Möbiusstrook

Möbiusstrook, also known as the Möbius strip, is a non-orientable surface embedded in three-dimensional space. It can be formed by taking a rectangular strip of material, giving it a half twist (180 degrees), and joining the ends to form a loop. The resulting object has exactly one boundary component—a single closed edge—and it possesses only one side. If a person were to travel along the surface, they would return to their starting point having traversed both “faces” without ever crossing an edge, illustrating its non-orientability.

Topologically, the Möbiusstrook deformation retracts onto a circle and has Euler characteristic 0. It is also described as the total space of the nontrivial line bundle over the circle, making it a fundamental example in the study of non-orientable surfaces with boundary. Although it can be embedded in ordinary three-dimensional space, its non-orientable nature means the surface cannot be given a consistent outward normal everywhere.

Historically, the figure is named after August Ferdinand Möbius, who described its properties in 1858, with Johann Benedict Listing independently describing a similar construction in the same period. The Möbiusstrook serves as a simple, tangible model for introducing concepts of orientation, boundary, and non-orientability in topology.

In education and art, Möbius strips are frequently used to illustrate basic topological ideas, to demonstrate how a single surface can have only one side, and to inspire designs in sculpture and jewelry. Extensions of the idea lead to other non-orientable surfaces and to applications in mathematics, computer graphics, and pedagogy.