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in the original Möbius strip, and the other two come from the way the two halves of the thinner strip wrap around each other. The result is not a Möbius strip, but instead is topologically equivalent to a cylinder. Cutting this double-twisted strip again along its centerline produces two linked double-twisted strips. If, instead, a Möbius strip is cut lengthwise, a third of the way across its width, it produces two linked strips. One of the two is a central, thinner, Möbius strip, while the other has two
342:: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image. In particular, a curved arrow pointing clockwise (↻) would return as an arrow pointing counterclockwise (↺), implying that, within the Möbius strip, it is impossible to consistently define what it means to be clockwise or counterclockwise. It is the simplest non-orientable surface: any other surface is non-orientable if and only if it has a Möbius strip as a
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2368:-axis at right angles. Take the subset of the upper half-plane between any two nested semicircles, and identify the outer semicircle with the left-right reversal of the inner semicircle. The result is topologically a complete and non-compact Möbius strip with constant negative curvature. It is a "nonstandard" complete hyperbolic surface in the sense that it contains a complete hyperbolic
1086:
1305:
350:, the Möbius strip has only one side. A three-dimensional object that slides one time around the surface of the strip is not mirrored, but instead returns to the same point of the strip on what appears locally to be its other side, showing that both positions are really part of a single side. This behavior is different from familiar
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288:, as a band with only a single twist. There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip. Some other ancient depictions of the
3225:. One such trick, known as the Afghan bands, uses the fact that the Möbius strip remains a single strip when cut lengthwise. It originated in the 1880s, and was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such as
1937:(allowing the surface to cross itself in certain restricted ways). A Klein bottle is the surface that results when two Möbius strips are glued together edge-to-edge, and – reversing that process – a Klein bottle can be sliced along a carefully chosen cut to produce two Möbius
461:
Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with four half-twists in it (relative to an untwisted annulus or cylinder) rather than two separate strips. Two of the half-twists come from the fact that this thinner strip goes two times through the half-twist
354:
in three dimensions such as those modeled by flat sheets of paper, cylindrical drinking straws, or hollow balls, for which one side of the surface is not connected to the other. However, this is a property of its embedding into space rather than an intrinsic property of the Möbius strip itself: there
306:
wear half as quickly when they form Möbius strips, because they use the entire surface of the belt rather than only the inner surface of an untwisted belt. Additionally, such a belt may be less prone to curling from side to side. An early written description of this technique dates to 1871, which is
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Möbius strips have been a frequent inspiration for the architectural design of buildings and bridges. However, many of these are projects or conceptual designs rather than constructed objects, or stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a
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Geometrically Lawson's Klein bottle can be constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the
Sudanese Möbius strip is obtained by sweeping a semicircle instead of a circle, or equivalently by slicing the Klein bottle along a circle that is perpendicular
381:
A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, includes all boundary points of the Möbius strip in a single continuous curve. For a Möbius strip formed by gluing and twisting a rectangle, it has twice the length of the centerline of the strip. In
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instead of constant
Gaussian curvature. The Sudanese Möbius strip was constructed as a minimal surface bounded by a great circle in a 3-sphere, but there is also a unique complete (boundaryless) minimal surface immersed in Euclidean space that has the topology of an open Möbius strip. It is called
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272:. Therefore, whether the ribbon is a Möbius strip may be coincidental, rather than a deliberate choice. In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not
136:, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can form
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The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in the plane that it rotates within, and the Möbius strip that it generates forms a slice through the
2307:
The resulting metric makes the open Möbius strip into a (geodesically) complete flat surface (i.e., having zero
Gaussian curvature everywhere). This is the unique metric on the Möbius strip, up to uniform scaling, that is both flat and complete. It is the
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The most symmetric projection is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles, but produces an unbounded embedding with the projection point removed from its
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Beyond the already-discussed applications of Möbius strips to the design of mechanical belts that wear evenly on their entire surface, and of the Plücker conoid to the design of gears, other applications of Möbius strips include:
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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
3002:, for which a building was planned in the shape of a thickened Möbius strip but refinished with a different design after the original architects pulled out of the project. One notable building incorporating a Möbius strip is the
1081:{\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}}
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Möbius strips with odd numbers of half-twists greater than one, or that are knotted before gluing, are distinct as embedded subsets of three-dimensional space, even though they are all equivalent as two-dimensional topological
5091:
Hinz, Denis F.; Fried, Eliot (2015). "Translation of
Michael Sadowsky's paper "An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem"".
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the space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of
Euclidean lines, punctures this space of projective
7053:
Bauer, Thomas; Banzer, Peter; Karimi, Ebrahim; Orlov, Sergej; Rubano, Andrea; Marrucci, Lorenzo; Santamato, Enrico; Boyd, Robert W.; Leuchs, Gerd (February 2015). "Observation of optical polarization Möbius strips".
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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
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In many cases these merely depict coiled ribbons as boundaries. When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent to
1958:
2976:(IMPA) uses a stylized smooth Möbius strip as their logo, and has a matching large sculpture of a Möbius strip on display in their building. The Möbius strip has also featured in the artwork for
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For the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity of the plane's rotation. This can be described as a
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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
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For strips too short to apply this method directly, one can first "accordion fold" the strip in its wide direction back and forth using an even number of folds. With two folds, for example, a
2752:, a form of dual-tracked roller coaster in which the two tracks spiral around each other an odd number of times, so that the carriages return to the other track than the one they started on
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Möbius strips have also been used to analyze many other canons by Bach and others, but in most of these cases other looping surfaces such as a cylinder could have been used equally well.
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appear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in
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More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot and they have the same number of twists as each
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in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a
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2417:. Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimal
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of the subgroup to points produces a space with the same topology as the underlying homogenous space. In the case of the symmetries of
Euclidean lines, the stabilizer of the
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The Möbius strip can be continuously transformed into its centerline, by making it narrower while fixing the points on the centerline. This transformation is an example of a
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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
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319:
Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto a
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The
Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself. Another form of the Möbius strip, called the
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is in the shape of an 'N' and would remain an 'N' after a half-twist. The narrower accordion-folded strip can then be folded and joined in the same way that a longer strip
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A Möbius strip of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the
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The Möbius strip can be cut into six mutually-adjacent regions, showing that maps on the surface of the Möbius strip can sometimes require six colors, in contrast to the
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For instance, if the front and back faces of a cube are glued to each other with a left-right mirror reflection, the result is a three-dimensional topological space (the
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One way to embed the Möbius strip in three-dimensional
Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of its
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An open strip with zero curvature may be constructed by gluing the opposite sides of a plane strip between two parallel lines, described above as the tautological line
1420:
774:
6829:
Yamashiro, Atsushi; Shimoi, Yukihiro; Harigaya, Kikuo; Wakabayashi, Katsunori (2004). "Novel electronic states in graphene ribbons: competing spin and charge orders".
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from the top of a hemisphere, orienting the edges of the quadrilateral in alternating directions, and then gluing opposite pairs of these edges consistently with this
1842:
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
3006:, which is surrounded by a large twisted ribbon of stainless steel acting as a façade and canopy, and evoking the curved shapes of racing tracks. On a smaller scale,
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by a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a
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from the 1940s. Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with a twist; these include
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The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is
1347:, and attaching the ends. The shortest strip for which this is possible consists of three equilateral triangles, folded at the edges where two triangles meet. Its
5725:
5556:(1954). "Le plongement isométrique de la bande de Möbius infiniment large euclidienne dans un espace sphérique, parabolique ou hyperbolique à quatre dimensions".
4378:
Proceedings of the Eugène Strens
Memorial Conference on Recreational Mathematics and its History held at the University of Calgary, Calgary, Alberta, August 1986
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For a form of the Klein bottle known as Lawson's Klein bottle, the curve along which it is sliced can be made circular, resulting in Möbius strips with circular
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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
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There are many different ways of defining geometric surfaces with the topology of the Möbius strip, yielding realizations with additional geometric properties.
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The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle
2664:, a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbius
264:
However, it had been known long before, both as a physical object and in artistic depictions; in particular, it can be seen in several Roman mosaics from the
1550:
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
6597:
3445:, because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle to have a different set of vertices.
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Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to the
1700:
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describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the
1830:
announced a proof that they do not exist, but this result still awaits peer review and publication. If the requirement of smoothness is relaxed to allow
2973:
2934:
1850:. Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean
416:, and its existence means that the Möbius strip has many of the same properties as its centerline, which is topologically a circle. In particular, its
4423:
3528:
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.
3190:, the same motif from two measures earlier. Because of this symmetry, this canon can be thought of as having its score written on a Möbius strip. In
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making it more convenient for attaching onto circular holes in other surfaces. In order to do so, it crosses itself. It can be formed by removing a
5002:
3564:. For a more fine-grained analysis of the smoothness assumptions that force an embedding to be developable versus the assumptions under which the
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199:
feature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.
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105:. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single
7907:
2182:
Instead, leaving the
Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the
363:
of a Möbius strip with an interval) in which the top and bottom halves of the cube can be separated from each other by a two-sided Möbius
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2173:
transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its
2161:
This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the
1838:
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
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62:
that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by
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8245:
4329:
Jablan, Slavik; Radović, Ljiljana; Sazdanović, Radmila (2011). "Nonplanar graphs derived from Gauss codes of virtual knots and links".
3194:, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, the
7757:. In Barrallo, Javier; Friedman, Nathaniel; Maldonado, Juan Antonio; Mart\'\inez-Aroza, José; Sarhangi, Reza; Séquin, Carlo (eds.).
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Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in
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of distinct points on a circle, the pairs of points at infinity of each line. This space, again, has the topology of an open Möbius
1278:
A line or line segment swept in a different motion, rotating in a horizontal plane around the origin as it moves up and down, forms
7364:
6251:. Pure and Applied Mathematics. Vol. 89. Translated by Goldman, Michael A. New York & London: Academic Press. p. 12.
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in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the
7362:
Walba, David M.; Richards, Rodney M.; Haltiwanger, R. Curtis (June 1982). "Total synthesis of the first molecular Moebius strip".
296:-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is
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2389:. However, in a sense it is only one point away from being a complete surface, as the open Möbius strip is homeomorphic to the
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The open Möbius strip also admits complete metrics of constant negative curvature. One way to see this is to begin with the
8772:
4955:
4858:
2806:
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Yoon, Zin Seok; Osuka, Atsuhiro; Kim, Dongho (May 2009). "Möbius aromaticity and antiaromaticity in expanded porphyrins".
6280:
López, Francisco J.; Martín, Francisco (1997). "Complete nonorientable minimal surfaces with the highest symmetry group".
8793:
7787:
829:
4752:
3263:, a fractal formed by repeatedly thickening a space curve to a Möbius strip and then replacing it with the boundary edge
3089:'s "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever inventions in multiple stories of
1313:
121:
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3636:
The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
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Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius
424:. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up to
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even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this
101:
centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are
252:
The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians
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of any point on the map can be found on the other printed side of the surface at the same point of the Möbius strip
1827:
1508:. A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a
2758:
projected onto a Möbius strip with the convenient properties that there are no east–west boundaries, and that the
2509:, a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called
6282:
466:
These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called
1558:
To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the
1554:
However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral
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8834:
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5051:
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3714:
2999:
2058:{\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )}
112:
Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a
140:. Certain highly symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip.
17:
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Mangahas, Johanna (July 2017). "Office Hour Five: The Ping-Pong Lemma". In Clay, Matt; Margalit, Dan (eds.).
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this sense, the Möbius strip is different from an untwisted ring and like a circular disk in having only one
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175:. Many architectural concepts have been inspired by the Möbius strip, including the building design for the
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8779:
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7644:
5675:
3260:
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8230:
6960:
6432:
3712:; González, Diego L. (2016). "Möbius strips before Möbius: topological hints in ancient representations".
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can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its
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7107:
Candeal, Juan Carlos; Induráin, Esteban (January 1994). "The Moebius strip and a social choice paradox".
6757:. World Scientific lecture notes in physics. Vol. 61 (2nd ed.). World Scientific. p. 269.
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Cantwell, John; Conlon, Lawrence (2015). "Hyperbolic geometry and homotopic homeomorphisms of surfaces".
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takes the shape of a Möbius strip. This conception, and generalizations to more points, is a significant
2414:
2348:, a geometry of constant curvature whose lines are represented in the model by semicircles that meet the
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Melikhov, Sergey A. (2019). "A note on O. Frolkina's paper "Pairwise disjoint Moebius bands in space"".
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with a compact design and a resonant frequency that is half that of identically constructed linear coils
1933:, a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can be
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3210:. Modern musical groups taking their name from the Möbius strip include American electronic rock trio
2964:. Some variations of the recycling symbol use a different embedding with three half-twists instead of
2534:
1091:
257:
67:
6385:
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2694:
ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism
2661:
2372:(actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard
2309:
2292:. The cases of negative and zero curvature form geodesically complete surfaces, which means that all
2258:
2189:
524:
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A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in
1925:, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly
1127:
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A single off-center cut produces a Möbius strip (purple) linked with a double-length two-sided strip
375:
copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously
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6792:. Encyclopaedia of Mathematical Sciences. Vol. 20. Springer-Verlag, Berlin. pp. 164–166.
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6571:. Aportaciones Mat. Notas Investigación. Vol. 8. Soc. Mat. Mexicana, México. pp. 67–79.
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of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a
367:
In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an
307:
after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a
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2425:, which defines a minimal surface uniquely from its boundary curve and tangent planes along this
1934:
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387:
253:
63:
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579:, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers
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With an even number of twists, however, one obtains a different topological surface, called the
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7754:
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Decker, Heinz; Stark, Eberhard (1983). "Möbius-Bänder: ...und natürlich auch auf Briefmarken".
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logo used a flat-folded three-twist Möbius strip, as have other similar designs. The Brazilian
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413:
137:
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5244:"Equilibrium shapes with stress localisation for inextensible elastic Möbius and other strips"
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A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a
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Bickel, Holger (1999). "Duality in stable planes and related closure and kernel operations".
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Because every line in the plane is symmetric to every other line, the open Möbius strip is a
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1272:
swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains
1230:
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Ticket To Ride: The Essential Guide to the World's Greatest Roller Coasters and Thrill Rides
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4579:. Wiley Series in Design Engineering. Vol. 3. John Wiley & Sons. pp. 135–137.
