Editing Möbius strip (section)

===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 {{nowrap|<math>\sqrt 3\approx 1.73</math>,}} the same ratio as for the flat-folded equilateral-triangle version of the Möbius {{nowrap|strip.{{r|sadowsky-translation}}}} This flat triangular embedding can lift to a smooth{{efn|This piecewise planar and cylindrical embedding has [[smoothness]] class <math>C^2</math>, and can be approximated arbitrarily accurately by [[infinitely differentiable]] {{nowrap|(class <math>C^\infty</math>)}} embeddings.{{r|bartels-hornung}}}} embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the {{nowrap|planes.{{r|sadowsky-translation}}}} Mathematically, a smoothly embedded sheet of paper can be modeled as a [[developable surface]], that can bend but cannot {{nowrap|stretch.{{r|bartels-hornung|starostin-vdh}}}} As its aspect ratio decreases toward <math>\sqrt 3</math>, all smooth embeddings seem to approach the same triangular {{nowrap|form.{{r|darkside}}}}

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 {{nowrap|folds.{{r|fuchs-tabachnikov}}}} 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 {{nowrap|<math>\pi/2\approx 1.57</math>,}} even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this {{nowrap|bound.{{r|fuchs-tabachnikov|halpern-weaver}}}} Without self-intersections, the aspect ratio must be at {{nowrap|least{{r|schwartz}}}}
<math display=block>\frac{2}{3}\sqrt{3+2\sqrt3}\approx 1.695.</math>

{{unsolved|mathematics|Can a <math>12\times 7</math> paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space?&thinsp;{{efn|12/7 is the simplest rational number in the range of aspect ratios, between 1.695 and 1.73, for which the existence of a smooth embedding is unknown.}}}}
For aspect ratios between this bound {{nowrap|and <math>\sqrt 3</math>,}} it has been an open problem whether smooth embeddings, without self-intersection, {{nowrap|exist.{{r|fuchs-tabachnikov|halpern-weaver|schwartz}}}} In 2023, [[Richard Schwartz (mathematician)|Richard Schwartz]] announced a proof that they do not exist, but this result still awaits peer review and publication.{{r|optimal|crowell}} 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 {{nowrap|becomes.{{efn|These surfaces have smoothness class <math>C^1</math>. For a more fine-grained analysis of the smoothness assumptions that force an embedding to be developable versus the assumptions under which the [[Nash–Kuiper theorem]] allows arbitrarily flexible embeddings, see remarks by {{harvtxt|Bartels|Hornung|2015}}, p. 116, following Theorem 2.2.{{r|bartels-hornung}}}}}} 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]].{{r|dundas}} Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean {{nowrap|space.{{r|blanusa}}}}

The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have a known analytic description, but can be calculated numerically, and has been the subject of much study in [[plate theory]] since the initial work on this subject in 1930 by [[Michael Sadowsky]].{{r|bartels-hornung|starostin-vdh}} It is also possible to find [[algebraic surface]]s that contain rectangular developable Möbius {{nowrap|strips.{{r|wunderlich|schwarz}}}}