We describe the exploration of the manifold novel shapes found in algebraic geometry and their application in architectural design. These surfaces represent the zero-sets of certain polynomials of varying degrees. They are therefore very structured, coherent and harmonious yet at the same time geometrically and topologically highly complex. Their application in design is mostly unprecedended as they have only recently begun to become accessible through novel software tools. We present and discuss experimental student design and research projects where shapes found in algebraic geometry were developed into pavilion designs. We describe historic precedents for the inspiration of art and architecture through mathematics and show how algebraic surfaces can be used to expand architectsi sculptural vocabulary, make the utmost of three-dimensional sculptural qualities, employ shapes that have a strong internal structure, transcend the imaginable and explore polynomials as a new kind of shape-making tool.