We all do it but we think we are bad at it

 kw: book reviews, nonfiction, mathematics, geometry

The book is Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, by Jordan Ellenberg, a professor of mathematics who happens to write very well, who has also advised in the production of several films. Thinking about what kind of image to use to illustrate this review, I tried a few prompts with DALL-E3, and settled on this one:

The old geometer in his workroom, surrounded by the tools of his craft, clip art

After several trials, this image in dark sepia appealed to me the most.

The term Geometry means "measuring the Earth", but in practical terms it means to measure earthly things. That's something we all do frequently. For example, my wife and I are in the process of getting a bathroom remodeled, because ancient pipes have given out and the walls and floor need to be destroyed to get at them. We need to buy tile and supplies for the new floor, and a different kind of tile (or other covering) for the walls. What do we do? We take measurements and calculate areas. That's a geometrical task.

When we were in seventh and eighth grade, most of us had Geometry one year and Algebra the next. However, I suspect few of us had an Algebra teacher who showed us that geometry and algebra tackle the same problems from different directions; they are equivalent. What made geometry class so hard for me was the proofs. The Proofs! One needs a certain kind of mind to formulate a Proof. I don't have it. With a lot of painful study, I can figure out all the steps in a proof I have been shown, but I could never have produced it myself, and I soon forget it.

This book is not about proofs (but a few simple ones are shown, perhaps as a kind of inoculation: "See, that isn't so bad, is it?" To which I answer, "Yup, it's bad."). No, it is about geometrical thinking. For example, about trees. Biologically, a tree is a branching structure with a root, trunk, branches, twigs, and leaves. Conceptually, a biological tree is a metaphor for a way of arranging information, such as a family tree, a categorization of machine tools, or the districts and blocks and buildings in a neighborhood. A tree has one specific, critical characteristic: branches don't join back to branches. Otherwise, you have a network.

In a chapter on genealogy (#8, You are your own negative-first cousin, and other maps) the author states that your parents "don't share a known ancestor (unless you are from a truly aristocratic clan)." Not so fast, Dr. Ellenberg. Generally, we consider that a "family tree" is a genuine tree. With you as the root, your parents as the first two branches, your grandparents as the next, and so forth, the tree branches and branches but branches don't re-combine. However, sooner or later, they have to, as author notes in passing. If we look deeper into the situation we can go one better than that. 

In most jurisdictions, it is legal for first cousins to marry. In some, only certain first cousins can marry: they can have a common grandparent as long as the siblings in their parents' generation are of opposite sex. Thus, my mother had a sister and a brother. It would not be legal to marry the daughter of my mother's sister, but is quite legal to marry the daughter of my mother's brother. Such "available" cousins are called "kissing cousins".

Among my ancestors is a certain Joseph Macy, born on Nantucket Island in 1765, son of Joseph Macy and Mary Starbuck. Among his great-great grandparents—there are 16 of them—the surname Coffin appears three times and the name Starbuck appears twice. So he was his own third cousin three different ways. There were ten families that settled Nantucket in the 1600's. Within just a couple of generations, marriages between second or third cousins became necessary, and marriages between first cousins were getting common. So the genealogy of the Nantucket settlers is a bit of a treelike network. There are other interesting cross-links elsewhere in my family "tree", but I'll leave them for a later discussion. I'll just leave you with this thought. Joseph Macy was descended from Charlemagne; and so, of course, am I. The relationship is distant: 39 generations. That many generations ago, I had, formally speaking, 549,755,813,764 ancestors. It's about 550 billion. That implies a lot of cross-linkages in everyone's family tree, once you get back one or two dozen generations. As it happens, according to the data I have at present, I have five ancestors descended from Charlemagne.

Dr. Ellenberg likes long chapter titles. One of the shorter titles is "His style was invincibility" (Ch 5), in which we are introduced to an unbeatable Checkers player named Marion Franklin Tinsley. His story is the backdrop to investigating combinatorial math, using games. Tic Tac Toe is a simple game with a simple strategy, and is a "known game" because of its simplicity. There are 765 possible positions (the "state space"), and about 27,000 possible games, or ways to pass through a series of states in the state space. If two players play perfectly, every game is a draw. What about Checkers? The state space of Checkers is about 500 billion billion, or a 5 followed by 20 other digits. With a state space this large, nobody can learn the whole of it, and neither can any computer so far produced. Just for the record, the state space of the positions in Chess is a 2 followed by 46 other digits. This means that it is possible for a human player to still win either Chess or Checkers against a computer program, but it is very hard, because there are various heuristics (rules of thumb based on experience) that measure the strength of one position relative to another, and a machine can check millions of possibilities while a human player is thinking over five or ten. Then there is Go, for which the number of legal board positions is a number with more than 170 digits; this is an estimate! Considering that the number of atoms in the Universe is a 100-digit number, you would need billions of billions of billions (string out four more "billions") of Universes to contain a computer memory big enough to encompass the state space of Go.

But the process of analyzing any of these games is to build a tree! The starting position of the game is the root of the tree. In a game of Checkers, the number of possible starting moves by the first player is 14. Similarly for the initial moves of the second player, so the two-move state space is 14x14 = 196. These moves vary in how "strong" they are, how likely to lead to a Win. If you have a way to evaluate the strength of each move, you can label the nodes in the tree accordingly.

With each move the state space for that move is another factor that's usually between 12 and 16, until pieces start getting captured (which reduces the branching), and it can exceed 16 when some pieces get longer multi-jump options. It doesn't take long for the number of twigs on this tree to become millions, billions, etc.

The last chapter has the title "How math broke democracy (and might still save it)". It is about Gerrymandering, the practice of setting up voting districts in a state or county so as to favor one political party. This graphic from Statista shows examples from 2020. Redistricting, and thus Gerrymandering, follows each census; it's in the Constitution (the redistricting part!). 

There are many definitions of "fair", and none of them is "fair enough" to satisfy everyone involved. There are many proposals for rules or laws to produce a more "fair" outcome, but again, every such proposal generates lots of heat but hardly any light. 

In 2019 the Supreme Court punted on a possible method to ameliorate the problem. The author was involved, as a co-creator of an Amicus brief. Gerrymandering cannot be eliminated, not so long as people have emotions and a lust for power. The chapter does not end with a conclusion, but with a few tantalizing possibilities. None of them is likely to receive the support of a large enough majority to become embalmed in law. It is clear from the passionate tone of the chapter that this matter is of great importance to the author. Here, Geometry is Power.

Geometry is all around us. We all do it. We all think we are bad at it. It takes a peculiar kind of mind to revel in the proofing process, but just as very few people design the cars that millions of us drive, or the phones we consult so compulsively, we know what we need to know, and as this book shows, we know more than we think we do.