Hexagon General /hg/

What makes hexagons the most superior storage device?

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Isn't six sides the greatest number you can have when you want a shape to be surrounded by identical shapes on each side? Like a hexagon surrounded by six hexagons in a beehive.

Its the best "stackable" shape to maximize storage, from a surface area point of view. Its also the most corners you can have before the structural integrity is compromised, ignoring circles, but those arent good for "stacking."

How is this true if we consider a rectangle is 100% space efficient

The tightest packing of something which requires some space (larva) is hexagonal close packed (hcp). If you then require them to be separated, you get a hexagonal lattice.

It's kind of interesting though. What do the bees want to maximize and minimize?

They want to maximize volume and minimize wax used since it is expensive to produce. It takes something 8 units of honey to make one unit of wax by volume. This is the shape that accomplishes that.

Hexagons have more area per unit of perimeter than squares. Thus, hexagonal buildings lose slightly less heat through their walls.

Wasn't it debunked that they don't actually make hexagons?

Benzene's aromaticity, duh.

It depends on the application.

Also you need to better define what disadvantages/advatages you are talking about.

Working in a warehouse I can say the 90 degrees walls of the building favor rectangles as the edges of hexagons and circles have wasted space. Granted you could rebuild the warehouse to better address that.

Basically there is no best universal storage device.
>in before bag of holding

Also hexagons only show on one axis, if you add more to compensate you introduce voids. This is why you see hexagonal tubes, but not hexagonal spheres.
(not the right terms, but I hope you get the idea, you got to think in 3D when designing a container)

From an common industrial view cubes are best.

No, if you look at tessellation sets you can have an infinite of interconnecting sides, however that becomes very impractical.

Some mix of dots, lines, triangles, rectangles and hexagons are what often form in reality.

The discovery of pentagons was actually a huge deal as they do exist, but are rare and not well understood yet, they also get insanely complex as they don't repeat. Many still don't believe it despite a Nobel prize being won for the discovery.

>The discovery of pentagons was actually a huge deal as they do exist, but are rare and not well understood yet, they also get insanely complex as they don't repeat.

Not sure what you mean, user. Are you talking specifically about regular pentagons? Or natural patterns?

Not that user, but him mentioning the fact that it doesn't repeat leads me to believe he is referring to Penrose tiling. There are examples of Penrose tilings that involve regular pentagons, such as pic related. I remember hearing about a type of Penrose tiling that occurs in nature, but I can't recall the specifics about it.

Quasicrystals?

en.m.wikipedia.org/wiki/Weaire–Phelan_structure

More in regards to natural patterns.
Like that pic you have can be redrawn using hexagons, as 4 pentagons make an elongated hexagon in it.
If you use basic crystal atomic packing models you will find it reduces to hexagons, thus it not a true stable pentagon structure. It gets really annoying at times as every time you think you found a new structure, it turns out to be a permutation of one of the existing predefined structures.

Yes, Dan Shechtman won the 2011 Nobel Prize in Chemistry for the discovery of quasicrystals. It is very interesting as it brings in a whole new frontier in material science when people were starting to get complacent in the field.

The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter. The conjecture was proven in 1999 by mathematician Thomas C. Hales.

arxiv.org/pdf/math/9906042v2

But doesn't that assume 2D space?

You can scale it to 3D why not

I do not understand... You fix the area M, and then the theorem assures that hexagonal honeycombs are the best subsurfaces of measure M to divide the surface ?

how did the bees know to use hexagons???

really makes you go hmmmmmm...

Im guessing they dont know, its just evolution.

For starters they used whatever random shape their instinct was telling them to use, probably a circle to store their shit and larva. It was not efficient and structurally optimal so they didnt prosper as good as a hive that had some bees that had different instincts that told them to use better shapes. Over time the good design survived with the instinct to create good designs.

Or it was aliens

Darwin spends a good amount of space discussing this unique instinct in Origin of Species. He tries to prove that this instinct developed from very small gradations in bees as they evolved, interesting stuff.

Hexagonal honeycombs use the least amount of perimeter to divide the plane into equal areas. In other words the lowest possible ratio of perimeter to area divided equally is 1/sqrt(3). If you divide the plane up into a square grid, then this ratio is 2.

Yes. The 3D version is called the "Kelvin Problem".

No, you can't.
The question "what space-filling arrangement of similar cells of equal volume has minimal surface area?" is an open problem.

>what space-filling arrangement of similar cells of equal volume has minimal surface area?
And then everyone started blowing bubbles for science reasons.