3245:, a shift register whose output bit is complemented before being fed back into the input bit
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2257:
The two parts of the surface formed by the two glued pairs of edges cross each other with a
1504:, which requires six triangles and six vertices, even when represented more abstractly as a
355:
exist other topological spaces in which the Möbius strip can be embedded so that it has two
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6406:
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The flat-folded Möbius strip formed from three equilateral triangles does not come from an
3226:
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3031:
3011:
2858:
2744:
2731:
2620:
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1785:
paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space?
1344:
572:
180:
160:
2479:
These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the
2245:, also has a circular boundary, but otherwise stays on only one side of the plane of this
1181:
describes the rotation angle of the plane around its central axis and the other parameter
8:
8717:
8698:
8459:
8453:
7887:
6138:
5401:
4854:"A polyhedral model in Euclidean 3-space of the six-pentagon map of the projective plane"
3257:, the mathematical theory of infinitesimally thin strips that follow knotted space curves
3230:
3178:
3081:
3054:
3003:
2990:
2812:
Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting by
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2285:
1921:. In common forms of the Möbius strip, it has a different shape from a circle, but it is
1910:
1611:
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184:
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117:
106:
8069:
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7069:
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7005:
6930:
6852:
6533:
6453:
4951:"Geometric realization of a triangulation on the projective plane with one face removed"
4372:
Larsen, Mogens Esrom (1994). "Misunderstanding my mazy mazes may make me miserable". In
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5458:
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5273:
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5127:
5101:
5019:
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4785:
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4380:. MAA Spectrum. Washington, DC: Mathematical Association of America. pp. 289–293.
4354:
4207:
4112:. Graduate Texts in Mathematics. Vol. 127. New York: Springer-Verlag. p. 49.
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59:
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Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
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7615:. MAA Spectrum. Mathematical Association of America, Washington, DC. pp. 31–35.
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7122:
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6693:
6667:
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An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity
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These symmetries also provide another way to construct the Möbius strip itself, as a
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projective plane, the surface obtained by removing any one point from the projective
2281:
1865:
532:
490:
417:
360:
303:
212:
144:
90:
6868:
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5277:
4986:
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3242:
2853:
2296:("straight lines" on the surface) may be extended indefinitely in either direction.
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Lie groups and Lie algebras I: Foundations of Lie Theory; Lie Transformation Groups
6659:
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5504:. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge. p.
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2891:(1976) is one of several pieces by Perry exploring variations of the Möbius strip.
2882:
2701:
2466:
2449:
2313:
2262:
2183:
1952:
1861:
1543:
1351: – the ratio of the strip's length to its width – is
312:
242:
172:
86:
from counterclockwise turns. Every non-orientable surface contains a Möbius strip.
8216:
7800:
6398:
5629:
4682:
1822:
it has been an open problem whether smooth embeddings, without self-intersection,
552:
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6703:
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6465:
6433:"Instability of a Möbius strip minimal surface and a link with systolic geometry"
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4123:
4076:
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3965:
3933:
3807:
3749:
3350:
3251:, an impossible figure whose boundary appears to wrap around it in a Möbius strip
3085:(1996) based on it. An entire world shaped like a Möbius strip is the setting of
2720:
2401:
2376:
Again, this can be understood as the quotient of the hyperbolic plane by a glide
1948:
1602:
the same ratio as for the flat-folded equilateral-triangle version of the Möbius
1389:
548:
368:
347:
129:
125:
102:
94:
6860:
4514:. Southwestern College, Winfield, Kansas: Bridges Conference. pp. 211–218.
3207:
8909:
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8174:
7604:
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5046:
4849:
4373:
4267:
Proceedings of the National Academy of Sciences of the United States of America
4257:
3062:
3015:
2945:
2944:
Because of their easily recognized form, Möbius strips are a common element of
2514:
2498:
2470:
2405:
2390:
2153:, gives a Möbius strip embedded in the hypersphere as a minimal surface with a
1481:
540:
148:
8869:
8158:
7434:
6797:
6203:
6161:
6104:
5558:
Bulletin International de l'Académie Yougoslave des Sciences et des Beaux-Arts
5260:
5243:
5204:
5152:
5115:
4969:
4950:
4906:(1948). "A non-singular polyhedral Möbius band whose boundary is a triangle".
4872:
4853:
4342:
4005:
3929:
3745:
498:
8929:
8662:
8480:
8436:
8422:
8320:
7998:
7302:
7298:
6428:
6420:
5671:
5648:
5191:
Bartels, Sören; Hornung, Peter (2015). "Bending paper and the Möbius strip".
4826:
4807:
4712:
4655:. Providence, Rhode Island: American Mathematical Society. pp. 199–206.
3793:
3602:
3266:
3254:
3165:
3094:
3061:
into which unwary victims may become trapped. Examples of this trope include
2977:
2817:
2813:
2526:
2437:
2330:
2250:
1527:
1283:
820:
555:, the boundaries of subdivisions of the Möbius strip into rectangles meeting
372:
339:
302:
Independently of the mathematical tradition, machinists have long known that
188:
113:
79:
8077:
7824:"'Norman said the president wants a pyramid': how starchitects built Astana"
7337:
7077:
6011:. Providence, Rhode Island: American Mathematical Society. pp. 99–100.
5346:"Inverting a cylinder through isometric immersions and isometric embeddings"
442:
Cutting the centerline produces a double-length two-sided (non-Möbius) strip
334:
A 2D object traversing once around the Möbius strip returns in mirrored form
330:
8914:
8844:
8616:
8327:
8260:
8093:
8023:
7828:
7409:
7085:
6946:
6903:
6882:
Rzepa, Henry S. (September 2005). "Möbius aromaticity and delocalization".
6477:
6002:
5870:
Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture
4639:
4315:
3775:
3416:
3203:
3191:
3157:
3035:
2969:
2919:
2867:
2821:
2510:
2325:
2154:
1930:
1914:
1857:
1516:
Möbius strip, one that is fully four-dimensional and for which all cuts by
1348:
563:
whose embedding into the Möbius strip shows that, unlike in the plane, the
544:
285:
223:
164:
71:
7219:; Tobler, Waldo R. (January 1991). "Three world maps on a Moebius strip".
6663:
6295:
5513:
4288:
4233:"Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen"
3784:. Outlooks. Washington, DC: Mathematical Association of America. pp.
1526:
Other polyhedral embeddings of Möbius strips include one with four convex
232:
8894:
8585:
6843:
4150:(2nd ed.). London & New York: Macmillan and co. pp. 53–54.
4059:
Kyle, R. H. (1955). "Embeddings of Möbius bands in 3-dimensional space".
3842:
Woll, John W. Jr. (Spring 1971). "One-sided surfaces and orientability".
3118:
2961:
2956:, designed in 1970, is based on the smooth triangular form of the Möbius
2783:
2518:
1520:
separate it into two parts that are topologically equivalent to disks or
1269:
187:
have based stage magic tricks on the properties of the Möbius strip. The
133:
43:
7377:
6992:
Pond, J. M. (2000). "Mobius dual-mode resonators and bandpass filters".
4072:
3781:
When Topology Meets Chemistry: A Topological Look at Molecular Chirality
3690:
810:
A Möbius strip swept out by a rotating line segment in a rotating plane
8889:
8722:
8647:
8085:
7460:
7272:
6624:
6569:
Differential Geometry Workshop on Spaces of Geometry (Guanajuato, 1992)
5799:
5598:
5371:
5319:
5023:
4927:
4773:
4446:
3865:
3455:
3169:
3131:
2837:
2716:
2675:
2369:
2225:
1751:
1610:
Mathematically, a smoothly embedded sheet of paper can be modeled as a
1535:
1517:
1497:
536:
308:
293:
289:
238:
192:
75:
7013:
6938:
6895:
6016:
5500:
Dundas, Bjørn Ian (2018). "Example 5.1.3: The unbounded Möbius band".
4660:
3904:
Frolkina, Olga D. (2018). "Pairwise disjoint Moebius bands in space".
8807:
8495:
8384:
8221:
5708:(2nd ed.). Wilmington, Delaware: Publish or Perish. p. 591.
4202:
4175:
3058:
2953:
2779:
2767:
2759:
2755:
2725:
2494:
2230:
156:
152:
83:
7452:
5946:
5929:
5791:
5726:"Tutorial 3: Lawson's Minimal Surfaces and the Sudanese Möbius Band"
5362:
5345:
5015:
4919:
4577:
The Kinematic Geometry of Gearing: A Concurrent Engineering Approach
4437:
4418:
3857:
3365:
2229:
Schematic depiction of a cross-cap with an open bottom, showing its
1743:{\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.}
82:
surface, meaning that within it one cannot consistently distinguish
34:
8621:
8372:
7892:
5463:
5415:
5000:
Brehm, Ulrich (1983). "A nonpolyhedral triangulated Möbius strip".
3996:
3920:
3728:
3221:
Möbius strips and their properties have been used in the design of
3173:
3151:
3027:
3018:
whose base and sides have the form of a Möbius strip. As a form of
2845:
2829:
2802:
2775:
2691:
2597:
consists of all symmetries that take the axis to itself. Each line
2317:
2293:
1501:
425:
277:
219:
168:
7311:"Soap-film Möbius strip changes topology with a twist singularity"
7027:
Rohde, Ulrich L.; Poddar, Ajay; Sundararajan, D. (November 2013).
6828:
6152:
5878:
5106:
1884:
5296:
Schwarz, Gideon E. (1990). "The dark side of the Moebius strip".
3667:
Larison, Lorraine L. (1973). "The Möbius band in Roman mosaics".
2735:
2566:
of these Lie groups. A group model consists of a Lie group and a
1896:
483:
448:
436:
7580:. Phoenix, Arizona: Tessellations Publishing. pp. 153–158.
5872:. Phoenix, Arizona: Tessellations Publishing. pp. 103–110.
4609:. New York: Thomas Y. Crowell Company. pp. 40–49, 200–201.
2490:
The affine transformations and Möbius transformations both form
8672:
8280:
8200:
8004:
Reprinted from an American Mathematical Society Feature Column.
7576:. In Swart, David; Séquin, Carlo H.; Fenyvesi, Kristóf (eds.).
6008:
Knots, Molecules, and the Universe: An Introduction to Topology
5706:
A Comprehensive Introduction to Differential Geometry, Volume I
5399:(2021). "An improved bound on the optimal paper Moebius band".
4147:
Mathematical Recreations and Problems of Past and Present Times
2980:
from countries including Brazil, Belgium, the Netherlands, and
2715:, a strip of conductive material covering the single side of a
1922:
1918:
1509:
798:
281:
7485:
Charles Olson and American Modernism: The Practice of the Self
6786:
Gorbatsevich, V. V.; Onishchik, A. L.; Vinberg, È. B. (1993).
6419:
5904:
5812:
See Section 7, pp. 350–353, where the Klein bottle is denoted
2233:. This surface crosses itself along the vertical line segment.
8286:
6057:. Princeton, New Jersey: Princeton University Press. p.
6054:
Euler's Gem: The Polyhedron Formula and the Birth of Topology
5868:. In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.).
4707:. Cambridge, UK: Cambridge University Press. pp. 33–36.
3047:
3043:
3039:
3023:
2899:
2571:
2321:
1642:, all smooth embeddings seem to approach the same triangular
315:
from 1206 depicts a Möbius strip configuration for its drive
7970:
7958:
7707:
6785:
6328:(1981). "The classification of complete minimal surfaces in
5583:
Wunderlich, W. (1962). "Über ein abwickelbares Möbiusband".
4649:
Mathematical Omnibus: Thirty Lectures on Classic Mathematics
2444:
One way to see this is to extend the Euclidean plane to the
1512:. This four-dimensional polyhedral Möbius strip is the only
7761:. Granada, Spain: University of Granada. pp. 353–360.
7511:
Figuring It Out: Entertaining Encounters with Everyday Math
5506:
https://books.google.com/books?id=7a1eDwAAQBAJ&pg=PA101
3881:
Elementary Topology: A Combinatorial and Algebraic Approach
3374:
3371:
3327:
3312:
3309:
3303:
1388:
For a strip of nine equilateral triangles, the result is a
1304:
1292:
576:
489:
Subdivision into six mutually-adjacent regions, bounded by
7694:
Para quem é fã do IMPA, dez curiosidades sobre o instituto
7555:
7553:
7293:
7029:"Printed resonators: Möbius strip theory and applications"
5229:, pp. 113–136. See in particular Section 5.2, pp. 129–130.
2112:{\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi }
428:) only by the number of times they loop around the strip.
163:. In popular culture, Möbius strips appear in artworks by
93:, the Möbius strip can be embedded into three-dimensional
8160:
The Möbius Strip in Magic: A Treatise on the Afghan Bands
7946:"Pasta Graduates From Alphabet Soup to Advanced Geometry"
7613:
Mathematical Treks: From Surreal Numbers to Magic Circles
7384:
6973:
3825:
3823:
3821:
2521:, and that not every solvmanifold can be factored into a
1534:
and one using the vertices and center point of a regular
523:
Six colors are always enough. This result is part of the
338:
The Möbius strip has several curious properties. It is a
7361:
5241:
3186:
symmetry in which each voice in the canon repeats, with
2708:
aligned along the cycle in the pattern of a Möbius strip
1694:
Without self-intersections, the aspect ratio must be at
1384:
and the same folding method works for any larger aspect
1299:
527:, which states how many colors each topological surface
8261:
Compact topological surfaces and their immersions in 3D
7550:
7249:(1940). "Soap film experiments with minimal surfaces".
7026:
6491:
Mira, Pablo (2006). "Complete minimal Möbius strips in
5086:
5084:
4328:
3568:
allows arbitrarily flexible embeddings, see remarks by
2998:
functional part of the architecture. An example is the
2778:
with a Möbius strip shape, and the formation of larger
2617:
corresponds to a coset, the set of symmetries that map
1392:, which can be flexed to reveal different parts of its
7759:
Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings
7052:
5480:"Mathematicians solve 50-year-old Möbius strip puzzle"
4087:
4040:
4028:
3818:
3202:
of two unordered points on a circle, the space of all
2840:-shaped Möbius strip. It is also a popular subject of
1955:
of 4-dimensional space, the set of points of the form
1530:
as faces, another with three non-convex quadrilateral
1436:
195:
have been analyzed using Möbius strips. Many works of
151:
whose carriages alternate between the two tracks, and
8211:
7927:"How to make a mathematically-endless strip of bacon"
6692:. Basel: Birkhäuser Verlag. pp. 83–88, 157–163.
