A bridge between Algebra and Geometry "via" proprtions

This paper is focused on the geometric interpretation of some algebraic relations defining metallic and plastic numbers. These numbers are considered in Architecture and Art as system of measurement and proportions started by dividing a segment in two parts. The variability of an integer coefficient, connected with the measure of the two parts, brings to different algebraic equations. The real positive solution of several second or third degree equations is a metallic or plastic number respectively. Among these, the first plastic number plays in 3D the well known important aesthetic role of the golden number in 2D. The paper provides also some geometrical interpretations in terms of aesthetic proportions related to architectural projects, in particular to two different vaults. The natural correspondences between geometric entities and algebraic formulas bring us to consider this work as a bridge between Algebra and Geometry.