# Shape ### Quotes > A lot of math is figuring out what we can, temporarily or for all time, get away with not caring about > This is how geometry always works. We start from our intuitions about shapes in the physical world (where else could we start?), we analyze closely our sense of the way those shapes look and move, so precisely that we can talk about them without relying on our intuition if we need to. Because when we rise up from the shallow waters of three-dimensional space, *we will need to*. (pg. 48) > "Mathematics is the art of given the same name to different things" - Poincare, pg 53 > To talk about the geometry of space, whether that space is a vial of fluid, the space of market conditions, or a mosquito-ridden marsh, is to talk about how one *moves through it*. > Once we start thinking this way, we're making judgments about which strings of letters are "close to" which other strings of letters, which means we're thinking about a *geometry* of latter strings. > We can prove things about games, just as we can prove things about geometry, because games *are* geometry. > Geometric objects are interesting to a broad spread of humanity just insofar as they resonate with real things we encounter in our lives with some frequency. If the only triangular things in our universe were the little metal percussion instruments, we wouldn't care as much as we do about triangles. > The tree represents the geometry of hierarchy. (pg. 106) > We thought we were screwing around with spinning bracelets, but in fact we were using the geometry of the circle and its rotations to prove a fact about the prime numbers, which on the face of it you'd never have thought geometric at all. Geometry is hiding everywhere, deep in the gears of things. > Everywhere there is a notion of distance, there is a notion of geometry too, and an concomitant idea of a circle. (pg. 190) > There is almost no context so abstract that we can't invent a notion of distance, and with it a notion of geometry. (pg. 191) > Rene Descartes was the first to make really systematic the idea that points on the plane can be thought of as pairs of numbers, an x and y coordinate, allowing us to transform a *geometric object* like a circle (the set of points a fixed distance from a given center) into an algebraic one (say, the set of pairs (x,y) such that $x^2 + (y-5)^2 = 25$) > We are thinking about how distant they are from one another. We are taking geometry into account. (pg. 300) > To assign a patch of land, or a set of people, or a set of horses a *geometry* is, at bottom, to assign a number to any two points, which we interpret as the distance between them. And a fundamental insight of modern geometry is that there are many different ways to do this, and a different choice means a different geometry. (pg. 303) > So the geometry of card shuffling is big, but somehow, like a world with a lot of direct intercontinental flights, also small; there are a lot of different places, but it doesn't take you many steps to get from one position to any another (pg. 330). > Abstract geometries, like the geometry of shuffled cards, are typically really fast to explore, much faster than geometries drawn from physical space. (pg. 333) > TODO: go through a rumple in space card example and touch on connected components (see pg 167 in Algorithm Design Manual) ### TODO * pg. 48 mid way * chapter 3 (Symmetry) * pg. 77 (geometry of statistics, correlations) -> this is crucial for AI * pg 84 * pg 202 (interested in the property) * chapter 10 (All of it) --- Date: 20211006 Links to: Tags: References: * []()