6497:
6363:
6334:
5853:
5851:
5818:
5755:
3543:
3496:
3464:
3377:
3336:
3315:
2644:
2623:
2603:
2581:
2537:
2483:, and the symmetries of hyperbolic lines include the
2354:
2192:
2127:
2072:
1961:
1802:
1765:
1703:
1660:
1624:
1576:
1428:
1402:
1358:
1322:
1233:
1209:
1188:
1167:
1130:
1094:
858:
744:
723:
703:
683:
645:
625:
605:
585:
420:
is the same as the fundamental group of a circle, an
7666:"Did Google Drive Copy its Icon From a Chinese App?"
7405:"Chemical origami used to create a DNA Möbius strip"
6994:
IEEE Transactions on Microwave Theory and Techniques
6687:
6233:
6231:
5630:"A pretender to the title 'canonical Moebius strip'"
5242:
Starostin, E. L.; van der Heijden, G. H. M. (2015).
5081:
4634:
4632:
4630:
4628:
4626:
4240:
Jahresbericht der Deutschen Mathematiker-Vereinigung
3485:, and can be approximated arbitrarily accurately by
3454:
This piecewise planar and cylindrical embedding has
3368:
3306:
1490:
Five-vertex polyhedral and flat-folded Möbius strips
7752:
7509:Crato, Nuno (2010). "Escher and the Möbius strip".
5664:
5047:"On geometrically realizable Möbius triangulations"
4512:
Renaissance Banff: Mathematics, Music, Art, Culture
3362:
3359:
3324:
3321:
3300:
3297:
551:on the Möbius strip, but not on the plane, are the
6512:
6375:
6349:
6099:. Universitext. Springer, Cham. pp. 152–153.
6088:
6086:
5848:
5837:
5768:
5670:
5186:
5184:
5182:
5180:
4948:
4808:"Tight topological embeddings of the Moebius band"
3708:
3556:
3509:
3477:
2650:
2629:
2609:
2587:
2552:
2360:
2209:
2145:
2111:
2057:
1860:since the initial work on this subject in 1930 by
1812:
1777:
1742:
1680:
1634:
1592:
1449:
1414:
1374:
1335:
1257:
1218:
1194:
1173:
1151:
1115:
1080:
835:swept out by a different motion of a line segment
768:
729:
709:
689:
669:
631:
611:
591:
6228:
5973:(Revised ed.). Springer-Verlag. p. 57.
5857:
5350:Transactions of the American Mathematical Society
5339:
5337:
4623:
4424:Transactions of the American Mathematical Society
4061:Proceedings of the Royal Irish Academy, Section A
2275:
2269:the same topological structure seen in Plücker's
1929:One such example is based on the topology of the
531:The edges and vertices of these six regions form
8927:
8545:
7845:"NASCAR Hall of Fame 'looks fast sitting still'"
7753:Thulaseedas, Jolly; Krawczyk, Robert J. (2003).
6969:. Vol. 84, no. 13. September 25, 1964.
6688:Ramírez Galarza, Ana Irene; Seade, José (2007).
6596:: CS1 maint: bot: original URL status unknown (
6237:
6198:. Universitext. Cham: Springer. pp. 96–98.
5237:
5235:
5044:
5003:Proceedings of the American Mathematical Society
4949:Bonnington, C. Paul; Nakamoto, Atsuhiro (2008).
4638:
4461:
2974:Instituto Nacional de Matemática Pura e Aplicada
38:A Möbius strip made with paper and adhesive tape
7908:"Cut Your Bagel The Mathematically Correct Way"
7316:Proceedings of the National Academy of Sciences
7106:
6658:. Princeton University Press. pp. 85–105.
6129:
6083:
5177:
4474:. Boca Raton, Florida: CRC Press. p. 430.
2960:as was the logo for the environmentally-themed
2766:Scientists have also studied the energetics of
2328:) is one of only five two-dimensional complete
1890:Gluing two Möbius strips to form a Klein bottle
1874:
1753:
1564:
143:The many applications of Möbius strips include
7435:"Visual art and mathematics: the Moebius band"
7221:Cartography and Geographic Information Systems
6135:
6092:
5962:
5334:
5190:
5045:Nakamoto, Atsuhiro; Tsuchiya, Shoichi (2012).
4746:
4510:. In Sarhangi, Reza; Moody, Robert V. (eds.).
4262:"Solution of the Heawood map-coloring problem"
3569:
2704:whose molecular structure forms a cycle, with
559:These include the utility graph, a six-vertex
8531:
8246:
7746:
6714:
6681:
5968:
5343:
5232:
4564:
4542:. Springer-Verlag, New York. pp. 81–83.
4538:Francis, George K. (1987). "Plücker conoid".
4250:
1343:angles so that its center line lies along an
7987:
7985:
7719:
7599:
7597:
7215:
7046:
6916:
6910:
6875:
6720:
6656:Office Hours with a Geometric Group Theorist
6273:
6194:(1992). "4.6 Classification of isometries".
5934:Notices of the American Mathematical Society
5283:
5226:
5137:
4570:
4256:
3983:Journal of Knot Theory and Its Ramifications
3907:Journal of Knot Theory and Its Ramifications
2719:Möbius strip, in a way that cancels its own
1868:that contain rectangular developable Möbius
7874:
7725:
7488:. Oxford University Press. pp. 77–78.
6779:
6754:Modern Differential Geometry for Physicists
6279:
5680:(2nd ed.). Chelsea. pp. 315–316.
5391:
5389:
4419:"Note on the unilateral surface of Moebius"
4322:
3176:) discovered in 1974 in Bach's copy of the
3038:, Möbius strips have been used for slicing
2517:, showing that not every solvmanifold is a
2121:Half of this Klein bottle, the subset with
1496:The Möbius strip can also be embedded as a
1316:Möbius strip in the plane by folding it at
147:that wear evenly on both sides, dual-track
8538:
8524:
8253:
8239:
7821:
7020:
6579:. Archived from the original on 2016-03-13
5730:DDG2019: Visualization course at TU Berlin
5582:
5576:
5457:(2023). "The optimal paper Moebius band".
4742:
4740:
4533:
4531:
4508:"Splitting tori, knots, and Moebius bands"
4496:
4164:
4140:
788:
124:; the flattened Möbius strips include the
8108:
8013:
7982:
7883:"Making a Mobius a matter of mathematics"
7864:"Pedro Reyes Makes an Infinite Love Seat"
7594:
7428:
7426:
7355:
7346:
7336:
6985:
6842:
6678:See in particular Project 7, pp. 104–105.
6500:
6337:
6190:
6184:
6151:
5945:
5930:"The Klein bottle: variations on a theme"
5877:
5858:Schleimer, Saul; Segerman, Henry (2012).
5741:
5739:
5647:
5495:
5493:
5462:
5414:
5361:
5259:
5141:The Mechanics of Ribbons and Möbius Bands
5105:
5090:
5064:
4968:
4902:
4896:
4871:
4825:
4646:(2007). "Lecture 14: Paper Möbius band".
4436:
4305:
4287:
4223:
4221:
4201:
4134:
3995:
3961:"Möbius strips defy a link with infinity"
3919:
3727:
3704:
3702:
3700:
3639:. Thunder's Mouth Press. pp. 28–29.
2836:(1963), depicting ants crawling around a
2540:
2497:, topological spaces having a compatible
1947:Lawson's Klein bottle is a self-crossing
1902:A projection of the Sudanese Möbius strip
8185:. New York: Dover Books. pp. 70–73.
8167:
8129:
8043:
7991:
7976:
7964:
7775:
7713:
7663:
7603:
7565:
7559:
7525:
7502:
7390:
7365:Journal of the American Chemical Society
7156:
7129:
6979:
6953:
6822:
6653:
6647:
6484:
6413:
6047:
6041:
5969:Huggett, Stephen; Jordan, David (2009).
5927:
5921:
5896:
5453:
5395:
5386:
5289:
4848:
4842:
4467:
4093:
4046:
4034:
3979:
3973:
3903:
3897:
3878:
3845:The Two-Year College Mathematics Journal
3829:
3768:
3629:
3420:
3419:rather than a Möbius strip, is given by
3208:application of orbifolds to music theory
2985:
2793:
2674:
2513:, and the Möbius strip can be used as a
2288:of constant positive, negative, or zero
2224:
1303:
639:of vertices, edges, and regions satisfy
329:
70:in 1858, but it had already appeared in
33:
8173:
8037:
7905:
7836:
7822:Wainwright, Oliver (October 17, 2017).
7815:
7657:
7637:
7571:
7531:
7245:
7239:
7135:
7100:
6093:Godinho, Leonor; Natário, José (2014).
5723:
5627:
5621:
5552:
5546:
5502:A Short Course in Differential Topology
5477:
5471:
5447:
5295:
5038:
4942:
4753:"The 9-vertex complex projective plane"
4737:
4598:
4596:
4537:
4528:
4413:
4407:
4170:
3872:
3666:
3662:
3660:
3658:
3656:
3623:
3394:. As is common for words containing an
1789:(more unsolved problems in mathematics)
1593:{\displaystyle {\sqrt {3}}\approx 1.73}
1450:{\displaystyle 1\times {\tfrac {1}{3}}}
1375:{\displaystyle {\sqrt {3}}\approx 1.73}
547:. Another family of graphs that can be
171:, and others, and in the design of the
14:
8928:
8156:
8150:
8135:
8007:
7937:
7924:
7918:
7899:
7880:
7861:
7855:
7781:
7475:
7423:
7189:
6727:Topological Modeling for Visualization
6610:
6604:
6562:
6556:
6001:
5995:
5749:(1970). "Complete minimal surfaces in
5745:
5736:
5719:
5717:
5715:
5700:
5694:
5499:
5490:
5478:Crowell, Rachel (September 12, 2023).
4802:
4796:
4571:Dooner, David B.; Seireg, Ali (1995).
4502:
4371:
4365:
4227:
4218:
4105:
4099:
3774:
3697:
2404:are described as having constant zero
1286:in the form of a self-crossing Möbius
567:can be solved on a transparent Möbius
8519:
8234:
8212:
8114:
7943:
7842:
7682:
7508:
7481:
7432:
7402:
7209:
7183:
7142:. Bloomsbury Publishing. p. 43.
6881:
6750:
6744:
6324:
6318:
5147:. Springer, Dordrecht. pp. 3–6.
5138:Fosdick, Roger; Fried, Eliot (2016).
4999:
4993:
4956:Discrete & Computational Geometry
4859:Discrete & Computational Geometry
3601:
3537:These surfaces have smoothness class
3389:
3114:Six Characters in Search of an Author
3030:into Möbius strips since the work of
2789:
1794:For aspect ratios between this bound
1618:As its aspect ratio decreases toward
1300:Polyhedral surfaces and flat foldings
1290:It has applications in the design of
8773:Geometric Exercises in Paper Folding
7992:Phillips, Tony (November 25, 2016).
7881:Thomas, Nancy J. (October 4, 1998).
7287:
7162:
6991:
6690:Introduction to Classical Geometries
6490:
5902:
4701:"4.2: The trihexaflexagon revisited"
4698:
4692:
4602:
4593:
4109:A Basic Course in Algebraic Topology
4058:
4052:
3958:
3952:
3841:
3653:
3415:Essentially this example, but for a
3214:and Norwegian progressive rock band
2807:Middelheim Open Air Sculpture Museum
1542:Every abstract triangulation of the
29:Non-orientable surface with one edge
8794:A History of Folding in Mathematics
7994:"Bach and the musical Möbius strip"
7788:Journal of Mathematics and the Arts
7664:Millward, Steven (April 30, 2012).
7396:
5712:
3835:
2146:{\displaystyle 0\leq \phi <\pi }
677:. For instance, Tietze's graph has
539:on this surface for the six-vertex
24:
8163:. Kangaroo Flat: Third Hemisphere.
7944:Chang, Kenneth (January 9, 2012).
7925:Miller, Ross (September 5, 2014).
7862:Gopnik, Blake (October 17, 2014).
7785:(January 2018). "Möbius bridges".
6357:with total curvature greater than
3959:Lamb, Evelyn (February 20, 2019).
3502:
3198:. Because the Möbius strip is the
2734:patterns in light emerging from a
2431:
2194:
1681:{\displaystyle \pi /2\approx 1.57}
545:drawn without crossings on a plane
25:
8962:
8193:
8115:Parks, Andrew (August 30, 2007).
8019:"Music reduced to beautiful math"
7755:"Möbius concepts in architecture"
7403:Gitig, Diana (October 18, 2010).
7252:The American Mathematical Monthly
5903:Gunn, Charles (August 23, 2018).
5299:The American Mathematical Monthly
4331:Journal of Mathematical Chemistry
2342:upper half plane (Poincaré) model
128:. The Sudanese Möbius strip is a
8199:
8136:Lawson, Dom (February 9, 2021).
8049:"The geometry of musical chords"
7532:Kersten, Erik (March 13, 2017).
6513:{\displaystyle \mathbb {R} ^{n}}
6350:{\displaystyle \mathbb {R} ^{3}}
5344:Halpern, B.; Weaver, C. (1977).
4813:Journal of Differential Geometry
3607:Longman Pronunciation Dictionary
3575:
3572:, p. 116, following Theorem 2.2.
3355:
3293:
2968:and the original version of the
2927:
2912:
2898:
2553:{\displaystyle \mathbb {R} ^{n}}
2316:, and (together with the plane,
1895:
1883:
1510:four-dimensional regular simplex
1480:
1471:
1116:{\displaystyle 0\leq u<2\pi }
819:
797:
780:
497:
482:
447:
435:
276:Another mosaic from the town of
231:
211:
8694:Alexandrov's uniqueness theorem
7906:Pashman, Dan (August 6, 2015).
7195:"A world map on a Möbius strip"
7169:. Chartwell Books. p. 20.
6522:Journal of Geometry and Physics
6283:American Journal of Mathematics
5674:; Cohn-Vossen, Stephan (1952).
3883:. Academic Press. p. 195.
3531:
3522:
3448:
3435:
3426:
3409:
3068:s "No-Sided Professor" (1946),
2832:biting each others' tails; and
2828:(1961), depicting three folded
2670:
2570:of its action; contracting the
2210:{\displaystyle \mathrm {O} (2)}
1754:Unsolved problem in mathematics
74:mosaics from the third century
8183:Mathematics, Magic and Mystery
7513:. Springer. pp. 123–126.
7433:Emmer, Michele (Spring 1980).
7265:10.1080/00029890.1940.11990957
6542:10.1016/j.geomphys.2005.08.001
6462:10.1103/PhysRevLett.114.127801
5724:Knöppel, Felix (Summer 2019).
