This thesis results from the following work: (1) We have carried out a comprehensive study of Islamic geometrical patterns. More then 300 patterns have been studied. We have given a critique of the work of previous authors on this subject and have discussed our own Ideas on the evolution of Islamic geometrical designs. (2) We have performed symmetry analysis on the patterns and classified them according to their symmetry groups. (3) We have extracted numerical data for efficient generation of the patterns based on the analysis In (2). The data for more than 300 patterns Is provided on the disk. (4) We have developed a mathematical formalism based on group theory and constructed algorithms suitable for the generation of the patterns using computer graphics. (5) the algorithms have been proved by writing an Interactive computer graphic program called Islamic Geometrical Patterns ' IGP'. A library of geometric Islamic patterns has been constructed. (6) At Lite end of this thesis, In an Appendix, we have provided suggestions for further extension of this work.