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 twodimensional 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.
Acknowledgments
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.
