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         9. Conclusion

The root 2 system of architecture had its roots in the architecture of ancient Rome and the Italian Renaissance. We have found that a root 3 geometry was also used during the Renaissance while a system of proportion based on the golden mean was used by the twentieth century architect, Le Corbusier. The first two systems can be shown to be based on the symmetry groups of the plane while the third system, based on the golden mean, is the basis of quasicrystal geometries depicted in two-dimensional projections as Penrose tilings of the plane. We have also shown that these proportional systems give rise to numbers of importance to the study of dynamical systems.


This paper owes much to Kim Williams who first called my attention to the root 3 sequence. I am also indebted to Asim Riaz for (his) help in preparing the text.