A faixa de Möbius: um estudo sob as perspectivas histórica, geométrica e aplicações

Abstract

This study investigates the Möbius strip and its key geometric properties from theoretical, historical, and mathematical perspectives, highlighting its applications across various fields of knowledge. The research begins with a historical overview—tracing the surface’s early representations to its 19th-century mathematical formalization by August Ferdinand Möbius and Johann Benedict Listing—and emphasizes its significance to the development of modern topology. The primary objective is to analyze the fundamental characteristics of the Möbius strip and understand how its properties manifest in both mathematical constructions and practical, everyday situations. To this end, a qualitative, descriptive, and bibliographic study was conducted, drawing on books, scientific articles, dissertations, and academic materials related to the subject. The study examined properties such as one-sidedness, non-orientability, parameterization, and the surface’s behavior when subjected to longitudinal cuts. The results revealed that the Möbius strip possesses characteristics that break with traditional geometric concepts, exhibiting behaviors that defy intuitive spatial perception. Furthermore, it was observed that these properties extend beyond the theoretical realm, finding applications in fields such as mechanical engineering, chemistry, nanotechnology, art, architecture, and mathematics education. The study concludes that the Möbius strip serves as a significant example of the relationship between mathematical abstraction and practical application, underscoring the importance of studying geometric structures that contribute to understanding phenomena across diverse scientific and everyday contexts.