Thread by @RadishHarmers, Grains of pollen from various plants as viewed under a scanning electron [...]

Grains of pollen from various plants as viewed under a scanning electron microscope. Observe, when we zoom in on one in particular (morning glory pollen), that it appears to be made up of six-sided faces, with three of them meeting at each spike.

But you can't actually make a ball using only six-sided faces. It's logically impossible (the reason why will be explained below). Indeed, if you look more carefully, you'll see that there are some five-sided faces visible as well.

Can you make a ball using only six-sided faces (hexagons) and five-sided faces (pentagons)? Sure. There's one very familiar way of doing so. It's sometimes called a "truncated icosahedron", but it's more familiar as a soccer ball.

On a soccer ball, we have 20 hexagons and 12 pentagons. On our morning glory pollen grain, we clearly have many more faces. Indeed, 110 hexagons and 12 pentagons.

There's another famous way of making a ball using only hexagons and pentagons; perhaps the most famous. Actually, it doesn't use any hexagons. Just 12 pentagons. It's called the dodecahedron, "dodeca" for those 12 faces. Pyrite crystals often take this form (albeit squashed).

And here's another way to make a ball out of hexagons and pentagons. A hexagon on top with six pentagons next to it, and then the same pattern again in mirror image on the bottom. So 2 hexagons and 12 pentagons.

We've now seen four different ways of making a ball using only hexagons and pentagons, three meeting at each corner. The number of hexagons could vary quite a bit, but the number of pentagons was always 12. Coincidence? No, clearly not, or else I wouldn't have made this thread.

There are infinitely many ways to make a ball using only hexagons and pentagons, three meeting at each corner, but any way you do it, you will have to use exactly 12 pentagons.

[We can also consider what happens with more than three faces meeting at some corners. And we can consider other kinds of faces. All in due time. But just three faces meeting at each corner has a simplicity that causes it to come up in nature often.]

So, why 12 pentagons? The only reason this story is any good is to actually understand the why, but I'll let you whet your appetite thinking about it for yourself for a bit before explaining it.

(Not that any of you care; this thread is write-only, I'm sure.)

In the meanwhile, enjoy this diagram of an adenovirus. 240 hexagons, 12 pentagons.

I wish it were adenoviruses destroying the Earth, and not boring coronaviruses.

One dilemma with math is that often one can give a very quick explanation to a particular question, vs. a slightly slower explanation for the one particular question, that builds up more machinery that makes further questions much easier to understand. I always prefer the latter.

However, in this case, given the frustration of writing on Twitter with its character count restrictions, lack of editing, etc, I will, with a sigh, outline a quick, lazy explanation first, for those whose patience is limited. I'll give a better discussion later.

The explanation comes in two parts.

First, let's introduce another mystery:

There's a certain formula that always holds whenever you make a ball out of faces (or tiles, or regions, or whatever you want to call them).

Specifically, the number of vertices (aka, corners, points, …) - the number of edges + the number of faces will always equal 2.

Don't worry, I will explain why this V - E + F = 2 (called "Euler's formula", though a dozen other things are also called that) holds later in this thread.

This will be the most enlightening part of the story, that generalizes in many directions to other wonderful math.

But first go ahead, see for yourself that it works this way, thinking about cubes and pyramids and all the example shapes above and whatever else you can think about. Vertices - edges + faces always equals 2, on anything ball-like.

It's worth my emphasizing, nothing in this depends on the angles or sizes or even straightness of the faces. Make them all kinds of squashed and squiggly, but this formula will still hold. It's purely a matter of considering the structure of how the faces connect to each other.

Once we have this formula, it's not hard to see why we need twelve 5-sided faces, if we're going to tile a ball with 6- and 5-sided faces, three meeting at each vertex. Indeed, many readers who were already familiar with the above formula saw it for themselves.

Each hexagon has 6 vertices and 6 edges. Each pentagon has 5 vertices and 5 edges. Each edge touches 2 of these, and we're also supposing for now that each vertex touches 3 of these.

