This paper serves as an introduction to a type of geometric construction discovered by Raffi J. Kasparian while writing a polyhedral construction tool, named “Archimedean.” Archimedean's unique exploratory approach to polyhedral construction led to the discovery of several objects where the faces of the polyhedron traverse the surface more than once before joining. Alice Petillo has used the Archimedean program in teaching geometry to undergraduate students at Marymount University in Arlington, Virginia for the last several years. Kasparian and Petillo have collaborated to bring this discovery to the larger mathematical community. Archimedean may be found at http://www.quantimegroup.com/solutions/pages/Archimedean2.0/Archimedean.html.