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Thinking about the whales really helped me understand why the natural spline doesn't have locality: if we're thinking about the spline as a physical piece of wood or metal with a lot of tension on it and these whales setting the points that we want to interpolate, it seems very plausible that moving a single whale around (if we think of the whales as being attached to the spline loosely, allowing rotation of the rod perhaps) could cause the whole rod to flip around into a different configuration and therefore a different series of piecewise polynomials.