5635:Pacific Journal of Mathematics
5312:10.1080/00029890.1990.11995680
4761:The Mathematical Intelligencer
3715:The Mathematical Intelligencer
3595:
3279:
3046:, and creating new shapes for
3000:National Library of Kazakhstan
2501:describing the composition of
2413:after its 1982 description by
2276:Surfaces of constant curvature
2204:
2198:
2052:
1962:
1864:. It is also possible to find
1252:
1234:
1152:{\displaystyle -1\leq v\leq 1}
1038:
1026:
958:
946:
878:
866:
346:Relatedly, when embedded into
241:with a Möbius drive chain, by
138:surfaces of constant curvature
13:
1:
8632:Regular paperfolding sequence
7801:10.1080/17513472.2017.1419331
6724:; Kunii, Tosiyasu L. (2013).
6399:10.1215/S0012-7094-81-04829-8
4908:American Mathematical Monthly
4376:; Woodrow, Robert E. (eds.).
3588:
2770:shaped as Möbius strips, the
2448:by adding one more line, the
2280:The open Möbius strip is the
2219:the group of symmetries of a
848:defined by equations for the
325:
8780:Geometric Folding Algorithms
8547:Mathematics of paper folding
8117:"Mobius Band: Friendly Fire"
7653:. March 12, 1972. p. 1.
7519:10.1007/978-3-642-04833-3_29
7123:10.1016/0165-1765(94)90045-0
6961:"Making resistors with math"
5677:Geometry and the Imagination
4144:(1892). "Paradromic rings".
3879:Blackett, Donald W. (1982).
3570:Bartels & Hornung (2015)
2265:at each end of the crossing
1875:Making the boundary circular
1565:Smoothly embedded rectangles
1312:A strip of paper can form a
1308:Trihexaflexagon being flexed
1282:or cylindroid, an algebraic
7:
7843:Muret, Don (May 17, 2010).
6861:10.1016/j.physe.2003.12.100
6563:Parker, Phillip E. (1993).
6520:and the Björling problem".
5928:Franzoni, Gregorio (2012).
5838:{\displaystyle \tau _{1,2}}
4260:; Youngs, J. W. T. (1968).
4106:Massey, William S. (1991).
3510:{\displaystyle C^{\infty }}
3443:abstract simplicial complex
3398:, it is also often spelled
3236:
2816:(memorialized in a poem by
2750:Möbius loop roller coasters
2415:William Hamilton Meeks, III
1832:continuously differentiable
1813:{\displaystyle {\sqrt {3}}}
1635:{\displaystyle {\sqrt {3}}}
1614:, that can bend but cannot
1336:{\displaystyle 60^{\circ }}
10:
8967:
8830:Margherita Piazzola Beloch
7891:. p. aa3 – via
7645:"Expo '74 symbol selected"
7233:10.1559/152304091783786781
6567:. In Del Riego, L. (ed.).
5586:Monatshefte für Mathematik
5425:10.1007/s10711-021-00648-5
5284:Fosdick & Fried (2016)
5227:Fosdick & Fried (2016)
5066:10.1016/j.disc.2011.06.007
4468:Junghenn, Hugo D. (2015).
3172:, the fifth of 14 canons (
3020:mathematics and fiber arts
2465:The space of lines in the
1778:{\displaystyle 12\times 7}
1226:-plane and is centered at
202:
8817:
8764:
8743:
8686:
8640:
8609:
8601:Yoshizawa–Randlett system
8553:
8473:
8445:
8410:
8401:
8347:
8302:
8273:
8266:
7572:Brecher, Kenneth (2017).
6798:10.1007/978-3-642-57999-8
6730:. Springer. p. 269.
6386:Duke Mathematical Journal
6204:10.1007/978-1-4612-0929-4
6162:10.1007/s10711-014-9975-1
6105:10.1007/978-3-319-08666-8
5261:10.1007/s10659-014-9495-0
5205:10.1007/s10659-014-9501-6
5153:10.1007/978-94-017-7300-3
5116:10.1007/s10659-014-9490-5
4970:10.1007/s00454-007-9035-9
4873:10.1007/s00454-007-9033-y
4540:A Topological Picturebook
4471:A Course in Real Analysis
4343:10.1007/s10910-011-9884-6
4006:10.1142/s0218216519710019
3930:10.1142/S0218216518420051
3746:10.1007/s00283-016-9631-8
3609:(3rd ed.). Longman.
3487:infinitely differentiable
2870:Möbius strip was used in
1415:{\displaystyle 1\times 1}
769:{\displaystyle 12-18+6=0}
8941:Recreational mathematics
8801:Origami Polyhedra Design
7678:– via Yahoo! News.
7136:Easdown, Martin (2012).
6751:Isham, Chris J. (1999).
6427:; Alexander, Gareth P.;
5649:10.2140/pjm.1990.143.195
5628:Schwarz, Gideon (1990).
4713:10.1017/CBO9780511543302
3794:10.1017/CBO9780511626272
3710:Cartwright, Julyan H. E.
3272:
3261:Smale–Williams attractor
2469:can be parameterized by
2438:topologically equivalent
2171:Stereographic projection
1915:topologically equivalent
1848:tautological line bundle
561:complete bipartite graph
103:topologically equivalent
78:. The Möbius strip is a
8392:Sphere with three holes
8078:10.1126/science.1126287
7849:Sports Business Journal
7338:10.1073/pnas.1015997107
7199:Surveying & Mapping
7078:10.1126/science.1260635
6441:Physical Review Letters
4606:Experiments in Topology
3034:in the early 1980s. In
1913:, of a Möbius strip is
1258:{\displaystyle (0,0,0)}
789:Sweeping a line segment
670:{\displaystyle V-E+F=0}
575:of the Möbius strip is
565:three utilities problem
506:three utilities problem
258:August Ferdinand Möbius
254:Johann Benedict Listing
179:. Performers including
68:August Ferdinand Möbius
64:Johann Benedict Listing
8591:Napkin folding problem
8157:Prevos, Peter (2018).
7163:Hook, Patrick (2019).
6514:
6377:
6376:{\displaystyle -8\pi }
6351:
6248:A Textbook of Topology
5971:A Topological Aperitif
5905:"Sudanese Möbius Band"
5839:
5770:
5397:Schwartz, Richard Evan
4827:10.4310/jdg/1214430493
4603:Barr, Stephen (1964).
4573:"3.4.2 The cylindroid"
3558:
3511:
3479:
3102:In Search of Lost Time
3079:" (1950) and the film
3042:, making loops out of
2994:
2842:mathematical sculpture
2809:
2683:
2652:
2631:
2611:
2589:
2554:
2486:Möbius transformations
2481:affine transformations
2362:
2234:
2211:
2147:
2113:
2059:
1844:unbounded Möbius strip
1814:
1779:
1744:
1682:
1636:
1594:
1451:
1416:
1376:
1337:
1309:
1259:
1220:
1196:
1175:
1153:
1117:
1082:
770:
731:
711:
691:
671:
633:
613:
593:
414:deformation retraction
390:object with right- or
340:non-orientable surface
335:
226:holding a Möbius strip
39:
8310:Real projective plane
8295:Pretzel (genus 3) ...
7728:Praxis der Mathematik
7295:Goldstein, Raymond E.
6664:10.1515/9781400885398
6565:"Spaces of geodesics"
6515:
6425:Goldstein, Raymond E.
6378:
6352:
6326:Meeks, William H. III
6296:10.1353/ajm.1997.0004
5840:
5779:Annals of Mathematics
5771:
5769:{\displaystyle S^{3}}
5747:Lawson, H. Blaine Jr.
5514:10.1017/9781108349130
5248:Journal of Elasticity
5193:Journal of Elasticity
5094:Journal of Elasticity
4289:10.1073/pnas.60.2.438
3631:Pickover, Clifford A.
3559:
3557:{\displaystyle C^{1}}
3512:
3480:
3478:{\displaystyle C^{2}}
3127:It's a Wonderful Life
3091:William Hazlett Upson
3077:A Subway Named Mobius
2989:
2844:, including works by
2820:), and two prints by
2797:
2679:Electrical flow in a
2678:
2653:
2632:
2630:{\displaystyle \ell }
2612:
2610:{\displaystyle \ell }
2590:
2555:
2446:real projective plane
2363:
2228:
2212:
2148:
2114:
2060:
1815:
1780:
1745:
1683:
1637:
1595:
1452:
1422:strip would become a
1417:
1377:
1338:
1307:
1260:
1221:
1197:
1176:
1154:
1118:
1083:
850:Cartesian coordinates
771:
732:
712:
692:
672:
634:
614:
594:
525:Ringel–Youngs theorem
422:infinite cyclic group
333:
280:(depicted) shows the
37:
8751:Fold-and-cut theorem
8707:Steffen's polyhedron
8571:Huzita–Hatori axioms
8561:Big-little-big lemma
8465:Euler characteristic
8208:at Wikimedia Commons
7650:The Spokesman-Review
7609:"Recycling topology"
7538:Escher in the Palace
7482:Byers, Mark (2018).
7139:Amusement Park Rides
6722:Fomenko, Anatolij T.
6495:
6361:
6332:
6196:Geometry of Surfaces
5816:
5753:
5052:Discrete Mathematics
4705:Flexagons Inside Out
3541:
3494:
3462:
3227:Harry Blackstone Sr.
3155:(1975) and the film
3139:Lost in the Funhouse
3070:Armin Joseph Deutsch
3032:Elizabeth Zimmermann
2782:Möbius strips using
2745:social choice theory
2642:
2621:
2601:
2579:
2535:
2352:
2190:
2166:to all of the swept
2125:
2070:
1959:
1800:
1763:
1701:
1658:
1622:
1574:
1538:, with a triangular
1426:
1400:
1356:
1345:equilateral triangle
1320:
1231:
1207:
1186:
1165:
1161:where one parameter
1128:
1092:
856:
742:
721:
701:
681:
643:
623:
603:
583:
573:Euler characteristic
218:Mosaic from ancient
181:Harry Blackstone Sr.
161:social choice theory
8951:Eponyms in geometry
8699:Flexible polyhedron
8070:2006Sci...313...72T
7979:, pp. 179–187.
7967:, pp. 174–177.
7888:The Times (Trenton)
7716:, pp. 156–157.
7696:. IMPA. May 7, 2020
7378:10.1021/ja00375a051
7329:2010PNAS..10721979G
7323:(51): 21979–21984.
7070:2015Sci...347..964B
7006:2000ITMTT..48.2465P
6931:2009NatCh...1..113Y
6853:2004PhyE...22..688Y
6613:Journal of Geometry
6534:2006JGP....56.1506M
6454:2015PhRvL.114l7801P
6139:Geometriae Dedicata
5484:Scientific American
5402:Geometriae Dedicata
4804:Kuiper, Nicolaas H.
4280:1968PNAS...60..438R
4194:1923Natur.111R.882B
3738:2016arXiv160907779C
3683:1973AmSci..61..544L
3566:Nash–Kuiper theorem
3391:[ˈmøːbi̯ʊs]
3231:Thomas Nelson Downs
3200:configuration space
3179:Goldberg Variations
3057:as the basis for a
3055:speculative fiction
3004:NASCAR Hall of Fame
2991:NASCAR Hall of Fame
2568:stabilizer subgroup
2499:algebraic structure
2440:to the open Möbius
2286:Riemannian geometry
1836:Nash–Kuiper theorem
1612:developable surface
1457:folded strip whose
352:orientable surfaces
197:speculative fiction
185:Thomas Nelson Downs
177:NASCAR Hall of Fame
118:developable surface
8880:Toshikazu Kawasaki
8703:Bricard octahedron
8678:Yoshimura buckling
8576:Kawasaki's theorem
8292:Number 8 (genus 2)
8214:Weisstein, Eric W.
8179:"The Afghan Bands"
7951:The New York Times
6625:10.1007/BF01229209
6510:
6373:
6347:
6243:Threlfall, William
6049:Richeson, David S.
5835:
5766:
5599:10.1007/BF01299052
4774:10.1007/BF03026567
4699:Pook, Les (2003).
4644:Tabachnikov, Serge
4176:"Paradromic rings"
3670:American Scientist
3554:
3507:
3475:
2995:
2810:
2790:In popular culture
2772:chemical synthesis
2706:molecular orbitals
2698:Möbius aromaticity
2684:
2648:
2627:
2607:
2585:
2550:
2454:projective duality
2382:Positive curvature
2358:
2337:Negative curvature
2290:Gaussian curvature
2235:
2207:
2143:
2109:
2055:
1866:algebraic surfaces
1810:
1775:
1740:
1678:
1632:
1590:
1506:simplicial complex
1498:polyhedral surface
1447:
1445:
1412:
1372:
1333:
1310:
1255:
1219:{\displaystyle xy}
1216:
1192:
1171:
1149:
1113:
1078:
1076:
846:parametric surface
766:
727:
710:{\displaystyle 18}
707:
690:{\displaystyle 12}
687:
667:
629:
609:
589:
517:four color theorem
336:
284:, held by the god
40:
8923:
8922:
8787:Geometric Origami
8658:Paper bag problem
8581:Maekawa's theorem
8513:
8512:
8509:
8508:
8343:
8342:
8204:Media related to
8138:"Ring Van Möbius"
7690:"Símbolo do IMPA"
7587:978-1-938664-22-9
7574:"Art of infinity"
7393:, pp. 52–58.
7372:(11): 3219–3221.
7309:(December 2010).
7303:Pesci, Adriana I.
7299:Moffatt, H. Keith
7110:Economics Letters
7064:(6225): 964–966.
7036:Microwave Journal
7014:10.1109/22.898999
7000:(12): 2465–2471.
6982:, pp. 45–46.
6939:10.1038/nchem.172
6896:10.1021/cr030092l
6890:(10): 3697–3715.
6699:978-3-7643-7517-1
6429:Moffatt, H. Keith
6421:Pesci, Adriana I.
6114:978-3-319-08665-1
6068:978-0-691-12677-7
6026:978-1-4704-2535-7
5980:978-1-84800-912-7
5889:978-1-938664-00-7
5782:. Second Series.
5687:978-0-8284-1087-8
5523:978-1-108-42579-7
5455:Schwartz, Richard
5162:978-94-017-7299-0
5059:(14): 2135–2139.
4904:Tuckerman, Bryant
4670:978-0-8218-4316-1
4481:978-1-4822-1927-2
4415:Maschke, Heinrich
4337:(10): 2250–2267.
4142:Rouse Ball, W. W.
3990:(7): 1971001, 3.
3914:(9): 1842005, 9.
