Intrinsic vs. Extrinsic Structure

At this point, I think it will be useful to introduce an idea from geometry that is very helpful in pure mathematics, and that I find helpful for understanding the geometry of data sets. This idea is difference between the intrinsic structure of an object (such as a data set) and its extrinsic structure. Have you ever gone into a building, walked down a number of different halls and through different rooms, and when you finally got to where you’re going and looked out the window, you realized that you had no idea which direction you were facing, or which side of the building you were actually on? The intrinsic structure of a building has to do with how the rooms, halls and staircases connect up to each other. The extrinsic structure is how these rooms, halls and staircases sit with respect to the outside world. So, while you’re inside…