This means, if we have H hexagons and 5 pentagons, the number of vertices is (6H + 5P)/3, the number of edges is (6H + 5P)/2, and the number of faces is just H + P. So our formula tells us (6H + 5P)/3 - (6H + 5P)/2 + H + P = 2.

All the H terms cancel out, and we're left with P/6 = 2, which is to say, P = 12. That's it. We need 12 pentagons.

The fact that the H terms cancel out is why we can build large sheets of as many hexagons as we want, without it making any difference to anything else.

But why does the formula V - E + F = 2 have to hold to begin with?

This is the jumping off point for a million things in math, and want to give a fuller, slower explanation of, that sets up some of those million other things.

But for now, the quick and dirty explanation is this.

Imagine you have a structure with V many marked vertices, and E many edges between vertices, and F many faces between loops of edges, filling up the surface of a ball.

When you see an edge between two different faces, you can erase it. This gets rid of an edge, and also turns two faces into one. So E goes down by 1 and F goes down by 1, but since they have opposite sign in our formula, this keeps V - E + F the same. It's unaffected.

Similarly, if you see a vertex connecting exactly two edges and nothing else, you can erase the vertex, and merge the two edges into one, bringing V and E each down by one, without affecting V - E + F.

If you keep merging faces by erasing edges, and merging edges by erasing useless vertices, eventually, you run out of things you can merge or erase in this way.

You'll reach a point where you've gotten down to just one vertex, and one looped edge from this vertex back to itself, and then two faces separated by that loop.

And if, in the throes of madness, you then knock out that edge, the result is one vertex, no edges, and just one big face comprising the whole ball. Well, now, there's really nothing left to do.

But also, now V - E + F = 2 clearly, and since it never changes, it must've been 2 to begin with.

That's it. That's the quick and dirty explanation, which I've lazily provided without pictures.

There are much better ways to think about Euler's formula, and there is much more to say about other math it connects to, but I will have to save that for later.

That's it for now! Enjoy the virus pictures! https://twitter.com/RadishHarmers/status/1238523013605273600

https://twitter.com/RadishHarmers/status/1238523013605273600

Ugh, there are a few typos above, in early-up posts. Madness to fix. This is not a well-designed medium.

Also, if anyone missed it, enjoy the pictures of dodecahedral fool's gold, wint's favorite. https://twitter.com/RadishHarmers/status/1238616853699846147

https://twitter.com/RadishHarmers/status/1238616853699846147

Incidentally, the same reasoning says that instead of 12/(6 - 5) = 12 pentagons, we could've used 12/(6 - 4) = 6 four-sided faces, or 12/(6 - 3) = 4 triangles. Or even 12/(6 - 2) = 3 two-sided faces! So why is it pentagons that so many viruses settled on using?

Of course, many things in nature do come in cubes, or use triangles, etc. But basically no viruses make capsids in this way. It's been theorized that this has to do with the superior efficiency of pentagonal faces in absorbing compression stress, vs. fewer-sided faces.

I don't know if that's correct or not. I wasn't there when viruses evolved. For that matter, I don't know why tetrapods settled on five fingers. I'm just some mathematician, there's a lot of things I don't know.

For some people, it might be easier to think of the Vertices - Edges + Faces = 2 reasoning in reverse from how I presented it above. Suppose you want to draw a particular diagram of faces, edges, and vertices onto a ball. Start by drawing just one of the vertices onto the ball.

At this moment, we have 1 vertex, 0 edges, and 1 face, so V - E + F = 2.

Then, on each turn, you're allowed to either draw a new edge between two vertices you've already drawn, or draw a new edge between one vertex you've already drawn and one new vertex.

If you draw a new edge between two vertices you've already drawn, you increase E by one, and since this edge splits one existing face into two, you also increase F by one. This keeps V - E + F the same.

If you draw a new edge between one vertex you've already drawn and one new vertex, you increase V and E by one each, keeping V - E + F the same.

Eventually, you'll draw the entire diagram you intend to, since everything in it is eventually reachable from the starting vertex. And V - E + F = 2 the whole time.

I've just learnt about another example of this kind of structure in nature, the carboxysome found in some bacteria.

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