3646:978-1-56025-826-1
3616:978-1-4058-8118-0
2805:, 1956, from the
2702:organic chemicals
2651:{\displaystyle x}
2588:{\displaystyle x}
2507:homogeneous space
2409:the Meeks Möbius
2361:{\displaystyle x}
2282:relative interior
1808:
1732:
1730:
1712:
1630:
1582:
1444:
1364:
1195:{\displaystyle v}
1174:{\displaystyle u}
1072:
1056:
1003:
987:
923:
907:
730:{\displaystyle 6}
632:{\displaystyle F}
612:{\displaystyle E}
592:{\displaystyle V}
508:on a Möbius strip
418:fundamental group
361:Cartesian product
266:third century CE.
91:topological space
16:(Redirected from
8958:
8860:David A. Huffman
8825:Roger C. Alperin
8728:Source unfolding
8596:Pureland origami
8540:
8533:
8526:
8517:
8516:
8428:Triangulatedness
8408:
8407:
8271:
8270:
8267:Without boundary
8255:
8248:
8241:
8232:
8231:
8227:
8226:
8203:
8187:
8186:
8171:
8165:
8164:
8154:
8148:
8147:
8133:
8127:
8126:
8112:
8106:
8105:
8053:
8047:(July 7, 2006).
8045:Tymoczko, Dmitri
8041:
8035:
8034:
8032:
8031:
8015:Moskowitz, Clara
8011:
8005:
8003:
7989:
7980:
7974:
7968:
7962:
7956:
7955:
7941:
7935:
7934:
7922:
7916:
7915:
7903:
7897:
7896:
7878:
7872:
7871:
7859:
7853:
7852:
7840:
7834:
7833:
7819:
7813:
7812:
7795:(2–3): 181–194.
7783:Séquin, Carlo H.
7779:
7773:
7772:
7750:
7744:
7743:
7723:
7717:
7711:
7705:
7704:
7702:
7701:
7686:
7680:
7679:
7677:
7676:
7661:
7655:
7654:
7641:
7635:
7634:
7601:
7592:
7591:
7569:
7563:
7557:
7548:
7547:
7545:
7544:
7534:"Möbius Strip I"
7529:
7523:
7522:
7506:
7500:
7499:
7479:
7473:
7472:
7430:
7421:
7420:
7418:
7417:
7400:
7394:
7388:
7382:
7381:
7359:
7353:
7352:
7350:
7340:
7291:
7285:
7284:
7247:Courant, Richard
7243:
7237:
7236:
7213:
7207:
7206:
7191:Tobler, Waldo R.
7187:
7181:
7180:
7160:
7154:
7153:
7133:
7127:
7126:
7104:
7098:
7097:
7050:
7044:
7043:
7033:
7024:
7018:
7017:
6989:
6983:
6977:
6971:
6970:
6957:
6951:
6950:
6919:Nature Chemistry
6914:
6908:
6907:
6884:Chemical Reviews
6879:
6873:
6872:
6846:
6844:cond-mat/0309636
6837:(1–3): 688–691.
6826:
6820:
6819:
6783:
6777:
6776:
6748:
6742:
6741:
6718:
6712:
6711:
6685:
6679:
6677:
6651:
6645:
6644:
6608:
6602:
6601:
6595:
6587:
6585:
6584:
6560:
6554:
6553:
6528:(9): 1506–1515.
6519:
6517:
6516:
6511:
6509:
6508:
6503:
6488:
6482:
6481:
6437:
6417:
6411:
6410:
6382:
6380:
6379:
6374:
6356:
6354:
6353:
6348:
6346:
6345:
6340:
6322:
6316:
6315:
6277:
6271:
6270:
6239:Seifert, Herbert
6235:
6226:
6225:
6188:
6182:
6181:
6155:
6133:
6127:
6126:
6090:
6081:
6080:
6045:
6039:
6038:
5999:
5993:
5992:
5966:
5960:
5959:
5949:
5940:(8): 1076–1082.
5925:
5919:
5918:
5916:
5915:
5900:
5894:
5893:
5881:
5865:
5855:
5846:
5844:
5842:
5841:
5836:
5834:
5833:
5811:
5775:
5773:
5772:
5767:
5765:
5764:
5743:
5734:
5733:
5721:
5710:
5709:
5698:
5692:
5691:
5668:
5662:
5661:
5651:
5625:
5619:
5618:
5580:
5574:
5573:
5550:
5544:
5543:
5497:
5488:
5487:
5475:
5469:
5468:
5466:
5451:
5445:
5444:
5418:
5393:
5384:
5383:
5365:
5341:
5332:
5331:
5293:
5287:
5281:
5263:
5239:
5230:
5224:
5199:(1–2): 113–136.
5188:
5175:
5174:
5146:
5135:
5109:
5088:
5079:
5078:
5068:
5042:
5036:
5035:
4997:
4991:
4990:
4972:
4946:
4940:
4939:
4900:
4894:
4893:
4875:
4846:
4840:
4839:
4829:
4800:
4794:
4793:
4757:
4744:
4735:
4734:
4696:
4690:
4689:
4687:
4681:. Archived from
4654:
4636:
4621:
4620:
4600:
4591:
4590:
4568:
4562:
4561:
4535:
4526:
4525:
4504:Séquin, Carlo H.
4500:
4494:
4493:
4465:
4459:
4458:
4440:
4411:
4405:
4402:Figure 7, p. 292
4399:
4369:
4363:
4362:
4326:
4320:
4319:
4309:
4291:
4254:
4248:
4247:
4237:
4229:Tietze, Heinrich
4225:
4216:
4215:
4205:
4203:10.1038/111882b0
4168:
4162:
4161:
4138:
4132:
4131:
4103:
4097:
4091:
4085:
4084:
4056:
4050:
4044:
4038:
4032:
4026:
4025:
3999:
3977:
3971:
3970:
3956:
3950:
3949:
3923:
3901:
3895:
3894:
3876:
3870:
3869:
3839:
3833:
3827:
3816:
3815:
3772:
3766:
3765:
3731:
3706:
3695:
3694:
3664:
3651:
3650:
3627:
3621:
3620:
3599:
3582:
3579:
3573:
3563:
3561:
3560:
3555:
3553:
3552:
3535:
3529:
3526:
3520:
3518:
3516:
3514:
3513:
3508:
3506:
3505:
3484:
3482:
3481:
3476:
3474:
3473:
3452:
3446:
3439:
3433:
3430:
3424:
3413:
3407:
3393:
3388:
3384:
3383:
3380:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3354:
3343:
3339:
3334:
3333:
3330:
3329:
3326:
3323:
3318:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3292:
3283:
3249:Penrose triangle
3196:chromatic circle
3148:
3144:Samuel R. Delany
3136:
3123:
3111:
3107:Luigi Pirandello
3099:
3087:Arthur C. Clarke
3074:
3067:
2983:
2967:
2959:
2950:three-arrow logo
2931:
2922:logo (2012–2014)
2916:
2906:Recycling symbol
2902:
2883:Charles O. Perry
2876:
2700:, a property of
2667:
2659:
2657:
2655:
2654:
2649:
2636:
2634:
2633:
2628:
2616:
2614:
2613:
2608:
2596:
2594:
2592:
2591:
2586:
2561:
2559:
2557:
2556:
2551:
2549:
2548:
2543:
2504:
2493:
2489:
2476:
2467:hyperbolic plane
2464:
2460:
2450:line at infinity
2443:
2428:
2423:Björling problem
2420:
2412:
2402:minimal surfaces
2396:
2387:projective plane
2379:
2375:
2367:
2365:
2364:
2359:
2346:hyperbolic plane
2334:
2314:glide reflection
2312:of a plane by a
2306:
2272:
2268:
2263:Whitney umbrella
2256:
2248:
2222:
2218:
2216:
2214:
2213:
2208:
2197:
2184:orthogonal group
2181:
2176:
2169:
2164:
2160:
2152:
2150:
2149:
2144:
2120:
2118:
2116:
2115:
2110:
2064:
2062:
2061:
2056:
1953:unit hypersphere
1944:
1940:
1928:
1899:
1887:
1871:
1862:Michael Sadowsky
1853:
1841:
1828:Richard Schwartz
1825:
1821:
1819:
1817:
1816:
1811:
1809:
1804:
1784:
1782:
1781:
1776:
1755:
1749:
1747:
1746:
1741:
1733:
1731:
1726:
1715:
1713:
1705:
1697:
1693:
1689:
1687:
1685:
1684:
1679:
1668:
1652:
1645:
1641:
1639:
1638:
1633:
1631:
1626:
1617:
1609:
1605:
1601:
1599:
1597:
1596:
1591:
1583:
1578:
1561:
1557:
1553:
1549:
1544:projective plane
1541:
1533:
1523:
1484:
1475:
1464:
1456:
1454:
1453:
1448:
1446:
1437:
1421:
1419:
1418:
1413:
1395:
1387:
1383:
1381:
1379:
1378:
1373:
1365:
1360:
1342:
1340:
1339:
1334:
1332:
1331:
1296:
1289:
1280:Plücker's conoid
1275:
1266:
1264:
1262:
1261:
1256:
1225:
1223:
1222:
1217:
1202:
1201:
1199:
1198:
1193:
1180:
1178:
1177:
1172:
1160:
1158:
1156:
1155:
1150:
1122:
1120:
1119:
1114:
1087:
1085:
1084:
1079:
1077:
1073:
1065:
1057:
1049:
1009:
1005:
1004:
996:
988:
980:
929:
925:
924:
916:
908:
900:
843:
833:Plücker's conoid
823:
801:
777:
775:
773:
772:
767:
736:
734:
733:
728:
716:
714:
713:
708:
696:
694:
693:
688:
676:
674:
673:
668:
638:
636:
635:
630:
618:
616:
615:
610:
598:
596:
595:
590:
570:
558:
530:
522:
504:Solution to the
501:
486:
475:
465:
451:
439:
409:
402:
398:
393:
392:left-handedness.
385:
378:
366:
358:
345:
322:
318:
313:Ismail al-Jazari
304:mechanical belts
299:
275:
267:
263:
243:Ismail al-Jazari
235:
215:
173:recycling symbol
155:printed so that
145:mechanical belts
21:
8966:
8965:
8961:
8960:
8959:
8957:
8956:
8955:
8926:
8925:
8924:
8919:
8905:Joseph O'Rourke
8840:Robert Connelly
8813:
8760:
8739:
8682:
8668:Schwarz lantern
8653:Modular origami
8636:
8605:
8549:
8544:
8514:
8505:
8469:
8446:Characteristics
8441:
8403:
8397:
8339:
8298:
8262:
8259:
8196:
8191:
8190:
8175:Gardner, Martin
8172:
8168:
8155:
8151:
8134:
8130:
8113:
8109:
8051:
8042:
8038:
8029:
8027:
8017:(May 6, 2008).
8012:
8008:
7990:
7983:
7977:Pickover (2005)
7975:
7971:
7965:Pickover (2005)
7963:
7959:
7942:
7938:
7923:
7919:
7904:
7900:
7879:
7875:
7860:
7856:
7841:
7837:
7820:
7816:
7780:
7776:
7769:
7751:
7747:
7724:
7720:
7714:Pickover (2005)
7712:
7708:
7699:
7697:
7688:
7687:
7683:
7674:
7672:
7662:
7658:
7643:
7642:
7638:
7623:
7605:Peterson, Ivars
7602:
7595:
7588:
7570:
7566:
7560:Pickover (2005)
7558:
7551:
7542:
7540:
7530:
7526:
7507:
7503:
7496:
7480:
7476:
7453:10.2307/1577979
7431:
7424:
7415:
7413:
7401:
7397:
7391:Pickover (2005)
7389:
7385:
7360:
7356:
7307:Ricca, Renzo L.
7292:
7288:
7244:
7240:
7217:Kumler, Mark P.
7214:
7210:
7188:
7184:
7177:
7161:
7157:
7150:
7134:
7130:
7105:
7101:
7051:
7047:
7031:
7025:
7021:
6990:
6986:
6980:Pickover (2005)
6978:
6974:
6959:
6958:
6954:
6915:
6911:
6880:
6876:
6827:
6823:
6808:
6784:
6780:
6765:
6749:
6745:
6738:
6719:
6715:
6700:
6686:
6682:
6674:
6652:
6648:
6609:
6605:
6589:
6588:
6582:
6580:
6561:
6557:
6504:
6499:
6498:
6496:
6493:
6492:
6489:
6485:
6435:
6418:
6414:
6362:
6359:
6358:
6341:
6336:
6335:
6333:
6330:
6329:
6323:
6319:
6278:
6274:
6259:
6236:
6229:
6214:
6192:Stillwell, John
6189:
6185:
6134:
6130:
6115:
6091:
6084:
6069:
6046:
6042:
6027:
6017:10.1090/mbk/096
6000:
5996:
5981:
5967:
5963:
5947:10.1090/noti880
5926:
5922:
5913:
5911:
5901:
5897:
5890:
5861:
5860:"Sculptures in
5856:
5849:
5823:
5819:
5817:
5814:
5813:
5792:10.2307/1970625
5760:
5756:
5754:
5751:
5750:
5744:
5737:
5722:
5713:
5702:Spivak, Michael
5699:
5695:
5688:
5669:
5665:
5626:
5622:
5581:
5577:
5554:Blanuša, Danilo
5551:
5547:
5524:
5498:
5491:
5476:
5472:
5452:
5448:
5394:
5387:
5363:10.2307/1997711
5342:
5335:
5306:(10): 890–897.
5294:
5290:
5254:(1–2): 67–112.
5240:
5233:
5189:
5178:
5163:
5144:
5089:
5082:
5043:
5039:
5016:10.2307/2045508
4998:
4994:
4947:
4943:
4920:10.2307/2305482
4901:
4897:
4850:Szilassi, Lajos
4847:
4843:
4801:
4797:
4755:
4749:Banchoff, T. F.
4745:
4738:
4723:
4697:
4693:
4685:
4671:
4661:10.1090/mbk/046
4652:
4637:
4624:
4617:
4601:
4594:
4587:
4569:
4565:
4550:
4536:
4529:
4522:
4501:
4497:
4482:
4466:
4462:
4438:10.2307/1986401
4412:
4408:
4388:
4374:Guy, Richard K.
4370:
4366:
4327:
4323:
4255:
4251:
4235:
4226:
4219:
4169:
4165:
4158:
4139:
4135:
4120:
4104:
4100:
4094:Pickover (2005)
4092:
4088:
4057:
4053:
4047:Pickover (2005)
4045:
4041:
4035:Pickover (2005)
4033:
4029:
3978:
3974:
3966:Quanta Magazine
3957:
3953:
3902:
3898:
3891:
3877:
3873:
3858:10.2307/3026946
3840:
3836:
3832:, pp. 8–9.
