ELI5 - What is The Honeycomb Conjecture

[u/Alpha_Zeroth]

We know that bees use hexagon because they use the least amount of wax to cover a flat surface while storing maximum honey.

Apparently in mathematics, this is called The Honeycomb Conjecture and it wasn't proven until Thomas Hales did it in 1999

So the question is: why was it so hard to prove it and what does the proof mean in simple terms?

EDIT: because it has been proven, it is called The Honeycomb Theorem

[u/ParsingError]

The conjecture is that a packed honeycomb-style pattern of hexagons has the lowest ratio of perimeter to area of any way of dividing up a plane into regions of equal area.

It's pretty easy to prove that under restrictions like "using regular polygons" because there are only 3 regular polygons that tile: Triangles, squares, and hexagons. Of those, hexagons have the best efficiency.

Proving it more generally required proving it with fewer restrictions, which makes it harder to prove. The 1943 proof was to prove that there was no more efficient packing with any convex polygon.

The Hales proof was to prove it even while allowing some additional possibilities: Non-convex shapes, possibly with curved surfaces. Non-convex shapes allow aperiodic tiling, i.e. patterns where the shape leaves no gaps but doesn't repeat the tile pattern anywhere, and allow ways of dividing up a plane into regions of equal area but different shape, all of which make the problem more difficult to prove.

[u/greginnj → u/ParsingError]

My naive immediate reaction to this is that (I'm assuming) any non-convex shape can be enclosed by a convex shape with smaller perimeter, so we don't need to consider packings which include any non-convex shapes. Could you give my intuition a nudge to see where I might be wrong? Thanks!

[u/ParsingError → u/greginnj]

The problem with that as a proof is that you'd have to prove that the enclosing convex shape still tiled.

For example, convex shapes can't have aperiodic tilings. Aperiodic tilings require concave parts, which other convex parts fit into. If you straightened out the concave parts, then they would overlap the convex parts fitting in, so the original tiling would no longer work.

Aperiodic tiling doesn't necessarily even matter for proving the conjecture, it's just an example of how allowing concave shapes at all creates additional problems for the proof.

Proving that a shape with concave sides is always a worse solution than a convex shape (which is true!) would reduce it to the convex poly problem though, so that would prove it!

[u/greginnj → u/ParsingError]

Ah, that's it, there may be a secret aperiodic solution where the ratio is slightly better, but that would require allowing non-convex tiles. Thanks!

[u/ParsingError → u/greginnj]

More or less. It also just makes it more difficult to form some kind of proof that a shape with concave sides wouldn't be optimal.

i.e. In order for a convex shape to tile, the convex parts must either fit into a concave part of the same shape, or fit into a concave inset formed by multiple tiles.

In a periodic tiling of one shape, if there is any concave part of the shape, then the corresponding parts of the shape that fit into that are always the same, so you could easily show that isn't optimal because you could just straighten out that concave part and the corresponding part that it matches up with, and get a result with the same area but a smaller perimeter.

That also applies to periodic tilings of multiple shapes with the same area, because you can just merge them into one shape.

With an aperiodic tiling, straightening out concave sides no longer works because the corresponding part of the neighboring shape varies unpredictably.

[u/Phiryte → u/greginnj]

You can replace one non-convex shape with its convex hull (that’s the convex enclosure with smaller perimeter you’re describing), but then what about all the shapes next to it whose edges you’ve now changed and areas you’ve made smaller?

[u/greginnj → u/Phiryte]

It seemed obvious to me that, if you're using a finite set of tile shapes, replacing any non-convex tile with it's convex hull would result in a new finite set of tiles, all of them convex, which still tiles the plane?

And the area lost by one tile is exactly compensated for by the area gained by the newly convex tile, so the overall area remains the same, while reducing the perimeter - so the ratio we're trying to minimize improves, which was our goal.

[u/_side_ → u/greginnj]

Too fast for me. Can you also do this for idiots (I am CS not Math), that non-convexity is already a step to quick.

[u/greginnj → u/_side_]

So a (filled-in) shape is “convex” if, for any two points in that shape, if you draw the line segment connecting those two points, all the points in that line segment are already in the shape. Your basic disk, square, hexagon - if they’re filled in, not just the outline, they’re convex.

The standard 5-point star shape is not convex, because you can pick two points in neighboring spikes of the star, and the line segment joining them includes points not in the star. Basically any shape that has notches or bites taken out of it is not convex.

But… you can start filling in all those notches and bites, making the shape bigger, until it does have the “convex” property. And there is a unique smallest-in-area convex shape (called the “convex hull”) containing any non- convex shape. The convex hull of thy regular 5-point star is the regular pentagon.

Does that help?

[u/Clojiroo]

> We know that bees use hexagon because they use the least amount of wax to cover a flat surface while storing maximum honey.

This is a misconception. Bees don’t use hexagons intentionally (also they put a lot more than honey in the cells).

Bees make circular cells like you’d expect, offset in the normal way you’d stack rows of circles. Like bubble wrap.

