We all say we hate it but we all do it

 kw: book reviews, nonfiction, science, mathematics, geometry

…No, my subject is not Sin (but that would be equally true), but Geometry. But before we go on, let me tell a story.

My brother and two close friends took several courses in college together, including a math class that emphasized proofs. I'll conceal identities here, and just represent my brother as Art, and his friends as Bob and Cal. They all did pretty well in their "math proofs" class. Art studied diligently and did well, while Bob struggled mightily to keep a "B" grade, but Cal did the best with the least work. Another fellow student told them one day, "Bob walked into a room and saw a big machine with a large gear on one end. He was told he had to make it run. He looked it over, then put his shoulder against the gear and heaved a great heave, making the gear turn. Art came in next. He nosed around and found a crank that fitted into the gear's shaft. He put it in, and turned the gear. Then Cal came in. He found a cord with a plug, plugged it in, pushed a button, and the machine began to run." When it comes to mathematics courses that require proofs (algebra) or demonstrations (geometry), I am definitely in Bob's league, at best.

Now, when I see a clear geometric demonstration, I can often comprehend it almost instantly. But I could never have produced that demonstration.

These two fellows (one an Arab, one a European), shown in a 15th Century drawing, are having a go at some demonstrations. The Westerner is trying his hand at squaring the circle (which is known to be impossible), while the Arab is, more practically, extending a demonstration of the Pythagorean Theorem.

In a memoir, we read that Abraham Lincoln said of himself that he "nearly mastered the six books of Euclid." In Shape, The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, by Jordan Ellenberg, we find that Honest Abe struggled for months to square a circle. Of course, he failed, and apparently he never came across a proof or demonstration that doing so is impossible.

All the demonstrations and constructions in Euclid's work must be done with compass and straightedge. The ancient compass was so constructed that once you set the two points you could scribe a circle about one of them, but the two legs collapsed when lifted from the paper; you cannot "set" that kind of compass. Further, the straightedge must have no markings on it; its only use is to draw straight lines through points already marked on the page. It so happens that if you are allowed to make a single mark on the straight edge, effectively turning it into a ruler, you can construct an extended radius, such as what we see in "b" in the illustration above, that has the required length of one side of the square. Otherwise, "No markee, no squaree." This point is not mentioned in Shape, but I suspect that Dr. Ellenberg knows it.

If Euclid were to drop into the office of any modern professor of geometry, he would recognize nearly nothing, except the (probably translated) books on a shelf, nearly out of sight, that he wrote about 2,300 years ago. Much of the "geometry" carried on these days is topology, which isn't about circles, squares or triangles, but about holes. Yes, holes. To a topologist, anything with no holes but with a defined edge is a "circle", including things we'd call squares or triangles. But if you punch a hole in it it's now something else. Most topological work is done in three (or more) dimensions. You may be familiar with the statement, "A topologist has trouble telling the difference between his cup of coffee and the donut he wants to dip into it." Both a cup with a handle, and a donut, are solid shapes with a single hole.

A fun discussion in Chapter 2 involves the title, "How Many Holes Does a Straw Have?" I asked my wife. She said, "One." That is a proper topological answer. If you shorten the straw, and make its wall thicker, it becomes a donut. Distort it some more, also pressing out a cavity (but not a hole!) in one section, and you get a coffee cup. This is easier to do with clay than with paper! But just for fun, the author shows how people defend the answer "Two", because many people would say, "It has a hole in each end"; and even the answer "None", because some would say you started with a flat sheet of paper and rolled it up.

I have to say, I was puzzled that the word "cavity" never appeared in that chapter. Topologically, a cave (source of the word "cavity") has no holes if it has no "other end". In topology, you only have a hole if you can go into one side (or end) and come out the other. So the animal known as Hydra has a cavity, but no holes, while most animals have a single hole called the Alimentary Canal, with a mouth at one end and an anus at the other. So we are donuts. Very lengthy donuts.

Well, that's not where this book is ultimately going. The earlier chapters help us get used to some geometrical ideas, and we soon get to maps. Here are two maps, and they are related:

These maps are the same (but for drawing idiosyncrasies) as on p. 394 of the book. The author calls the second (green) one a "chart", but it is also a map. It is the "first inversion" of the blue map; the green lines represent the relationships between the blue-outlined areas.