3830:Pickover (2005)
3828:
3819:
3804:
3773:
3769:
3707:
3698:
3665:
3654:
3647:
3628:
3624:
3617:
3600:
3596:
3591:
3586:
3585:
3580:
3576:
3548:
3544:
3542:
3539:
3538:
3536:
3532:
3527:
3523:
3501:
3497:
3495:
3492:
3491:
3489:
3469:
3465:
3463:
3460:
3459:
3453:
3449:
3440:
3436:
3431:
3427:
3421:Blackett (1982)
3414:
3410:
3386:
3358:
3349:
3348:
3341:
3337:
3320:
3296:
3287:
3286:
3284:
3280:
3275:
3239:
3216:Ring Van Möbius
3204:two-note chords
3146:
3134:
3121:
3109:
3097:
3072:
3065:
2981:
2965:
2957:
2948:. The familiar
2942:
2941:
2940:
2939:
2938:
2932:
2924:
2923:
2917:
2909:
2908:
2903:
2874:
2868:trefoil-knotted
2792:
2721:self-inductance
2713:Möbius resistor
2681:Möbius resistor
2673:
2665:
2660:Therefore, the
2643:
2640:
2639:
2638:
2622:
2619:
2618:
2602:
2599:
2598:
2580:
2577:
2576:
2575:
2544:
2539:
2538:
2536:
2533:
2532:
2530:
2502:
2491:
2484:
2474:
2471:unordered pairs
2462:
2458:
2441:
2434:
2432:Spaces of lines
2426:
2418:
2410:
2394:
2377:
2373:
2353:
2350:
2349:
2329:
2304:
2278:
2270:
2266:
2261:like that of a
2254:
2246:
2220:
2193:
2191:
2188:
2187:
2186:
2179:
2174:
2167:
2162:
2158:
2126:
2123:
2122:
2071:
2068:
2067:
2066:
1960:
1957:
1956:
1949:minimal surface
1942:
1938:
1926:
1907:
1906:
1905:
1904:
1903:
1900:
1892:
1891:
1888:
1877:
1869:
1851:
1839:
1823:
1803:
1801:
1798:
1797:
1795:
1792:
1791:
1786:
1764:
1761:
1760:
1757:
1725:
1714:
1704:
1702:
1699:
1698:
1695:
1691:
1664:
1659:
1656:
1655:
1654:
1650:
1643:
1625:
1623:
1620:
1619:
1615:
1607:
1603:
1577:
1575:
1572:
1571:
1570:
1567:
1559:
1555:
1551:
1547:
1539:
1531:
1521:
1494:
1493:
1492:
1491:
1487:
1486:
1485:
1477:
1476:
1462:
1435:
1427:
1424:
1423:
1401:
1398:
1397:
1393:
1390:trihexaflexagon
1385:
1359:
1357:
1354:
1353:
1352:
1327:
1323:
1321:
1318:
1317:
1302:
1291:
1287:
1273:
1232:
1229:
1228:
1227:
1208:
1205:
1204:
1187:
1184:
1183:
1182:
1166:
1163:
1162:
1129:
1126:
1125:
1124:
1093:
1090:
1089:
1075:
1074:
1064:
1048:
1041:
1020:
1019:
995:
979:
972:
968:
961:
940:
939:
915:
899:
892:
888:
881:
859:
857:
854:
853:
852:of its points,
841:
838:
837:
836:
831:
826:
825:
824:
813:
812:
811:
809:
804:
803:
802:
791:
783:
743:
740:
739:
738:
722:
719:
718:
702:
699:
698:
682:
679:
678:
644:
641:
640:
624:
621:
620:
604:
601:
600:
584:
581:
580:
568:
556:
528:
520:
513:
512:
511:
510:
509:
502:
494:
493:
487:
470:
463:
459:
458:
457:
456:
455:
452:
444:
443:
440:
404:
400:
396:
391:
383:
376:
369:uncountable set
364:
356:
348:Euclidean space
343:
328:
320:
316:
297:
273:
270:untwisted rings
265:
261:
250:
249:
248:
247:
246:
236:
228:
227:
216:
205:
149:roller coasters
130:minimal surface
126:trihexaflexagon
95:Euclidean space
89:As an abstract
30:
23:
22:
15:
12:
11:
5:
8964:
8954:
8953:
8948:
8943:
8938:
8921:
8920:
8918:
8917:
8912:
8910:Tomohiro Tachi
8907:
8902:
8897:
8892:
8887:
8885:Robert J. Lang
8882:
8877:
8875:Humiaki Huzita
8872:
8867:
8862:
8857:
8855:Rona Gurkewitz
8852:
8850:Martin Demaine
8847:
8842:
8837:
8832:
8827:
8821:
8819:
8815:
8814:
8812:
8811:
8804:
8797:
8790:
8783:
8776:
8768:
8766:
8762:
8761:
8759:
8758:
8753:
8747:
8745:
8741:
8740:
8738:
8737:
8736:
8735:
8733:Star unfolding
8730:
8725:
8720:
8710:
8696:
8690:
8688:
8684:
8683:
8681:
8680:
8675:
8670:
8665:
8660:
8655:
8650:
8644:
8642:
8638:
8637:
8635:
8634:
8629:
8624:
8619:
8613:
8611:
8607:
8606:
8604:
8603:
8598:
8593:
8588:
8583:
8578:
8573:
8568:
8566:Crease pattern
8563:
8557:
8555:
8551:
8550:
8543:
8542:
8535:
8528:
8520:
8511:
8510:
8507:
8506:
8504:
8503:
8498:
8492:
8486:
8483:
8477:
8475:
8471:
8470:
8468:
8467:
8462:
8457:
8449:
8447:
8443:
8442:
8440:
8439:
8434:
8425:
8420:
8414:
8412:
8405:
8399:
8398:
8396:
8395:
8389:
8388:
8387:
8377:
8376:
8375:
8370:
8362:
8361:
8360:
8351:
8349:
8345:
8344:
8341:
8340:
8338:
8337:
8334:Dyck's surface
8331:
8325:
8324:
8323:
8318:
8306:
8304:
8303:Non-orientable
8300:
8299:
8297:
8296:
8293:
8290:
8284:
8277:
8275:
8268:
8264:
8263:
8258:
8257:
8250:
8243:
8235:
8229:
8228:
8217:"Möbius Strip"
8209:
8195:
8194:External links
8192:
8189:
8188:
8166:
8149:
8128:
8107:
8064:(5783): 72–4.
8036:
8006:
7981:
7969:
7957:
7936:
7917:
7898:
7873:
7854:
7835:
7814:
7774:
7767:
7745:
7734:(7): 207–215.
7718:
7706:
7681:
7656:
7636:
7621:
7593:
7586:
7564:
7549:
7524:
7501:
7494:
7474:
7447:(2): 108–111.
7422:
7395:
7383:
7354:
7286:
7259:(3): 167–174.
7238:
7227:(4): 275–276.
7208:
7182:
7175:
7155:
7148:
7128:
7117:(3): 407–412.
7099:
7045:
7019:
6984:
6972:
6952:
6925:(2): 113–122.
6909:
6874:
6821:
6806:
6778:
6763:
6743:
6736:
6713:
6698:
6680:
6672:
6646:
6603:
6555:
6507:
6502:
6483:
6448:(12): 127801.
6412:
6393:(3): 523–535.
6372:
6369:
6366:
6344:
6339:
6317:
6272:
6257:
6227:
6212:
6183:
6128:
6113:
6082:
6067:
6040:
6025:
5994:
5979:
5961:
5920:
5895:
5888:
5847:
5832:
5829:
5826:
5822:
5786:(3): 335–374.
5763:
5759:
5735:
5711:
5693:
5686:
5672:Hilbert, David
5663:
5642:(1): 195–200.
5620:
5593:(3): 276–289.
5575:
5545:
5522:
5489:
5470:
5446:
5385:
5333:
5288:
5231:
5176:
5161:
5080:
5037:
5010:(3): 519–522.
4992:
4963:(1): 141–157.
4941:
4914:(5): 309–311.
4895:
4866:(3): 395–400.
4841:
4820:(3): 271–283.
4795:
4736:
4721:
4691:
4688:on 2016-04-24.
4669:
4622:
4615:
4592:
4585:
4563:
4548:
4527:
4520:
4495:
4480:
4460:
4406:
4386:
4364:
4321:
4274:(2): 438–445.
4249:
4217:
4172:Bennett, G. T.
4163:
4156:
4133:
4118:
4098:
4086:
4051:
4039:
4027:
3972:
3951:
3896:
3889:
3871:
3834:
3817:
3802:
3767:
3696:
3677:(5): 544–547.
3652:
3645:
3622:
3615:
3603:Wells, John C.
3593:
3592:
3590:
3587:
3584:
3583:
3574:
3551:
3547:
3530:
3521:
3504:
3500:
3472:
3468:
3447:
3434:
3425:
3408:
3277:
3276:
3274:
3271:
3270:
3269:
3264:
3258:
3252:
3246:
3243:Möbius counter
3238:
3235:
3188:inverted notes
3166:musical canons
3063:Martin Gardner
3016:courting bench
2978:postage stamps
2946:graphic design
2933:
2926:
2925:
2918:
2911:
2910:
2904:
2897:
2896:
2895:
2894:
2893:
2854:José de Rivera
2850:Endless Ribbon
2834:Möbius Band II
2791:
2788:
2764:
2763:
2753:
2747:
2741:
2729:
2723:
2709:
2695:
2672:
2669:
2662:quotient space
2647:
2626:
2606:
2584:
2547:
2542:
2523:direct product
2515:counterexample
2433:
2430:
2406:mean curvature
2398:
2397:
2391:once-punctured
2383:
2380:
2357:
2338:
2335:
2331:flat manifolds
2310:quotient space
2301:
2300:Zero curvature
2277:
2274:
2206:
2203:
2200:
2196:
2142:
2139:
2136:
2133:
2130:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1901:
1894:
1893:
1889:
1882:
1881:
1880:
1879:
1878:
1876:
1873:
1834:surfaces, the
1807:
1787:
1774:
1771:
1768:
1758:
1752:
1739:
1736:
1729:
1724:
1721:
1718:
1711:
1708:
1677:
1674:
1671:
1667:
1663:
1629:
1589:
1586:
1581:
1566:
1563:
1560:triangulation.
1528:quadrilaterals
1489:
1488:
1479:
1478:
1470:
1469:
1468:
1467:
1466:
1443:
1440:
1434:
1431:
1411:
1408:
1405:
1371:
1368:
1363:
1330:
1326:
1301:
1298:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1215:
1212:
1191:
1170:
1148:
1145:
1142:
1139:
1136:
1133:
1112:
1109:
1106:
1103:
1100:
1097:
1071:
1068:
1063:
1060:
1055:
1052:
1047:
1044:
1042:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1021:
1018:
1015:
1012:
1008:
1002:
999:
994:
991:
986:
983:
978:
975:
971:
967:
964:
962:
960:
957:
954:
951:
948:
945:
942:
941:
938:
935:
932:
928:
922:
919:
914:
911:
906:
903:
898:
895:
891:
887:
884:
882:
880:
877:
874:
871:
868:
865:
862:
861:
828:
827:
818:
817:
816:
815:
814:
806:
805:
796:
795:
794:
793:
792:
790:
787:
782:
779:
765:
762:
759:
756:
753:
750:
747:
726:
706:
686:
666:
663:
660:
657:
654:
651:
648:
628:
608:
588:
553:Möbius ladders
543:but cannot be
541:complete graph
533:Tietze's graph
503:
496:
495:
491:Tietze's graph
488:
481:
480:
479:
478:
477:
453:
446:
445:
441:
434:
433:
432:
431:
430:
327:
324:
237:
230:
229:
217:
210:
209:
208:
207:
206:
204:
201:
107:boundary curve
80:non-orientable
28:
9:
6:
4:
3:
2:
8963:
8952:
8949:
8947:
8944:
8942:
8939:
8937:
8934:
8933:
8931:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8898:
8896:
8893:
8891:
8888:
8886:
8883:
8881:
8878:
8876:
8873:
8871:
8868:
8866:
8863:
8861:
8858:
8856:
8853:
8851:
8848:
8846:
8843:
8841:
8838:
8836:
8833:
8831:
8828:
8826:
8823:
8822:
8820:
8816:
8810:
8809:
8805:
8803:
8802:
8798:
8796:
8795:
8791:
8789:
8788:
8784:
8782:
8781:
8777:
8775:
8774:
8770:
8769:
8767:
8763:
8757:
8756:Lill's method
8754:
8752:
8749:
8748:
8746:
8744:Miscellaneous
8742:
8734:
8731:
8729:
8726:
8724:
8721:
8719:
8716:
8715:
8714:
8711:
8708:
8704:
8700:
8697:
8695:
8692:
8691:
8689:
8685:
8679:
8676:
8674:
8671:
8669:
8666:
8664:
8663:Rigid origami
8661:
8659:
8656:
8654:
8651:
8649:
8646:
8645:
8643:
8641:3d structures
8639:
8633:
8630:
8628:
8625:
8623:
8620:
8618:
8615:
8614:
8612:
8610:Strip folding
8608:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8577:
8574:
8572:
8569:
8567:
8564:
8562:
8559:
8558:
8556:
8552:
8548:
8541:
8536:
8534:
8529:
8527:
8522:
8521:
8518:
8502:
8499:
8497:
8493:
8491:
8487:
8485:Making a hole
8484:
8482:
8481:Connected sum
8479:
8478:
8476:
8472:
8466:
8463:
8461:
8458:
8455:
8451:
8450:
8448:
8444:
8438:
8437:Orientability
8435:
8433:
8429:
8426:
8424:
8421:
8419:
8418:Connectedness
8416:
8415:
8413:
8409:
8406:
8400:
8393:
8390:
8386:
8383:
8382:
8381:
8378:
8374:
8371:
8369:
8366:
8365:
8363:
8358:
8357:
8356:
8353:
8352:
8350:
8348:With boundary
8346:
8336:(genus 3) ...
