What is a simplex

The word simplex is a geometrical term which denotes a polytope with (k + 1) vertices ( comers ) in a k-dimensional space  

In two dimensions, k = 2, a simplex is a triangle in three dimensions, k = 3, it is a tetrahedron etc. see Fig. 11.1. Although we cannot imagine the shape of simplexes in higher dimensions than three, they exist as mathematical constructions. 

The important issue is We cannot draw a straight line which fits to all vertices of a triangle. There will always be one vertex which does not fit. We cannot fit a plane to all vertices of a tetrahedron.  

Assume that there are k experimental variables. Let each variable define a coordinate axis. If a simplex is placed in this fc-dimensional space, the coordinates of the vertices will define different settings of the experimental variables, see Fig. 11.2. 

As the vertices of a simplex are not linearly related to each other, an experimental design defined by a simplex forces the experimenter to change each variable in relation to each other in an uncorrelated manner over the set of k + 1) experiments. A consequence of this is that the experimental result will carry 

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