What shape honeycomb would a four dimensional bee make?

--Let's start with good old three dimensional honeybees.

Why is there a hexagonal structure in there honeycombs?

Why not squares or asymmetrical blobs?

In 36 BC, the Roman scholar, Marcus Terentius Varro, wrote about the two leading theories of the day.

First, bees have six legs, so they must obviously prefer six sided shapes.

But that charming piece of numerology did not fool the geometers of the day.

They provided a second theory.

Hexagons are the most efficient shape.

Bees used wax to build the honeycombs.

And producing that wax expends bee energy.

The ideal honeycomb structure is one that minimizes the amount of wax needed, while maximizing storage.

And the hexagonal structure does this best.

The Greek mathematician, Pappus of Alexandria, gave an argument for the efficiency of hexagons.

He believed that there are three possible honeycomb patterns-- triangles, squares, and hexagons.

Because bees like symmetry and they'd only use tilings of regular polygons-- shapes with straight sides of equal length and equal angle between them.

Triangles, squares and hexagons are the only regular polygons that tile the plane-- cover it without leaving any space.

Here's a fun geometry exercise.

Show that with a fixed amount of wax-- that is, a fixed perimeter length-- a single hexagon can hold more honey than one square or one triangle.

In fact, a circle would hold more than any of these.

It has the best perimeter to area ratio.

But you can't arrange equal sized circles to entirely fill the plane.

Among the options that tile the plane, hexagons are the most efficient.

But there are many possible honeycomb shapes besides squares, triangles, and hexagons.

How can we convert the question of efficient honeycomb structure into a general and rigorous math problem?

We'll draw lines on a two dimensional plane to represent the way bees use walls of wax to partition the honeycomb.

Let's assume the bees want honey pockets of equal area.

So we want to draw the lines on the plane, so that they enclose regions of area one square centimeter.

Remember, the bees want to minimize wax, while maximizing honey storage.

The mathematical analog of this is minimizing the length of the lines relative to the number of regions with area one we're creating.

Also, since we're mathematicians, we'll just assume the honeycomb is infinitely large, which means the lines will divide up the entire plane.

So putting this all together, here's the mathematical question.

How should one partition the two dimensional plane into infinitely many regions with area one to minimize the perimeter?

For example, we could partition the plane like this or this or this or this, or infinitely many other ways.

Which has the smallest perimeter?

Here's a technical caveat.

Since there are infinitely many regions with area one, the total perimeter will be infinity, no matter what shape we use.

We're really interested in minimizing perimeter relative to the number of honey pockets or cells enclosed.

To calculate that, look within a circle and divide the perimeter by the area of the honey pockets.

Then do the same thing with bigger and bigger circles, taking a limit.

We're looking at the ratio of the perimeter to area one cells enclosed.

And we want to minimize this quantity.

And just as Pappus predicted, the hexagonal tiling of the plane minimizes perimeter.

But Pappus had simplified the mathematics problem.

He assumed that the solution would be a regular polygon.

It took another 2,000 years before a mathematician, Thomas Hales, rigorously proved this fact.

That name might sound familiar.

In our very first episode, we highlighted his proof that this and this are the optimal ways to pack spheres in three dimensional space.

Improving on Pappus' results, mathematicians had already concluded that honeycombs were the best polygonal tilings-- better than this or this, for example.

But Hales finished the proof, eliminating tilings like this.

He showed that the optimal honeycomb wouldn't have bulging sides.

So two millennia later, Pappus' conjecture about the efficiency of hexagonal honeycombs was substantiated with rigorous mathematics.

Turns out bees are acting optimally.

Does that mean bees are little math geniuses?

Good question, Kelsey.

According to English biologist, D'Arcy Wentworth Thompson, everything is the way it is, because it got that way.

Hey, Joe.

Maybe you can help us with how a biologist would approach this problem.

Pappus wrote that the bees selected their hexagonal structure wisely and that they possessed a certain geometrical forethought.

Does that imply that there is geometry programmed in a bee's brain?

Well, Darwin would explain honeycombs by saying bees must have experimented with squares, triangles, and countless other shapes, and that bees that made hexagons had more energy left to find food and that somehow they passed this knowledge of measurement geometry on in their genes.

But Thompson had his own way of looking at evolution.

He believed Darwin's view, that an organism's traits could always and only be explained by selection, didn't always get at the root cause.

