Thirty four distinct composition series arising out of the 32 crystallographic double point groups are employed to re-derive in a simple and elegant fashion all the 169 distinct colour symmetry groups generated by the 32 double point groups, exploiting the idea of colour generators. The advantage of the method employed and some possible applications of these colour groups are discussed. The resulting colour groups are tabulated.

A flow chart (inverted ‘tree’) for generating and identifying the 58 magnetic and 18 polychromatic point groups using a classification for the 32 generating crystallographic point groups is suggested. The idea of colour generator is explored for generating the colour symmetry point groups. The advantages in presenting the identification of colour groups through a tree are discussed.

The maximum number of non-vanishing and independent second order photoelastic coefficients required by the seven pentagonal and the two icosahedral point groups 5(C₅),$$\bar 5$$(S₁₀),$$\overline {10} $$(C_5h),$$\overline {10} $$ m2(D_5h), 52(D₅), 5m(C_5v),$$\bar 5$$ 2m(D_5d); 235(I), 2/m$$\bar 3$$$$\bar 5$$(I_h)—that describe the quasicrystals symmetry groups in two and three dimensions—is obtained. The schemes of non-vanishing and independent coefficients have been calculated and listed. Finally the results of this group-theoretical study are briefly discussed.

The transport coefficients for the nine point groups$$5(C_5 ), \bar 5(S_{10} ), \overline {10} (C_{5h} ), \overline {10} m2(D_{5h} ), 52(D_5 ), 5m(C_{5\upsilon } ), \bar 5 2m(D_{5d} ), 235(I), 2/m \bar 3 \bar 5(I_h )$$ —which represent the symmetry groups of the quasicrystals in two and three dimensions—have been evaluated and tabulated in this work, employing group-theoretical methods.

Surface harmonics for the seven pentagonal point groups 5(C₅), {ie859-1}(S₁₀), {ie859-2}(C_5h), {ie859-3}m2(D_5h), {ie859-4} 2(D₅), 5m(C_5v) and 5 2m(D_5d) that represent the symmetries of quasicrystals in two and three dimensions are obtained, employing the projection operator method [11] and the simplified (authors) method. The results obtained are tabulated and are briefly discussed.