think nature usually finds the best way to develop itself, things like this take an incredible amount of years to finally become true.

Nature always proves its wisdom.

Speaking of honeycomb and efficiency, I can attest that the hexagonal honeycomb is the best as I used a honeycomb (artificial) in my flow visualization wind tunnel. Several literatures confirmed that they are the most suitable for my research. Although I could not use it. I had to improvise with a circular one instead due to the unavailability here in Nigeria.

For the bees, precision is one thing I have observed in the animal kingdom especially with insects. Making an hexagonal honeycomb is indeed most efficient as this will save them some effort. Nature has a lot for we humans to tap from.

Thanks @greenrun.

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The flow visualisation you mentioned sounded like something you'd see in aeronautics. You studying aeronautical engineering? When you say circular one, are you saying the beehives in Nigeria are not hexagonal? The one I saw last week was hexagonal, can't recall seeing one that wasn't. Thanks.

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Well, it is related to aeronautics because it cuts across aerodynamics but flow visualization is not strictly aerodynamics, instead it is an aspect of fluid dynamics in all.

I studied mechanical engineering.

And about the honeycomb I used, it is artificial. I made it myself. The natural honeycomb does not have the required strength.

Its pretty cool, isn't it, how things we tend to not really care about turns out to be so amazing? I really liked the article, especially the parts with the calculations :)
Keep up the good work!

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Thank you.

Wonderful. Just wonderful, its the ability to see the hidden things and reason fast that makes one a great mathematician.. This post is insightful and brilliant. Keep steeming

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If you look close enough you'd see a lot of more seemingly "hidden" details all hiding in plain sight. Thank you.

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Exactly. Thats why we have people who understand things very fast and some of us just look at them like how do they do it. Makes me remember the movie hidden figure. Thanks for showing us this. I love it.

Wow, i never knew there was a mathematical angle to this, maybe because i am not a fan of maths.
I

I would however propose the reason for the hexagonal hive as a way by which nature decides to baffle us. Nature in her beauty does that all the time

I have sincerely learnt something today!

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You'd be surprised at the mathematical angle to the most seemingly ordinary random things in nature. Even the spider's webs, peacock's plumage display, etc. The geometry makes for an interesting read. Thank you.

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I agree with you on that one.
Our world is full of interesting geometric pieces. Even the world itself is a geometric wonder!

As a bee farmer and honey trader, I find it quite revealing. I have never even come across this info anywhere before now. Thanks.

Does this info hold true for all species of honeybees?

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The different species all have similar behaviour, so I guess the theorem should be true for all. Thanks.

Man, you are just a genius..😩😩

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Genius? Not really, but thank you.

This just wowed me ..
What a great mathematical expression.
Remembering when i was in college..

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Jesus christ... @greenrun i duff my hat for this post. Super insightful

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It's good to relive those great memories.

i have always thinked about this in my head but didn't do any search and what's better today you came up with a fascinating post wow now i know some great stuff :D

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Thank you.

Cool @greenrun . I was thinking of another interesting Hexagon in nature, Saturn's north polar hood

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If he can explain that one, I'll buy him a beer :D
Although there was a scientific explanation of it,some one simulated the lower atmosphere winds and it created that hexagon shape. It didn't make much sense to me but I didn't read deeper into it at the time.

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@alexdory, you owe me a six pack :)

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Looks pretty interesting. I'd read up on that. Thank you @terrylovejoy

wow this is fantastic awesome information !

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Thank you.

@greenrun i love every part of the post apart from the mathematics, it a very bad reminder...
I love honey for my hair and skin and seeing the mathematics behind it is simply amazing.

The science behind the dragonfly's eye sight is making me jealous, i mean such sharp senses. I'll have to develop a new strategy to catch one now, we used to put a bucket of water under a very bright light at night especially when it rains and by morning a large number of them are in the water...

Thank you so much for this education.
I should get a notepad already

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Well, you'd need to catch one while it's sleeping :)
Thank you.

I disagree with the last school of thought. The honeycomb is definitely hexadecimal for a reason which I believe is to cover more area as you rightly stated mathematically above. I've not really thought about this.

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I think so too, just that one should not be too sure of things of this nature as sometimes there may be some truth to it. Thank you.

this world is beautiful and amazing the more we explore the more we learn

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Thank you.

it has science behind it too wow that's something cool to learn today

What a great outlook....greenrun

interesting blogs and posting follow now have a nice day

The Efficiency of nature should be aspired to in all endeavors

Nature loves economy!