It is a great pleasure to write a paper about architecture and mathematics on the occasion of the conference, Bridges: Mathematical Connections in Art, Music and Science. It is architecture's intimate relationship to mathematics that underscores its ties to art, music and science. The subject is too vast to lie within the range of a single discussion; this paper will look at some facets of these various relationships with the aim of introducing the reader to ideas meriting further study. We are surrounded by architecture. It determines the myriad spaces within which our lives unfold: where we live, work, worship; where we are taught, where we are healed, and, eventually, where we are buried. On one level, the relationship between mathematics and architecture seems a practical one: architecture has dimension and can be measured, therefore, it relates to numbers. It has shape and volume, as do plane figures and solids, therefore it relates to geometry. It involves composition and relationships, and therefore has to do with ratio, proportion and symmetry. One essential quality of architecture is that it must be constructed. This means that the architect must be capable of describing the building and explaining how it is to be built. In ancient times this was no,mean feat: many people believe that ancient architects were often members of secret societies and were forbidden under threat of death to divulge secrets of the trade to outsiders, so workers were told only the most superficial details of what they were constructing. Additionally, workers were more often than not illiterate. It is known that in times as recent as 500 years ago architects merely explained their designs to workers, never producing the exhaustive set of construction documents that today's architects labor over. How were the designs explained? Very often they were explained through geometry. One key dimension was laid out on the site, and other dimensions determined by means of geometric constructions, often with strings, ropes or wires, but sometimes with actual giant compasses. It has been argued, however, that measuring length and width has very little to do with pure mathematics, that the act of counting must not be confused with the science of number theory . Pure mathematics has to do with ideas, not with measurable quantities. So how is one to describe the relationship between architecture and mathematics?