## Visualizing the 2d World with Cartograms

Space and time are favorite subjects of mine, since they are the root concepts for two of the most fundamental types of questions we can ask, where and when questions. I discussed three dimensions in detail in a previous post, so I am going to dive into the subject of cartograms and show why you should be careful about your two-dimensional thinking as well. I’ll give you a question to stick behind your ear before I begin: how do tiny island nations like Britain and Japan manage to dramatically influence the world, while huge continents like Africa and South America often don’t even register on the radar? Let me warn you right now, that’s a trick question.

## The Third Dimension is Not Simple

Ever since Einstein got us thinking about the fourth dimension and string theorists got us worried about ten and eleven dimensions, we have not really given serious thought to the mundane old third dimension. Several things, ranging from the emerging three-dimensional Internet over at Second Life, to the delightful modern religion of Parkour and the Nintendo Wii controller, have made me think seriously about the third dimension in recent weeks. It isn’t just badly-developed characters in movies and books that are two dimensional — you and I are as well, in fundamental ways.

## Harry Potter and the Concept of Magic

The upcoming end of the Harry Potter series demands piggyback attention, especially from a new blog like mine. Since I have been talking lately about concepts and definitions using toy examples from geometry, I thought I’d take on a more complex concept: magic. In this first of a series of posts aimed squarely at piggybacking the Potter phenomenon, I’ll attempt a definition of the concept of magic that explains why we delight in imagined realities that depend on it.

## How to Define Concepts

Let us say you are the sort of thoughtful (or idle) person who occasionally wonders about the meaning of everyday concepts. So there you are, at the fair, laughing at yourself in a concave mirror, when suddenly it hits you. You don’t really know what “concave” means. You just recall vague ideas of concave and convex lenses and mirrors from high school and using the term in general conversation to describe certain shapes. So you decide to figure out a definition.

What do you? How do you make up a definition? Let’s get you into some trouble.

## Concepts and Prototypes

We think about abstract concepts in terms of prototypical instances. These prototypical instances inform how we construct arguments using these concepts. At a more basic level, they determine how we go about constructing definitions themselves. Prototypes pop up in all sorts of conceptual domains, ranging from “war” to “airplane” to “bird.” So how do prototypes work in our thinking? Let’s start with an apparently simple example — the concept of triangle — that can get tricky really quickly.