You know, hexagons are the bestogons. Why? Because bees. Bees are the best and build only the bestagon the hexagon.

Now, I know what you're thinking. Bees build hexagons because they're hexapods with hexagon eyes.

How could they do otherwise? Excellent point. But the Humble Bumble has an engineering problem to solve.

She makes two things. Honey and

wax, the former to eat and the latter to contain the former. To make but a little honey,

she must visit a lot of flowers, and to make one unit of wax, she needs eight units of honey.

Wax is costly for bees in flower terms, and honey is drippy in food terms. So to make a hive

that contains the maximum honey while using the minimum wax is royally vital.

Thus, a honeycomb conjecture, which shape works best.

To answer, we need to talk tiles.

Tiling is covering a surface with a pattern of polygons. There's lots of options because there's lots of polygons.

Even the regulars go on and on a gone.

Now for bees picking patterns, the more complicated ones obviously use more lines than necessary.

That's what complicated means, and thus a honeycomb of that tile would use more wax per honey.

So sticking to the simple regulars, there are just three that tile tightly, triangle

square and hexagon. Pentagons are broken hexagons, leaving gaps, same with septagons.

Octagons are all right, but there are no hexagon, which leaves the tiling trio which tile differently.

A square is a square of squares, which is a square of squares and so on.

Squares tile tightly by basically cheating, covering an infinite plane with an infinite number of parallel lines.

Like, wow, that's what a plane is. Boring!

Triangles pull the same trick, dividing themselves down into infinite nothing.

But not the hexagon, the only regular polygon to tile a plane without resorting to debasing self-division.

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