8335:
8332:
8329:
8326:
8322:
8321:Roman surface
8319:
8317:
8316:Boy's surface
8313:
8312:
8311:
8308:
8307:
8305:
8301:
8294:
8291:
8288:
8285:
8282:
8279:
8278:
8276:
8272:
8269:
8265:
8256:
8251:
8249:
8244:
8242:
8237:
8236:
8233:
8224:
8223:
8218:
8215:
8210:
8207:
8206:Moebius Strip
8202:
8198:
8197:
8184:
8180:
8176:
8170:
8162:
8161:
8153:
8145:
8144:
8139:
8132:
8124:
8123:
8118:
8111:
8103:
8099:
8095:
8091:
8087:
8083:
8079:
8075:
8071:
8067:
8063:
8059:
8058:
8050:
8046:
8040:
8026:
8025:
8020:
8016:
8010:
8001:
8000:
7999:Plus Magazine
7995:
7988:
7986:
7978:
7973:
7966:
7961:
7953:
7952:
7947:
7940:
7932:
7928:
7921:
7913:
7909:
7902:
7894:
7890:
7889:
7884:
7877:
7869:
7865:
7858:
7850:
7846:
7839:
7831:
7830:
7825:
7818:
7810:
7806:
7802:
7798:
7794:
7790:
7789:
7784:
7778:
7770:
7768:84-930669-1-5
7764:
7760:
7756:
7749:
7741:
7737:
7733:
7729:
7722:
7715:
7710:
7695:
7691:
7685:
7671:
7667:
7660:
7652:
7651:
7646:
7640:
7632:
7628:
7624:
7622:0-88385-537-2
7618:
7614:
7610:
7606:
7600:
7598:
7589:
7583:
7579:
7575:
7568:
7562:, p. 13.
7561:
7556:
7554:
7539:
7535:
7528:
7520:
7516:
7512:
7505:
7497:
7495:9780198813255
7491:
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7470:
7466:
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7450:
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7436:
7429:
7427:
7412:
7411:
7406:
7399:
7392:
7387:
7379:
7375:
7371:
7367:
7366:
7358:
7349:
7344:
7339:
7334:
7330:
7326:
7322:
7318:
7317:
7312:
7308:
7304:
7300:
7296:
7290:
7282:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7253:
7248:
7242:
7234:
7230:
7226:
7222:
7218:
7212:
7204:
7200:
7196:
7192:
7186:
7178:
7176:9780785835776
7172:
7168:
7167:
7159:
7151:
7149:9781782001522
7145:
7141:
7140:
7132:
7124:
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7112:
7111:
7103:
7095:
7091:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7059:
7058:
7049:
7041:
7037:
7030:
7023:
7015:
7011:
7007:
7003:
6999:
6995:
6988:
6981:
6976:
6968:
6967:
6962:
6956:
6948:
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6940:
6936:
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6928:
6924:
6920:
6913:
6905:
6901:
6897:
6893:
6889:
6885:
6878:
6870:
6866:
6862:
6858:
6854:
6850:
6845:
6840:
6836:
6832:
6825:
6817:
6813:
6809:
6807:3-540-18697-2
6803:
6799:
6795:
6791:
6790:
6782:
6774:
6770:
6766:
6764:981-02-3555-0
6760:
6756:
6755:
6747:
6739:
6737:9784431669562
6733:
6729:
6728:
6723:
6717:
6709:
6705:
6701:
6695:
6691:
6684:
6675:
6673:9781400885398
6669:
6665:
6661:
6657:
6650:
6642:
6638:
6634:
6630:
6626:
6622:
6619:(1–2): 8–15.
6618:
6614:
6607:
6599:
6593:
6578:
6574:
6570:
6566:
6559:
6551:
6547:
6543:
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6523:
6505:
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6342:
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6313:
6309:
6305:
6301:
6297:
6293:
6289:
6285:
6284:
6276:
6268:
6264:
6260:
6258:0-12-634850-2
6254:
6250:
6249:
6244:
6240:
6234:
6232:
6223:
6219:
6215:
6213:0-387-97743-0
6209:
6205:
6201:
6197:
6193:
6187:
6179:
6175:
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6167:
6163:
6159:
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6140:
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6110:
6106:
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6098:
6097:
6089:
6087:
6078:
6074:
6070:
6064:
6060:
6056:
6055:
6050:
6044:
6036:
6032:
6028:
6022:
6018:
6014:
6010:
6009:
6004:
6003:Flapan, Erica
5998:
5990:
5986:
5982:
5976:
5972:
5965:
5957:
5953:
5948:
5943:
5939:
5935:
5931:
5924:
5910:
5906:
5899:
5891:
5885:
5880:
5875:
5871:
5867:
5864:
5854:
5852:
5830:
5827:
5824:
5820:
5809:
5805:
5801:
5797:
5793:
5789:
5785:
5781:
5780:
5761:
5757:
5748:
5742:
5740:
5731:
5727:
5720:
5718:
5716:
5707:
5703:
5697:
5689:
5683:
5679:
5678:
5673:
5667:
5659:
5655:
5650:
5645:
5641:
5637:
5636:
5631:
5624:
5616:
5612:
5608:
5604:
5600:
5596:
5592:
5588:
5587:
5579:
5571:
5567:
5563:
5559:
5555:
5549:
5541:
5537:
5533:
5529:
5525:
5519:
5515:
5511:
5507:
5503:
5496:
5494:
5485:
5481:
5474:
5465:
5460:
5456:
5450:
5442:
5438:
5434:
5430:
5426:
5422:
5417:
5412:
5408:
5404:
5403:
5398:
5392:
5390:
5381:
5377:
5373:
5369:
5364:
5359:
5355:
5351:
5347:
5340:
5338:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5300:
5292:
5286:, pp. 67–112.
5285:
5282:Reprinted in
5279:
5275:
5271:
5267:
5262:
5257:
5253:
5249:
5245:
5238:
5236:
5228:
5225:Reprinted in
5222:
5218:
5214:
5210:
5206:
5202:
5198:
5194:
5187:
5185:
5183:
5181:
5172:
5168:
5164:
5158:
5154:
5150:
5143:
5142:
5136:Reprinted in
5133:
5129:
5125:
5121:
5117:
5113:
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5103:
5099:
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5013:
5009:
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4996:
4988:
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4971:
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4962:
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4952:
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4937:
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4921:
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4909:
4905:
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4887:
4883:
4879:
4874:
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4865:
4861:
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4855:
4851:
4845:
4837:
4833:
4828:
4823:
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4815:
4814:
4809:
4805:
4799:
4791:
4787:
4783:
4779:
4775:
4771:
4767:
4763:
4762:
4754:
4750:
4743:
4741:
4732:
4728:
4724:
4722:0-521-81970-9
4718:
4714:
4710:
4706:
4702:
4695:
4684:
4680:
4676:
4672:
4666:
4662:
4658:
4651:
4650:
4645:
4641:
4640:Fuchs, Dmitry
4635:
4633:
4631:
4629:
4627:
4618:
4616:9780690278620
4612:
4608:
4607:
4599:
4597:
4588:
4586:9780471045977
4582:
4578:
4574:
4567:
4559:
4555:
4551:
4549:0-387-96426-6
4545:
4541:
4534:
4532:
4523:
4521:0-9665201-6-5
4517:
4513:
4509:
4505:
4499:
4491:
4487:
4483:
4477:
4473:
4472:
4464:
4456:
4452:
4448:
4444:
4439:
4434:
4430:
4426:
4425:
4420:
4416:
4410:
4403:
4397:
4393:
4389:
4387:0-88385-516-X
4383:
4379:
4375:
4368:
4360:
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4325:
4317:
4313:
4308:
4303:
4299:
4295:
4290:
4285:
4281:
4277:
4273:
4269:
4268:
4263:
4259:
4253:
4245:
4241:
4234:
4230:
4224:
4222:
4213:
4209:
4204:
4199:
4195:
4191:
4188:(2800): 882.
4187:
4183:
4182:
4177:
4174:(June 1923).
4173:
4167:
4159:
4157:9780608377803
4153:
4149:
4148:
4143:
4137:
4129:
4125:
4121:
4119:0-387-97430-X
4115:
4111:
4110:
4102:
4096:, p. 11.
4095:
4090:
4082:
4078:
4074:
4070:
4066:
4062:
4055:
4049:, p. 12.
4048:
4043:
4037:, p. 52.
4036:
4031:
4023:
4019:
4015:
4011:
4007:
4003:
3998:
3993:
3989:
3985:
3984:
3976:
3968:
3967:
3962:
3955:
3947:
3943:
3939:
3935:
3931:
3927:
3922:
3917:
3913:
3909:
3908:
3900:
3892:
3890:9781483262536
3886:
3882:
3875:
3867:
3863:
3859:
3855:
3851:
3847:
3846:
3838:
3831:
3826:
3824:
3822:
3813:
3809:
3805:
3803:0-521-66254-0
3799:
3795:
3791:
3787:
3783:
3782:
3777:
3776:Flapan, Erica
3771:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3735:
3730:
3725:
3721:
3717:
3716:
3711:
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3703:
3701:
3692:
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3680:
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3608:
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3525:
3498:
3488:
3470:
3466:
3457:
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3438:
3429:
3422:
3418:
3412:
3405:
3401:
3397:
3392:
3382:
3352:
3346:
3345:
3332:
3290:
3282:
3278:
3268:
3267:Umbilic torus
3265:
3262:
3259:
3256:
3255:Ribbon theory
3253:
3250:
3247:
3244:
3241:
3240:
3234:
3232:
3228:
3224:
3219:
3217:
3213:
3209:
3205:
3201:
3197:
3193:
3189:
3185:
3184:glide-reflect
3182:, features a
3181:
3180:
3175:
3171:
3167:
3162:
3160:
3159:
3154:
3153:
3145:
3141:
3140:
3133:
3129:
3128:
3120:
3116:
3115:
3108:
3105:(1913–1927),
3104:
3103:
3096:
3095:Marcel Proust
3092:
3088:
3084:
3083:
3078:
3071:
3064:
3060:
3056:
3051:
3049:
3045:
3041:
3037:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3008:Moebius Chair
3005:
3001:
2992:
2988:
2984:
2979:
2975:
2971:
2963:
2955:
2951:
2947:
2937:logo on stamp
2936:
2930:
2921:
2915:
2907:
2901:
2892:
2890:
2889:
2884:
2880:
2873:
2872:John Robinson
2869:
2865:
2862:, 1967), and
2861:
2860:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2826:Möbius Band I
2823:
2819:
2818:Charles Olson
2815:
2814:Corrado Cagli
2808:
2804:
2800:
2799:Endless Twist
2796:
2787:
2785:
2781:
2777:
2773:
2769:
2761:
2757:
2754:
2751:
2748:
2746:
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2740:
2738:
2733:
2730:
2727:
2724:
2722:
2718:
2714:
2710:
2707:
2703:
2699:
2696:
2693:
2690:
2689:
2688:
2682:
2677:
2668:
2663:
2645:
2624:
2604:
2582:
2573:
2569:
2565:
2545:
2529:solvmanifold
2528:
2524:
2520:
2516:
2512:
2511:solvmanifolds
2508:
2500:
2496:
2492:6-dimensional
2487:
2482:
2477:
2472:
2468:
2455:
2451:
2447:
2439:
2429:
2424:
2416:
2407:
2403:
2392:
2388:
2384:
2381:
2371:
2355:
2347:
2343:
2339:
2336:
2332:
2327:
2323:
2319:
2315:
2311:
2302:
2299:
2298:
2297:
2295:
2291:
2287:
2283:
2273:
2264:
2260:
2252:
2251:quadrilateral
2244:
2240:
2232:
2227:
2223:
2201:
2185:
2172:
2156:
2140:
2137:
2134:
2131:
2128:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1954:
1950:
1945:
1936:
1932:
1924:
1920:
1916:
1912:
1909:The edge, or
1898:
1886:
1872:
1867:
1863:
1859:
1854:
1849:
1845:
1837:
1833:
1829:
1805:
1790:
1772:
1769:
1766:
1750:
1737:
1734:
1727:
1722:
1719:
1716:
1709:
1706:
1675:
1672:
1669:
1665:
1661:
1646:
1627:
1613:
1587:
1584:
1579:
1562:
1545:
1537:
1529:
1524:
1519:
1515:
1511:
1507:
1503:
1499:
1483:
1474:
1465:
1460:
1459:cross section
1441:
1438:
1432:
1429:
1409:
1406:
1403:
1391:
1369:
1366:
1361:
1350:
1346:
1328:
1324:
1315:
1306:
1297:
1294:
1285:
1284:ruled surface
1281:
1276:
1271:
1249:
1246:
1243:
1240:
1237:
1213:
1210:
1189:
1168:
1146:
1143:
1140:
1137:
1134:
1131:
1110:
1107:
1104:
1101:
1098:
1095:
1069:
1066:
1061:
1058:
1053:
1050:
1045:
1043:
1035:
1032:
1029:
1023:
1016:
1013:
1010:
1006:
1000:
997:
992:
989:
984:
981:
976:
973:
969:
965:
963:
955:
952:
949:
943:
936:
933:
930:
926:
920:
917:
912:
909:
904:
901:
896:
893:
889:
885:
883:
875:
872:
869:
863:
851:
847:
834:
830:
822:
808:
800:
786:
781:Constructions
778:
763:
760:
757:
754:
751:
748:
745:
724:
704:
684:
664:
661:
658:
655:
652:
649:
646:
626:
606:
586:
578:
574:
566:
562:
554:
550:
546:
542:
538:
535:, which is a
534:
526:
518:
507:
500:
492:
485:
476:
473:
469:
450:
438:
429:
427:
423:
419:
415:
410:
407:
389:
379:
374:
370:
362:
353:
349:
341:
332:
323:
314:
311:in a work of
310:
305:
300:
295:
291:
287:
283:
279:
271:
259:
255:
244:
240:
234:
225:
221:
214:
200:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
141:
139:
135:
131:
127:
123:
119:
115:
114:ruled surface
110:
108:
104:
100:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
36:
32:
27:
19:
8915:Eve Torrence
8845:Erik Demaine
8806:
8799:
8792:
8785:
8778:
8771:
8765:Publications
8627:Möbius strip
8626:
8617:Dragon curve
8554:Flat folding
8380:Möbius strip
8379:
8328:Klein bottle
8220:
8182:
8169:
8159:
8152:
8141:
8131:
8120:
8110:
8061:
8055:
8039:
8028:. Retrieved
8024:Live Science
8022:
8009:
7997:
7972:
7960:
7949:
7939:
7930:
7920:
7911:
7901:
7886:
7876:
7867:
7857:
7848:
7838:
7829:The Guardian
7827:
7817:
7792:
7786:
7777:
7758:
7748:
7731:
7727:
7721:
7709:
7698:. Retrieved
7693:
7684:
7673:. Retrieved
7670:Tech in Asia
7669:
7659:
7648:
7639:
7612:
7577:
7567:
7541:. Retrieved
7527:
7510:
7504:
7484:
7477:
7444:
7438:
7414:. Retrieved
7410:Ars Technica
7408:
7398:
7386:
7369:
7363:
7357:
7320:
7314:
7289:
7256:
7250:
7241:
7224:
7220:
7211:
7202:
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7165:
7158:
7138:
7131:
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7102:
7061:
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7048:
7039:
7035:
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6993:
6987:
6975:
6964:
6955:
6922:
6918:
6912:
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6883:
6877:
6834:
6830:
6824:
6788:
6781:
6753:
6746:
6726:
6716:
6689:
6683:
6655:
6649:
6616:
6612:
6606:
6581:. Retrieved
6568:
6558:
6525:
6521:
6486:
6445:
6439:
6415:
6390:
6384:
6320:
6290:(1): 55–81.
