Shape by Jordan Ellenberg

Whew, what a book. About 100 or so pages too long, but what a book. Ellenberg is a Professor of Mathematics at the University of Wisconsin-Madison, he is a heck of a writer, and he is ALL … ABOUT … GEOMETRY! Like, seriously all about geometry: theorems, charts, graphs, logic, reasoning, equations, letters, numbers … ALL the jazz. I wrote above that the book was about 100 pages too long (clocks in at a cool 422) because he spends a TON of those pages analyzing a shit-ton of charts, graphs, theorems, logic, and reasoning, but he is able to make the leap from that analysis to everyday life, especially when he writes about COVID-19 statistical modeling and gerrymandering, both of which warrant two of the longer chapters towards the end of the book. I’ll be honest: I skimmed a lot of this because it was so heavy on the logic and reasoning part, but I’m glad that I powered through until the end because the gerrymandering section is absolutely relevant to the present day, as I just listened to a podcast about how Democrats in New York were potentially thinking about redrawing the maps to their benefit — even after denouncing the same practice in southern GOP states such as North Carolina. None of the this is surprising, obviously, because those that are in power will do whatever they need to do to stay there, but Ellenberg makes a clear and compelling case that we simply need to use the computerized tools to our advantage by eliminating large outliers in legislative map drawing, and leaving the rest to (hopefully) non partisan bodies that can continue to draw these maps by the imperfect human hand. Ellenberg still believes in logic and reasoning to understand how things work and how we can do them better — even if our political and, oftentimes, daily discourse with one another doesn’t allow it. But geometry can show us the way, because, as Ellenberg states: “It is the way we measure the world … and the way we measure ourselves.”

  • What Lincoln took from Euclid was the idea that if you were careful, you could erect a tall, rock solid building of belief and agreement on rigorous deductive steps, story by story, on a foundation of axioms no one could doubt
  • truths one holds to be self-evident
  • Gettysburg: proposition that all men are created equal
  • proposition is a term Euclid uses for a fact that follows logically from the self-evident axioms, one you simply cannot rationally deny
  • Euclid collected the information from Greek math at the time in his book Elements — triangles, shapes, lines, angles, circles
  • ultimate reason for teaching kids to write a proof is no that the world is full of proofs, but that the world is full of non-proofs, and grown ups need to know the difference
  • one thing we can learn from the history of geometry — you shouldn’t admit a new axiom into your book until it really proves its worth
  • what made Lincoln special was that it was morally impossible for him to argue dishonestly
  • a proof is an incomprehensible demonstration of a fact that you already knew
  • we teach geometry because it’s a formal system that’s not just a formal system
  • it’s built into the way we think about space, location, and motion
  • quite unlikely that 200 poll respondents will differ substantially from Wisconsonites as a whole
  • assuming there is no bias lurking in our choice of whom to poll
  • in real life, we never know a poll is perfectly unbiased
  • 2018 paper found that actual election results typically deviated from polls about twice as much as the margin of error would suggest
  • anti-averages: the more heads you get in a row, the more you should start to expect heads in the future
  • Nate Silver saw things differently — law of anti averages in action, it means a Trump sweep of the swing states was more likely than you’d expect from the individual numbers
  • Silver’s assessment was that the race was very close and that either candidate could win
  • old axiom: open country is the most probably place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point — “random walk”
  • constant up and down wandering of a stock price looks like events are driving it, but it might as well be as random as the mosquito’s endless flitting
  • park your money in an index fund and forget about it
  • statistics of human behavior and action, like frequencies of various crimes and age of first marriages, tend to settle down to fixed averages too, as if people in the aggregate were just a bunch of mindless coins
  • Markov chain — order in which the variables appear matters a lot
  • if you want to know where the mosquito is likely to be tomorrow, it doesn’t matter where it was yesterday or the day before that, only where it is today
  • each variable is independent of all the earlier ones, conditionally on the most recent value
  • real math is a lot of trial and error, method that gets looked down on a lot, probably because it has the word “error” in it
  • in math we’re not afraid of errors, errors are good
  • an error is just another opportunity to run another trial
  • gradient descent: form of trial and error, you try a bunch of possible moves and pick the one that helps you out the most
  • gradient = how much does the height changes when you take one tiny step in that direction
  • it’s an algorithm, an explicit rule that tells you what to do in any situation you might encounter
  • consider all he moves you can make, figure out which one offers you the biggest gradient, and do that. repeat
  • ineffective strategies can be wrong in a lot of situations you’ve already encountered
  • or, the strategy is so precisely tailored to situations you’ve already encountered that it’s useless for new situations
  • underfitting (two notes up) — you haven’t used your experience enough when forming your strategy
  • overfitting — we’ve relied on our experience too much
  • linear regression — strategy to predict one variable given the value of another
  • single principle underlying all mathematical predictions: what will happen today will happen tomorrow (geometric progression)
