For a one-dimensional shape, a boundary point is one you can stand on and move in one direction, but not in the other. So for an open line segment 0 < x < 1, th

For a one-dimensional shape, a boundary point is one you can stand on and move in one direction, but not in the other. So for an open line segment 0 < x < 1, there is no boundary, since at every point you can move in both directions. For a half-open line segment 0 < x <= 1 the point x=1 is the boundary, since you can move left but not right. For a two-dimensional shape (or more formally "manifold"), an interior point is one which locally looks like the plane (you can move in all directions), and a boundary point is one which locally looks like a half-plane. All the boundary points taken together will be a one-dimensional manifold _without boundary_, which is pretty neat. For example, take an annulus (a 2D shape like a CD-rom). The boundary of the annulus is two one-dimensional manifolds (the boundary circles), and those circles have no boundary.