The flexing geometry of full second-order polynomial models

15 The flexing geometry of full second-order polynomial models 

But this simple explanation becomes inadequate when it is realized that the information is greater on the ring of hexagonal points that surround the center point 

Geometrically, there are an infinite number of paraboloids that can pass through a circle (e.g., the circle of equal responses at the hexagonal points in this example). Some of the paraboloids will be tall and elongated, some will be short and compressed, some will point up, some will point down, one of them will even be a degenerate flat plane [Rider (1947)]. Although the hexagonal points hold the sides 

At a very basic level, the shapes of the normalized uncertainty and normalized information surfaces for a given model are a result of the location of points in factor space simply because carrying out an experiment provides information - that is, information is greatest in the vicinity of the design. But at a more sophisticated and often more important level, the shapes of the normalized uncertainty and normalized information surfaces are caused by the geometric vibrations of the response surfaces themselves - the more rigidly the model is pinned down by the experiments and the less it can squirm and thrash about, then the less will be the uncertainty and the greater will be the information content. 

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