  • viruses, however, don’t spread due to geometric progression because the mechanics of the spread demand that the ratio between last week’s infections and this week’s infections is the same as that between this week’s infections and next week’s infections
  • geometric progression’s are interpreted by our minds as slow, steady, manageable growth followed up an abrupt and terrifying steepness
  • this week is like last week but twice as bad
  • magic number needs to drop (Ro) below 1, which means there are fewer and fewer new cases each week
  • you don’t have to stop all transmission, you just have to stop enough transmission
  • for COVID, we don’t need to eliminate every transmission of the disease, which is impossible; we just need to control, which is not perfectionism
  • natural Ro of 2 will start to fade out once half the population is infected — “herd immunity”
  • once enough people are impervious to a disease, an epidemic can’t sustain itself
  • how much “enough” depends on the original Ro
  • if we’re right that COVID has an Ro, between 2 and 3, then the current epidemic will start burning itself out on its own once half to 2/3 of the world has come down with it — that’s a lot
  • threshold might be lower, though probably not radically lower, for reasons having to do with heterogeneity — not everyone infects the same number of people
  • white people with COVID are more likely to die because old people with COVID are more likely to die, and white people, in the aggregate, are old
  • Simpson’s paradox: insist that we keep both the parts and the whole in mind at once
  • polymerase chain test that detects COVID multiplies even a tiny trace of viral RNA by a huge factor
  • that makes group testing feasible, and in situations where prevalence of the disease is very low and testers and equipment are in short supply, appealing
  • difference equation — tells us exactly the difference between the situation tomorrow and the situation today
  • models of disease can’t predict the future because they can’t predict what we’ll do
  • but they can help us decide what to do, and then we need to do it
  • Rinderpest is a disease of cattle, but eradicated in 2011 (only virus ever eradicated, with Smallpox)
  • COVID curve has been steadfastly asymmetric, rocketing in each region to its high point and then declining with a painful slowness
  • comes up the escalator and goes down the stairs
  • projection going to keep changing its tune as it encounters new data that conflicts with the predictions it’s committed to
  • if the range of laws we’re willing to consider is too narrow, we are going to jitter and glitch as we attempt to match our too rigid rule to reality — underfitting
  • Coronavirus Models Aren’t Supposed to Be Right
  • better goal is to make a much broader and qualitative assessment — is the pandemic spiraling out of control right now, growing but flattening, or dying out?
  • give us good advice about which of the many paths before us is most likely to have victory at the end
  • in an epidemic, you make a model based on certain facts about who’s transmitting to whom and when, and then those facts suddenly change, by mass human action or by government decree
  • real modeling is always a dance between predicable dynamics and our unpredictable responses
  • all models are wrong — but some are useful
  • reverse engineering — start with the facts known about the spread of disease, and from there reason your way to the differential equations that the curve of the epidemic must necessarily satisfy
  • “curve fitting” mode of prediction is to look for regularities in the past and guess that those patterns will persist to the future, without troubling yourself too much as to why
  • what happened yesterday will happen today
  • google has seen billions of sentences, enough to obverse statistical regularities concerning which
  • combinations of words are likely to be meaningful sentences and which are not — which are most likely
  • number of infections in each state always trend towards a geometric progression, at least at first, because the models rely on the early part of the epidemic, where the virus hasn’t started running out of susceptible hosts
  • eigenvalue — captures something deep and global about a system’s behavior
  • computing an eigenvalue of a matrix — the latent number that governs the growth of the many part system described by the box of numbers — has come to be seen as one of the fundamental calculations
  • most mathematicians calculate eigenvalues daily
  • gives you a much more refined picture of a pandemic’s progress and its expected future than the basic models
  • stoastically, which means that you’re not just assigning each individual their own precise Ro, but a random variable
  • greater the extent to which infection is dominated by super spread, the more dumb luck there is in which populations suffer and which are spared
  • Google was eigenvalues
  • agent based model — keeping track of the monumental data of each individual’s interactions with everyone else
  • agent based models haven’t dominated COVID models because we don’t have anything like the fine grained data about individual contacts we’d need to make it work
  • Law of Long Walks: if a mosquito has a finite set of bogs it can land in, and if each bog has a fixed set of bogs it is connected to, and if the mosquito, at each moment, chooses a destination at random from the bogs it can reach from its current bog, then each bog has a limiting probability
  • Google — limiting probability for a website gives you a score, which they called the Page Rank, which captures the true geometry of the internet as no one had before
  • non commutativity — doing one thing and then doing a different thing doesn’t always have the same result as doing the latter thing followed by the former (try to put on your shoes before your socks
  • geometry means measuring the earth in greek
  • to assign a number to any two points, which we are to interpret as the distance between them
  • pandemics travel both fast and slow, in whichever vehicle they can hitch a ride on
  • a straight line is NOT the shortest distance between two points
  • a line is not a distance!