Thompson believed, especially when it came to forms and patterns, you had to bring the laws of physics and math into the mix.

Thompson said, sure, efficiently tiled honeycomb gives bees a survival advantage.

But if nature's so economical, why insist on building complex systems and patterns through millions of years of strenuous trial and error when math or physics gives a much simpler answer?

So bees aren't really doing math at all.

It's more like physics is doing math, which is less surprising.

Now so far we've found a mathematical framework that gives us the form we see in nature.

We just need a physical process that might produce that framework.

And we can see that in a familiar place-- bubbles-- and the physics that controls the shape.

In a bubble, the force of air pressure inside is balanced by surface tension pulling on the liquid.

Soap films meeting at junctions will always seek out the most mechanically stable arrangement, which the forces acting on the films are all in balance.

For a flat array of identical bubbles, the most mechanically stable pattern that minimizes surface tension is a honeycomb array.

The intersection of four bubble walls is unstable compared to three.

And they will always rearrange to the lower energy triad.

See?

A physical process that seeks to reach a stable, low-energy state is enough to produce this algorithm.

This rule of minimal surfaces extends to three dimensional bubble stacks as well.

But there, things start to get more interesting.

Let's apply this idea to higher dimensions.

Time to explore a 4D bee.

Assume that a four-dimensional bee makes four-dimensional honeycombs, but they're arranged in three-dimensional sheets.

This is analogous to the fact that regular three-dimensional bees essentially make two-dimensional sheets of hexagonal wax patterns.

Ignoring the base, if you take a two dimensional slice along the face of a honeycomb, it would show a hexagonal tiling.

Similarly, if you took a three-dimensional slice along the four dimensional honeycomb, it would show 3D space partitioned into equal volume pockets.

The three-dimensional bees created hexagons, because they were the most efficient.

They minimized the perimeter relative to the number of area one cells created.

In four dimensions, the question becomes, how do we partition three-dimensional space into pockets with volume one to minimize the surface area between them?

Motivated by soap foam, Lord Kelvin, eponym of the temperature unit, Kelvin, asked this question in 1887.

And he conjectured this solution.

Kelvin's proposed solution is comprised of truncated octahedron, which are formed by taking in octahedron and chopping off all six of its corners, but with a slight curve on the hexagonal faces.

They neatly pack together with four truncated octahedron meeting at each vertex.

It's a beautifully simple honeycomb, but unfortunately suboptimal.

In 1993, physicists, Dennis Weaire and Robert Phelan, discovered a way to partition three-dimensional space into packets with volume one that has a slightly smaller surface area than Kelvin's structure.

The Weaire-Phelan structure is more complicated than Kelvin's.

It uses two different equal volume shapes-- the truncated hexagonal trapezohedron and the pyritohedron, each with a slight curve on some of the faces.

The Weaire-Phelan structure has less surface area than the Kelvin structure by a whopping 0.3%.

But no one knows whether the Weaire-Phelan structure is the most efficient.

Just like its two dimensional counterpart hexagonal bubble arrays, the Weaire-Phelan structure can be found in something as common as soap foams.

They're a great example of how, despite our tendency to assume that making something complex requires special intelligence, a simple physical process or mathematical rule can form an ordered structure.

But exactly how does physics guide a bubble's shape?

Can we really prove bees use the same physics to build honeycomb?

And what does this tell us about all the other amazing places we see hexagons in nature?

We're going to explore that and a lot more over on It's OK to be Smart.

And Kelsey's going to join me.

And you should too.

during the sign up process.

Wow.

We asked you all to send in ideas for topics you'd like to see covered, and you certainly delivered.

There were tons of great ideas, covering a wide range of mathematics.

It kind of looked like a math library exploded inside our email, in a good way.

Several of you-- Kram, Brendan, and Zenten-- mentioned type theory and even homotopy type theory.

That could be a really fun topic to explore and is definitely one I'd like to learn more about.

There were actually a lot of topics that sit at the intersection of mathematics and computer science.

Morton suggested, P equals NP.

And Daniel suggested.

Kolmogorov complexity.

Yuval and several others suggested topics in the foundations of mathematics.

I love this topic.

It has math, philosophy, and human drama.

It's definitely something we'd like to tackle in a few episodes.

There were a ton of other great ideas-- way too many for us to mention.

If you encounter other things that you'd like an Infinite Series perspective on, email us at pbsinfiniteseries@gmail.com.. [MUSIC PLAYING]

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