Design in the service of topology and the classification theorem

Abstract

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object that provides a glimpse into the construction of surfaces as defined by the classification theorem. This allows any user to explore different options and discover possible surfaces with three holes or fewer. This tactile experience of exploration grants the user intuition about the meaning of the theorem, without needing the rigorous proof; thus, it simplifies the depth of the classification theorem and its respective proof for non-mathematical audiences. We strive to accurately illustrate these esoteric concepts by giving them a form and to describe the modeling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. We believe that this playful object will be exciting for a variety of communities: university students, graduate students, teachers, professional mathematicians, and especially designers, who can use it to produce new tools.