Build Boy's Surface

TL;DR

The paper presents a hands-on, visualization-light construction of Boy's surface, an immersed copy of the projective plane in $\mathbb{R}^3$, by decomposing the boundary into an octacross and crossbridges to form an immersed Möbius band $M$, then cone the boundary to infinity to obtain a disk $P$ in $S^3$, yielding $M\cup P$ as Boy's surface. It provides a detailed, piecewise-linear and topological framework (octahedral graph, octacross, crossbridges) plus an explicit assembly kit and a rectilinear model, emphasizing symmetry and cone-friendliness. The work connects to a YouTube airplane construction, discusses intrinsic flatness of the resulting Moebius band, and poses optimization questions about curvature under ambient homeomorphisms. Overall, it offers an accessible, replicable pathway for exploring an iconic nonorientable surface in a classroom or computational setting, with potential educational and geometric-topology implications.

Abstract

This article describes Boy's surface in a nice way that does not make many demands on three-dimensional visualization. The article includes a kit that you can print out onto card stock and assemble with scissors and tape.