The crossmetric tensor and the geometrical meaning of the imaginary numbers

Abstract

The product of two Cartesian quaternions can be written as a quadratic form based on a 4x4 matrix made of the symbols 1,i,j,k where i,j,k are imaginary numbers introduced by Hamilton. We generalized this matrix to non-Cartesian bases, and showed that the matrix is made of the metric and the cross tensors. We called it crossmetric tensor. Its symbols are s,a,b,c; they are the elementary crystallographic quaternions. We determined the crossmetric tensors for the six crystal families. We also showed that any unit crystallographic quaternion can be geometrically represented by an infinity of pairs of oriented planes intersecting along the vectorial component of the quaternion such that the angle between them is the semiangle of the rotation. The composition of quaternions follows the intuitive source-target rule. The elementary quaternions a,b,c are geometrically represented by pair of perpendicular and oriented planes intersecting along the axis a,b,c, respectively. Other complementary quaternions noted a',b',c' were also introduced. They are the pairs of planes ma, mb, mc. The elementary quaternions a,b,c follow Hamilton rules on the squares of imaginary numbers. The complementary quaternions follow Hamilton rules on the bi and tri-products. For Cartesian basis, the quaternions i,j,k appear as a specific case of crystallographic quaternions. They are formed by the pairs of perpendicular faces of the cube mx, my, mz. Since these planes intersect along the x, y and z axis, respectively, the elementary and complementary quaternions are equal, i.e. i=i',j=j',k=k', which explains why the square, bi and tri-product Hamilton rules are satisfied all together with only three quaternions i,j,k.