Book: Shape

Subtitled "The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything else," this 2021 book by Jordan Ellenberg shows how math, and in particular geometry, underlies just about everything.  

There is statistics of course, and recursive growth patterns, and information theory, and lots of other applications which are obvious with a moment's reflection.  But the author, being the good teacher he is, moves the discussion into surprising and non-obvious realms using amusing and absurd examples.  He uses humor to defuse the dry mathematical precepts, using memorable and hilarious examples.

For instance, how many holes are there in a straw?  Is it one hole that goes all the way through, or a different hole at each end? The author states that you can put the straw into a milkshake and suck, and the milkshake that goes in the bottom hole comes out the top hole, so it's obviously just one hole.  Or rather, a "plus 1 hole" and a "minus 1 hole."  What if you folded the straw in half? The middle would be blocked by the crimp, so now you'd have two "plus 1" holes. If you opened up the crimp by carefully trimming the plastic on the top of the crimp, you'd end up with two shorter straws joined at the top.  You'd have two "plus 1" holes and one "minus 1."  This topology, by the way, is exactly the same as a pair of pants (he points out) so you could drink a milkshake with a pair or pants. But wait, if you slice a straw lengthwise with a knife and lay it out on the tabletop, you end up with a long rectangle of plastic.  Where did the hole go? You could say the hole was formed by the void in the original straw. If you shorten the straw to just a ribbon, you get a bagel.  Bagels have only one hole ... and you can't drink a milkshake through one. Even though, mathematically, they're the same topology.

It goes on like this, but better constructed than I have presented here.  Amusing and informative. And memorable.

The longest chess game ever played had 269 moves and took over 20 hours.  The theoretical maximum for a legal game is 5,898 moves -- only possible if neither player is trying to win.  The rules of grandmaster chess state that if fifty moves go by without anyone capturing any pieces, the game is declared a draw. This prevents an oscillating game where the same moves are repeated over and over to avoid losing.

The number of possible moves in a chess game has been calculated to be something like one with 120 zeroes after it. Nobody has bothered to calculate it exactly.

The author saw this book in an airport bookstore and noted the math illiteracy it displayed.

Pi to 40 digits is enough accuracy to define the radius of the Milky Way to within the width of a proton.

Ellenberg is a mathematician first, an author second, so he keeps diving down rabbit holes about weird little properties of certain numbers & constants, and different ways to graph numbers, and "eigensequences" and other such ephemera of interest only to mathematicians who masturbate to numbers all day.

It all reminds me of divinity students, who spend years debating the number of angels that can dance on the head of a pin.

Jordan Ellenberg wrote:

Perhaps the simplest way of diluting the inequality of the Electoral Colllege would be to increase the size of the House of Representatives. There were 435 representatives in 1912 and here are 435 representatives today, in a country more than three times as large.

I did not realize this. Apparently the Reps each represent three times as many people now?

Chapter 14, about gerrymandering, is fascinating.  

We all know that gerrymandering, the creation of voting districts that favor one party or the other, is a bad thing.  But how do you avoid it, what's the alternative?

You can't divide counties by any kind of regular mathematical grid, because people don't live in equal-population grids (and the one "given" is that voting districts should have roughly equal numbers of voters).  Liberal Democrats tend to congregate in cities, and conservative Republicans in rural areas -- but creating 100% Republican or 100% Democratic districts wouldn't be very democratic.

There is an art, and increasingly a science, to creating districts that just barely favor one party.  You need 51% or 50.5% of the favored party.  If your district is 100% one party, you're essentially "wasting" 49.5% of those voters, that could be apportioned elsewhere to ameliorate the opposing party.

All states except Nevada require that legislating districts be contiguous.  But they don't have to be geometric shapes.

Physical barriers like rivers and mountains and highways often define district boundaries, as they do population boundaries. City limits form another boundary, though more porous.

Mapmakers also have to try to project ten years of growth, since their districts are only redrawn after the census. Where will the suburbs grow, where will farmland become bedroom communities? Will districts change makeup in the next decade?

The party in power when the maps are drawn try to favor their own party, but in most cases it's nearly impossible to define what a "neutral" map would look like. There are laws on the book to prevent districting to exclude voters of a particular race, but there is no law that says you can't favor one party over the other, or one economic class over another.  In fact, it's arguable that a coherent voting district, where the representative works to solve endemic problems of that district, would ideally have the least amount of diversity in that district. Diversity actually works against a coherent district.

Fascinating dilemmas.

In the final chapter Ellenberg speculates on the near future, when AI can be used to prove or disprove theorems which are today beyond mathematicians' reach.

If the proofs are too thorny, too elaborate, too convoluted for the human mind to grasp, he says, the "proofs" would be useless. Mathematical proofs are only useful when they open up new understandings, new areas of research, new insights into how he universe works. Already mathematicians are well beyond what normal mortal human beings can comprehend, with their 28-dimensional strings and convex space. If, nay when machines supersede human comprehension, their "insights" would not have the same earth-shattering results of Einstein or Poincaré or Euclid.

Math is about to exit the realm of human understanding.