O'Reilly logo

The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

III.17 Dimension

What is the difference between a two-dimensional set and a three-dimensional set? A rough answer that one might give is that a two-dimensional set lives inside a plane, while a three-dimensional set fills up a portion of space. Is this a good answer? For many sets it does seem to be: triangles, squares, and circles can be drawn in a plane, while tetrahedra, cubes, and spheres cannot. But how about the surface of a sphere? This we would normally think of as two dimensional, contrasting it with the solid sphere, which is three dimensional. But the surface of a sphere does not live inside a plane.

Does this mean that our rough definition was incorrect? Not exactly. From the perspective of linear algebra, the set {(x, y, z) : x ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required