Have you ever played Nim? You begin by stacking an arbitrary number of coins in two or more piles. In one version, each player can take either one, two, or three coins, all from one pile. Other versions exist with different "taking" rules. Players alternate taking coins until one coin is left. The player who must then take that last coin loses. The author shows a simple proof that the first player will always lose if the other player makes no mistakes.

These two maps (or "map" and "chart") illustrate something about electoral districts in a state. One method for detecting Gerrymandering (where it isn't obvious, and I'll explain more anon) involves playing Nim with the line segments in the green map until a player can't remove any segment without breaking the map into two pieces. One such game ends after four moves, to look like this:

The green map from before is now a Tree, a connected map with branching but no loops and no holes.

I won't take this further here, except to say this Nim game brings about several possible ways to break up a district made of smaller units into two districts. It is one step in the process of making an electoral map having districts that are "more fair", but then we get into a discussion of what "fair" means.

For example, "proportional representation" is often talked about. To get away from R vs D or L vs R (or X vs Y, which could be sexist), I'll refer to the two "interested" parties as H and O (Lionel fans, take note). Consider a state that has ten districts, and having 60% H voters and 40% O voters. Assume they are all pretty evenly spread throughout the state. How would you draw district boundaries to "ensure" that 6 H and 4 O district representatives will be elected? If you could, would that be "fair"? That would simply guarantee that the H's in the legislature would always win every vote, unless the O's could sometimes convince a couple of H's to vote their way. So, effectively, the O citizens in the state would be without representation.

Gerrymandering, so-called because a proposed district map drawn by one Eldridge Gerry included a district shaped like a lizard, is thought of as unfair mapmaking designed to ensure that a certain party will always win. When you know where the voters live, and voter registration is how you know, you can wiggle the boundaries around to get most opponents into the smallest number of districts, and create many more districts that will just barely elect your own members. The illustration below was cropped from an article titled "The most Gerrymandered districts in America":

Even though the author's passion clearly lies in "dealing with Gerrymandering", he acknowledges that while it is often visible, it is dramatically hard to quantify. The more so because we cannot yet clearly define what "fair" means.

I have an idea: a Federal law that requires one election in four to be automatically reversed the day after the poll results are revealed. To reduce the amount of "gaming the system" that would be indulged in, a pair of fair coins would be flipped on the day in question. This ceremony would be conducted in public, with much publicity. If both coins come up Heads, all elections are reversed. Otherwise, their results stand. Of course, that means that sometimes two Reversal Years might occur in a row, and perhaps even three. It would also happen that five, ten, or more years may pass with no Reversal Years. That's OK. The added uncertainty might not make for better legislating, but it would definitely make it more interesting! I have about as much confidence that such a procedure could become law as I have of the Sun setting in the East tomorrow evening.

Now I want to back up to the idea of the Tree. The author has an interesting statement about trees and related graphs in a footnote on p. 106: "…there's a more general notion than a tree, called a directed acyclic graph…a DAG is like a tree where some branches are allowed to fuse together… Think of a particularly aristocratic family where your parents may share a great-grandparent or two." This isn't as rare as he thinks. Inbreeding happens whenever the "breeding pool" gets too small. 

For example, I have an ancestor, a Quaker, whose parents left Nantucket in the fifth generation after its settlement by ten families. Four generations back, her father could have been descended from eight of the ten families if no marriages between cousins or second cousins occurred, as could her mother. However, her father is descended from only seven, and her mother is descended from six. Between the two, going back to the settlers' generation, the Starbuck couple appears three times, the Coffin couple appears three times, and the Garner couple appears twice. And the couple themselves were second cousins (or a little closer than that, considering). First-cousin marriage is legal in seven U.S. states, and second-cousin marriage is legal in all. First-cousin-once-removed marriage is legal in 42 states. Thus, many family "trees" are really "family directed acyclic graphs". Fortunately the software at sites such as Ancestry.com is written to accommodate relationships of all kinds, perhaps even including the one described in the song "I'm My Own Grandpa."

Also, realistically speaking, when you go back more than a dozen generations or so, you'll find all kinds of links between relatives. Anyone living today who is descended from Charlemagne (crowned in the year 800), is a 38th or 39th or 40th generation descendant. Take a "tree" back 38 generations, and there are theoretically almost 275 billion ancestors. But the population of Europe in 800 AD was around 25-30 million. Think that over…

When I started the book, I had no idea it would go in these directions. It is too much fun to think about all these things. This is an author I'll keep in a tickler file.