Quaternion groups

The above analysis shows that the set of turns To Tn E, E-, i = 1,2,3, provides a geometric realization of the quaternion group and thus establishes the connection between the quaternion units and turns through rc/2, and hence rotations through % (binary rotations). This suggests that the whole set of turns might provide a geometric realization of the set of unit quaternions. Section 12.5 will not only prove this to be the case, but will also provide us with the correct parameterization of a rotation. 

Example A2.4-2 Deduce the character table for the quaternion group Q defined in Chapter 12. 

Table A2.3. Multiplication table for the Dirac characters of the quaternion group Q.

Table A2.4 Characteristic determinants obtained in the diagonalization of the Dirac characters for the quaternion group Q.

Table A2.6. Character table for the quaternion group Qfound by the diagonalization of the MRs of the Dirac characters. ...

The applicability of homotopic theory becomes much less obvious for liquid crystal phases with more complicated order parameters such as biaxial nematics and cholesterics, which are both locally defined by three directors forming a tripod. This gives rise to a description of the line singularities in terms of the quaternion group, Q. This is particularly interesting because the quaternion group Q is non-Abelian, a property that... 

The only remaining case to consider is that of the quaternion groups with N odd. The Hamiltonian matrix may be blocked by the quaternion transformation given previously. 

The concept of quaternion groups i.s useful for the understanding of icosahedral quasicrystals and shows how the existence of icosahedral quasicrystals is a natural consequence of the use of quaternions to represent symmetry groups. In this connection a real quaternion is defined as an ordered quadruple of four real numbers w,x, y, z). subject to the following rules of addition and multiplication where q = (w, x, y, z) and q = (w, r. /, c ) ... 

The quaternions thus form a ring which meets the mathematical requirements for a noncommutative field. The subring of the type (w, 0, 0, 0) is isomorphic with the field of real numbers and the subring of the type (w, x, 0, 0) (or (w, 0, y, 0) or (w, 0, 0, z)) is isomorphic with the complex numbers a + bi. Quaternion groups arise from applying group structure (Section 3) to sets of quaternions. 

The following shows how symmetry point groups (Section 3) can be described by quaternion groups ... 

Crystallographic and Quasicrystallographic Lattices from Finite Quaternion Groups... 

Figure 7 Crystal and quasicrystal lattices generated from the dihedral Dj, the tetrahedral Tj, and the icosahedral //, finite quaternion groups...

Quasicrystals may thus be regarded as a special type of incommensurate system, which may be described by space groups of dimension larger than three in a similar way to modulated crystal phases and incommensurate composite structures. In the case of icosahedral quasicrystals the above model based on the icosahedral quaternion group //, fits well into the idea of 6D space groups. Each of the three standard coordinates, namely x, y, and z, corresponds to two coordinates in 6D space, namely a rational and an irrational coordinate corresponding to the rational and irrational portions of variables, of the form a -H o V5, where a and d are integers. Projection of the lattice points of this 6D space of icosahedral symmetry into conventional 3D space leads to the icosahedral quasicrystal lattice. 

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