Of course, if all the relations were totally symmetric, i.e., invariant under all permutations of the relata, there would
be no need to introduce permutations of the relations. But most geometries involve some order relations that are not
symmetric. Suppose, for example, two points on a straight line obey the relation: . Then permuting the two points
will violate the relation “”, so we must introduce the permuted relation “,” thus preserving the validity of the
permuted order relation .