a triangular Every abstract triangulation of the projective plane can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its an example is the six-vertex projective plane obtained by adding one vertex to the five-vertex Möbius strip, connected by triangles to each of its boundary However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the Smoothly embedded rectangles A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than the same ratio as for the flat-folded equilateral-triangle version of the Möbius This flat triangular embedding can lift to a smooth embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the Mathematically, a smoothly embedded sheet of paper can be modeled as a developable surface, that can bend but cannot As its aspect ratio decreases toward , all smooth embeddings seem to approach the same triangular The lengthwise folds of an accordion-folded flat Möbius strip prevent it from forming a three-dimensional embedding in which the layers are separated from each other and bend smoothly without crumpling or stretching away from the Instead, unlike in the flat-folded case, there is a lower limit to the aspect ratio of smooth rectangular Möbius strips. Their aspect ratio cannot be less than even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this Without self-intersections, the aspect ratio must be at For aspect ratios between this bound it is unknown whether smooth embeddings, without self-intersection, If the requirement of smoothness is relaxed to allow continuously differentiable surfaces, the Nash–Kuiper theorem implies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratio The limiting case, a surface obtained from an infinite strip of the plane between two parallel lines, glued with the opposite orientation to each other, is called the unbounded Möbius strip or the real tautological line bundle. Although it has no smooth embedding into three-dimensional space, it can be embedded smoothly into four-dimensional Euclidean The minimum-energy shape of a smooth Möbius strip glued