  • we simply DEFINE a straight line to be the shortest path
  • Rubiks Cube has 43 quintillion positions, but you can get from any of them back to the original setting in just 20 moves
  • Milgram — Six Degrees of Separation
  • pick two users at random anywhere on the globe, average length of the shortest chain of Facebook friends between them is 4.57
  • world’s geometry is even smaller now than it was years ago, more connected, and more prone to exponential spread
  • Wisconsin — same electoral facts now translate into many more Republican seats than they did ten years ago
  • nothing suddenly changes in WI politics between 2010 and 2012 — maps changed
  • practice of drawing district lines to secure advantage for yourself or your co-partisans is gerrymandering
  • in WI the GOP enjoys a bigger legislative lower house majority than they do in more conservative states like Iowa or Kentucky
  • U.S. states are semiautonomous governments, each with it’s own particular interests
  • districts within states, on the other hand, are patches of land without much meaning
  • segments = districting
  • the way you cut up a state into districts has an enormous effect on who ends up in the statehouse making laws
  • representatives are often choosing voters
  • Reynolds v Sims — court threw out Alabama state legislature districts
  • boundaries that are constitutional when drawn become unconstitutional when the next census rolls in
  • 26 smallest states, whose 52 reps make up a majority of the Senate, speak for just 18% of the population
  • Electoral College and all those compromises were arrived at reluctantly and wearily, with no one coming up with a better solution
  • 435 reps in the house in 1912 and 435 reps in the house today, in a country more than 3X as large
  • bigger house would mean more representative house, and an electoral college that better represented the people voting, without changing a single jot of the founders plan
  • gerrymandering goes back well before Gerry, and the word
  • used to be an art, advanced computation has made it a science
  • gerrymander feeds itself
  • just as the maps can be tuned to produce substantial partisan advantage, they can be jiggered to reduce risk to incumbents at the same time
  • we are pretty predictable, getting more so (voting)
  • invariant — shouldn’t change when a region is moved, or rotated or enlarged, or shrunk
  • Population Polygon score of a district is the ratio between the number of people who live in the district and the number of people who live in its convex hull
  • fewer choices map drawers have, the less likely they are able to find options that are grievously rigged
  • we need a measure of a map to tell how gerrymandered they are
  • what makes a gerrymander work is that your party wins a lot of districts by a little and a few districts by a lot
  • efficiency gap — difference between the number of votes wasted by the two parties expressed as a percentage of total number of voters cast
  • objective mature, easy to calculate, reams of evidence show that it jumps in maps we know were gerrymandered, like WI
  • does have flaws, severe ones — discontinuous, claims are murkier, over rigid
  • a false figure can be corrected; a true one chosen to make the wrong impression is much harder poodle to nuzzle
  • proportional representation isn’t what happens when districts are neutrally drawn
  • when we ask whether a district map is fair, what we really want to ask is does this districting tend to produce outcomes similar to a map randomly selected from the set of all legally permissible maps?
  • elected officials and voters hate the idea of a computerized map; also impossible to do, just too many options
  • ensemble, set of maps generated at random by a computer
  • WI map, only in Dem leaning environments that the gerrymander really kicks in, acting as a firewall to maintain the Republican majority against prevailing popular sentiment
  • ReCom geometry — recombination
  • goal isn’t to enforce an impossible absolute neutrality, but to block the very worst offenses
  • in practice, if you start with the NC map that their legislature made, and mess with it, it gets less republican no matter what you do to it
  • experiment proves a strong indication, in any statistical sense, that the map was cooked
  • ensemble method is the best we have
  • proportional representation is a lousy criteria for fairness
  • Supreme court case — Allison Riggs tried to explain that her client was only asking the court to throw out the most outlandish gerrymanders, whose partisan performance sets them apart from all but a handful of neutral alternatives
  • states would then still have a lot of breathing room to choose from the other 99% of all maps with a free hand, taking whatever nonpartisan criteria they liked into account
  • Kagan: What’s not allowed is deviation from whatever the state would have come up with, absent these partisan considerations
  • 5–4 decision court ruled (2019) that it was outside the scope of the federal courts to decide whether a partisan gerrymander was constitutional or not — states can gerrymander with wanton abandon
  • majority ruled that gerrymandering was a political question, Supreme Court is forbidden to intervene
  • in geometry, you make your own knowledge, power is in your hands
  • represents an alternative source of authority
  • point is to understand things — we want not just the facts but the souls of the facts
  • measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics
  • understanding — here to teach us things, to make the house expand
  • everyone does geometry differently — everyone does it
  • the way we measure the world, and the way we measure ourselves

I love books, I have a ton of them, and I take notes on all of them. I wanted to share all that I have learned and will continue to learn. I hope you enjoy.