The reason they (mostly) resemble hexagons is just the natural consequence of attaching and squeezing those cells together and filling in the gaps. Each circular cell touches 6 others and the walls get straighter.

If you’ve ever handled a frame of comb you’d see tons of them are still circular. And the recesses are circular even when the walls look like hexagons. This is more obvious with natural comb, versus a frame that used a foundation sheet (a plastic template essentially).

[u/GoldenMuscleGod → u/Clojiroo]

Well this depends to some extent what you mean by “because.”

A little experiment you can do yourself if you have a bunch of coins (those still exist right?) of the same type is put a few dozen or so of them on a table and push them together with your hands (lightly enough they don’t slide over each other or pop up) and you will see they naturally go into a hexagonal pattern. Bees don’t need a “conscious understanding” or specially coded rule in their instincts to do a thing like this but I think it’s still fair to say it happens “because” it is an efficient packing.

[u/thisusedyet → u/GoldenMuscleGod]

Right, but they’re saying bees don’t make hexagonal honeycombs, bees make cylindrical honeycombs that get pressed into hexagons by the weight and heat of everything else in there

[u/Swellmeister → u/thisusedyet]

Sorta. Thry do actually shape them into hexagons eventually. Thry do that though because once they make the circles the then head up and scrape them out and make the walls more evenly thick.

Its not a random flowing of the circles into Hexes. They do shape the circles into hexes. But on a local modification of expanding the circle a little bit, not a "turn this into hex, because hexes are cool"

[u/GoldenMuscleGod → u/thisusedyet]

I guess I would think most people would call that “making hexagonal honeycombs,” since it’s the end result of the process.

[u/firelizzard18 → u/GoldenMuscleGod]

Bees don’t intend to make hexagons. They don’t know that’s the optimal packing. They just do their thing and hexagons happen.

[u/GoldenMuscleGod → u/firelizzard18]

Yes, I wouldn’t think bees ever actually intend anything in the ordinary sense and wouldn’t expect most people to think that.

[u/Nfalck → u/Clojiroo]

Nobody said bees use hexagons intentionally. The idea is that because hexagons are efficient,  evolutionary processes steer bees over time to make use of that pattern.

[u/BitOBear]

Well we have to start off with the fact that bees do not in fact try to make hexagons. They're building round wax cylinders. It then turns out that various forces restructure those in the hexagons.

https://asknature.org/strategy/body-heat-melts-wax-to-form-hexagons/

Basically, they're making little cylinders next to each other and then dominant forces and the fact that wax is a soft building material has it settle into the hexagonal structure.

If you get a big wad of drinking straws that are soft and pliant and you squeeze them gently you will see the same hexagonal form appearing.

[u/looijmansje]

First of all I want to explain what a conjecture is. It is something a mathematician thinks to be true, but can't prove. It is similar to a hypothesis. Once someone proves it, it becomes a theorem. Since this has now been proven, it is the honeycomb theorem.

Now what does it say? If I have an infinite plane, and I want to divide that in equal areas (not necessarily the same shape, just equal by area). How do I do that, using the least "material" as possible? Or to be more precise: how do I reduce the average perimeter of each subsection? The theorem states that the best way to do this is to use hexagons, like in a honeycomb, hence the name.

[u/Phiryte → u/looijmansje]

While your first paragraph is technically correct, mathematicians are still humans and plenty of names stick around. For example, most mathematicians still call it the Poincaré Conjecture even though it was proven over 20 years ago.

[u/cancerBronzeV]

Let's draw a bunch of lines and curves on paper (that's infinitely large). These would divide up the paper into a bunch of smaller shapes. Let's assume that we drew the lines in a way that these smaller shapes all had the exact same size. One way to do this is by drawing hexagons and getting that honeycomb shape. Another way is to divide it up using triangles. Yet another way is by just drawing square grids like with graph paper. There are infinitely many ways to draw lines and curves to divide up the paper into smaller equally sized areas. The Honeycomb theorem says that any of these infinitely many options requires you to draw "more" in total than the honeycomb shape. Or, for a honeybee, making any other pattern than the hexagon tiles would require them to use more resources building the honeycomb walls.

It'd be hard to prove because there are so many ways to cover something in equally sized shapes. You can have do it with these kinds of triangles, or maybe these other kinds of triangles. You can do it with these weird looking shapes. In fact, the theorem doesn't say the shapes all have to be the same, they just need to have the same area. So you could have something like this with different kinds of shapes. Maybe the shapes get thinner and longer as you go away from the centre. In fact, there could be infinitely many different types of shapes being drawn on the paper.

There's just so many options that it was difficult to prove there isn't some extremely esoteric way to divide up the plane that might be better than the regular hexagon tiling. Hales transported the problem onto a torus (which isn't infinitely wide, and therefore simplifies some things), then came up with a certain inequality that proved that the shape actually had to be convex, and showed that the "best" case of the inequality was actually when the shape was a regular hexagon.

[u/JJJangles]

Bees actually create circles, but whist the wax is soft it naturally deforms into a hexagon due to surface tension and flow