6287:
6281:
6275:
6247:
6195:
6186:
6143:
6137:
6131:
6095:
6053:
6043:
6007:
5997:
5970:
5964:
5937:
5933:
5923:
5912:. Retrieved
5908:
5898:
5869:
5862:
5783:
5777:
5729:
5705:
5696:
5676:
5666:
5639:
5633:
5623:
5590:
5584:
5578:
5561:
5557:
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5501:
5483:
5473:
5449:
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5400:
5353:
5349:
5303:
5297:
5291:
5251:
5247:
5196:
5192:
5140:
5100:(1–2): 3–6.
5097:
5093:
5056:
5050:
5040:
5007:
5001:
4995:
4960:
4954:
4944:
4911:
4907:
4898:
4863:
4857:
4844:
4817:
4811:
4798:
4768:(3): 11–22.
4765:
4759:
4747:Kühnel, W.;
4704:
4694:
4683:the original
4648:
4605:
4576:
4566:
4539:
4511:
4498:
4470:
4463:
4428:
4422:
4409:
4377:
4367:
4334:
4330:
4324:
4271:
4265:
4252:
4243:
4239:
4185:
4179:
4166:
4146:
4136:
4108:
4101:
4089:
4064:
4060:
4054:
4042:
4030:
3987:
3981:
3975:
3964:
3954:
3911:
3905:
3899:
3880:
3874:
3849:
3843:
3837:
3780:
3770:
3722:(2): 69–76.
3719:
3713:
3674:
3668:
3635:
3625:
3606:
3597:
3577:
3533:
3524:
3450:
3437:
3428:
3417:Klein bottle
3411:
3403:
3399:
3281:
3220:
3192:music theory
3177:
3163:
3158:Donnie Darko
3156:
3150:
3137:
3125:
3112:
3100:
3080:
3052:
3036:food styling
3007:
2996:
2982:Switzerland.
2970:Google Drive
2943:
2920:Google Drive
2886:
2878:
2857:
2849:
2833:
2825:
2822:M. C. Escher
2811:
2798:
2765:
2736:
2732:Polarization
2685:
2671:Applications
2563:
2478:
2435:
2399:
2326:Klein bottle
2279:
2255:orientation.
2242:
2238:
2236:
2155:great circle
1946:
1931:Klein bottle
1908:
1858:plate theory
1855:
1846:or the real
1843:
1793:
1647:
1568:
1525:
1513:
1495:
1349:aspect ratio
1311:
1277:
839:
784:
514:
471:
467:
464:half-twists.
460:
411:
380:
337:
301:
294:figure-eight
251:
165:M. C. Escher
142:
111:
88:
55:
51:
48:Möbius strip
47:
41:
31:
26:
8900:Kōryō Miura
8895:Jun Maekawa
8870:Kôdi Husimi
8586:Map folding
8423:Compactness
7868:Artnet News
5409:: 255–267.
4067:: 131–136.
3852:(1): 5–18.
3519:embeddings.
3285:Pronounced
3223:stage magic
3212:Mobius Band
3164:One of the
3119:Frank Capra
3012:Pedro Reyes
2879:Immortality
2784:DNA origami
2564:group model
2519:nilmanifold
2503:symmetries.
2378:reflection.
2259:pinch point
2180:centerline.
1518:hyperplanes
1270:solid torus
717:edges, and
557:end-to-end.
134:hypersphere
122:folded flat
56:Möbius loop
52:Möbius band
44:mathematics
18:Möbius loop
8930:Categories
8890:Anna Lubiw
8723:Common net
8648:Miura fold
8474:Operations
8456:components
8452:Number of
8432:smoothness
8411:Properties
8359:Semisphere
8274:Orientable
8030:2022-03-21
7700:2022-03-27
7675:2022-03-27
7543:2022-04-17
7416:2022-03-28
6583:2022-03-21
5914:2022-03-17
5464:2308.12641
5416:2008.11610
4258:Ringel, G.
4246:: 155–159.
3997:1810.04089
3921:2212.02983
3729:1609.07779
3589:References
3456:smoothness
3170:J. S. Bach
3132:John Barth
3026:have been
3010:(2006) by
2838:lemniscate
2768:soap films
2756:World maps
2726:Resonators
2717:dielectric
2495:Lie groups
2370:half-plane
2231:level sets
1536:octahedron
1274:connected.
697:vertices,
537:dual graph
468:paradromic
326:Properties
309:chain pump
290:ourobouros
239:Chain pump
222:depicting
193:J. S. Bach
153:world maps
8808:Origamics
8687:Polyhedra
8501:Immersion
8496:cross-cap
8494:Gluing a
8488:Gluing a
8385:Cross-cap
8330:(genus 2)
8314:genus 1;
8289:(genus 1)
8283:(genus 0)
8222:MathWorld
7931:The Verge
7809:216116708
7469:123908555
7094:206562350
6831:Physica E
6641:122209943
6592:cite book
6371:π
6365:−
6312:121366986
6178:119640200
6153:1305.1379
6146:: 27–42.
5879:1204.4952
5821:τ
5615:122215321
5564:: 19–23.
5540:125997451
5441:220279013
5356:: 41–70.
5221:119782792
5132:119733903
5107:1408.3034
4790:120926324
4431:(1): 39.
4359:121332704
4022:119179202
3946:126421578
3762:119587191
3503:∞
3340:-bee-əs,
3059:time loop
2954:recycling
2888:Continuum
2864:Sebastián
2852:, 1953),
2780:nanoscale
2776:molecules
2625:ℓ
2605:ℓ
2419:surfaces.
2374:surfaces.
2294:geodesics
2239:cross-cap
2175:boundary.
2159:boundary.
2141:π
2135:ϕ
2132:≤
2107:π
2098:ϕ
2095:≤
2086:π
2080:θ
2077:≤
2050:ϕ
2047:
2041:θ
2035:
2026:ϕ
2023:
2017:θ
2011:
2002:ϕ
1999:
1993:θ
1990:
1981:ϕ
1978:
1972:θ
1969:
1927:circular.
1923:unknotted
1826:In 2023,
1770:×
1735:≈
1673:≈
1662:π
1585:≈
1540:boundary.
1463:would be.
1433:×
1407:×
1367:≈
1329:∘
1314:flattened
1144:≤
1138:≤
1132:−
1111:π
1099:≤
1062:
1014:
993:
934:
913:
749:−
737:regions;
650:−
397:surfaces.
384:boundary.
377:embedded.
274:match up.
157:antipodes
84:clockwise
8946:Surfaces
8936:Topology
8865:Tom Hull
8835:Yan Chen
8718:Blooming
8622:Flexagon
8454:boundary
8373:Cylinder
8177:(1956).
8094:16825563
7912:The Salt
7893:NewsBank
7607:(2002).
7440:Leonardo
7193:(1961).
7086:25636796
6947:21378823
6904:16218564
6869:17102453
6478:25860771
6431:(2015).
6245:(1980).
6051:(2008).
6005:(2016).
5704:(1979).
5278:53462568
4987:10887519
4890:38606607
4852:(2008).
4806:(1972).
4751:(1983).
4506:(2005).
4417:(1900).
4316:16591648
4231:(1910).
4073:20488581
3778:(2000).
3691:27843983
3633:(2005).
3605:(2008).
3237:See also
3174:BWV 1087
3161:(2001).
3152:Dhalgren
3142:(1968),
3130:(1946),
3117:(1921),
2993:entrance
2962:Expo '74
2881:(1982).
2859:Infinity
2846:Max Bill
2830:flatfish
2803:Max Bill
2760:antipode
2692:Graphene
2318:cylinder
2267:segment,
2243:crosscap
2168:circles.
1935:immersed
1911:boundary
1840:becomes.
1616:stretch.
1556:surface.
1522:circles.
1502:cylinder
1394:surface.
549:embedded
519:for the
426:homotopy
373:disjoint
321:garment.
298:unclear.
278:Sentinum
220:Sentinum
169:Max Bill
8404:notions
8402:Related
8368:Annulus
8364:Ribbon
8102:2877171
8086:3846592
8066:Bibcode
8057:Science
7740:0720681
7631:1874198
7461:1577979
7348:3009808
7325:Bibcode
7281:0001622
7273:2304225
7066:Bibcode
7057:Science
7002:Bibcode
6927:Bibcode
6849:Bibcode
6816:1306737
6773:1698234
6708:2305055
6633:1675956
6577:1304924
6550:2240407
6530:Bibcode
6470:3447638
6450:Bibcode
6407:0630583
6304:1428058
6267:0575168
6222:1171453
6170:3370020
6123:3289090
6077:2440945
6035:3443369
5989:2483686
5956:2985809
5808:0270280
5800:1970625
5658:1047406
5607:0143115
5570:0071060
5532:3793640
5433:4330341
5380:0474388
5372:1997711
5328:1079975
5320:2324325
5270:3326186
5213:3326187
5171:3381564
5124:3326180
5075:2921579
5032:0715878
5024:2045508
4979:2429652
4936:0024138
4928:2305482
4882:2443291
4836:0314057
4782:0737686
4731:2008500
4679:2350979
4558:0880519
4490:3309241
4455:1500522
4447:1986401
4396:1303141
4351:2846715
4298:0228378
4276:Bibcode
4212:4099647
4190:Bibcode
4128:1095046
4081:0091480
4014:3975576
3938:3848635
3866:3026946
3812:1781912
3754:3507121
3734:Bibcode
3679:Bibcode
3490:(class
3404:Moebius
3387:German:
3082:Moebius
3024:scarves
2637:to the
2527:compact
2344:of the
2305:bundle.
2271:conoid.
2247:circle,
2221:circle.
2157:as its
1951:in the
1939:strips.
1870:strips.
1608:planes.
406:annulus
344:subset.
203:History
99:knotted
60:surface
8818:People
8673:Sonobe
8490:handle
8281:Sphere
8122:Magnet
8100:
8092:
8084:
7914:. NPR.
7807:
7765:
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7271:
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4445:
4400:. See
4394:
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4349:
4314:
4307:225066
4304:
4296:
4210:
4181:Nature
4154:
4126:
4116:
4079:
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4020:
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3643:
3613:
3458:class
3400:Mobius
3396:umlaut
3135:'s
3110:'s
3098:'s
3040:bagels
2958:strip,
2739:-plate
2666:strip.
2658:-axis.
2572:cosets
2475:strip.
2463:strip.
2459:lines.
2442:strip.
2427:curve.
2411:strip,
2395:plane.
2324:, and
2163:1970s.
1943:edges.
1919:circle
1852:space.
1824:exist.
1759:Can a
1738:1.695.
1692:bound.
1651:folds.
1604:strip.
1552:edges.
1548:faces;
1532:faces,
1386:ratio.
1288:strip.
842:lines.
619:, and
569:strip.
529:needs.
521:plane.
401:other.
388:chiral
365:strip.
357:sides.
317:chain.
292:or of
282:zodiac
245:(1206)
189:canons
120:or be
8460:Genus
8287:Torus
8098:S2CID
8082:JSTOR
8052:(PDF)
7805:S2CID
7465:S2CID
7457:JSTOR
7269:JSTOR
7090:S2CID
7042:(11).
7032:(PDF)
6865:S2CID
6839:arXiv
6637:S2CID
6436:(PDF)
6308:S2CID
6174:S2CID
6148:arXiv
5909:Vimeo
5874:arXiv
5796:JSTOR
5611:S2CID
5536:S2CID
5459:arXiv
5437:S2CID
5411:arXiv
5368:JSTOR
5316:JSTOR
5274:S2CID
5217:S2CID
5145:(PDF)
5128:S2CID
5102:arXiv
5020:JSTOR
4983:S2CID
4924:JSTOR
4886:S2CID
4786:S2CID
4756:(PDF)
4686:(PDF)
4653:(PDF)
4443:JSTOR
4355:S2CID
4236:(PDF)
4208:S2CID
4069:JSTOR
4018:S2CID
3992:arXiv
3942:S2CID
3916:arXiv
3862:JSTOR
3786:82–83
3758:S2CID
3724:arXiv
3687:JSTOR
3273:Notes
3147:'
3122:'
3073:'
3066:'
3048:pasta
3044:bacon
3014:is a
2875:'
2595:-axis
2531:with
2525:of a
2452:. By
2322:torus
1917:to a
1696:least
1644:form.
1514:tight
1293:gears
472:rings
262:1858.
132:in a
72:Roman
58:is a
54:, or
8355:Disk
8143:Prog
8090:PMID
7763:ISBN
7617:ISBN
7582:ISBN
7490:ISBN
7171:ISBN
7144:ISBN
7082:PMID
6966:Time
6943:PMID
6900:PMID
6802:ISBN
6759:ISBN
6732:ISBN
6694:ISBN
6668:ISBN
6598:link
6474:PMID
6253:ISBN
6208:ISBN
6109:ISBN
6063:ISBN
6021:ISBN
5975:ISBN
5884:ISBN
5682:ISBN
5518:ISBN
5157:ISBN
4717:ISBN
4665:ISBN
4611:ISBN
4581:ISBN
4544:ISBN
4516:ISBN
4476:ISBN
4382:ISBN
4312:PMID
4152:ISBN
4114:ISBN
3885:ISBN
3798:ISBN
3641:ISBN
3611:ISBN
3229:and
3028:knit
2966:one,
2952:for
2935:IMPA
2866:. A
2711:The
2400:The
2138:<
2101:<
2083:<
2065:for
1796:and
1676:1.57
1588:1.73
1370:1.73
1123:and
1105:<
1088:for
577:zero
571:The
286:Aion
256:and
224:Aion
183:and
66:and
46:, a
8713:Net
8430:or
8394:...
8074:doi
8062:313
7797:doi
7515:doi
7449:doi
7374:doi
7370:104
7343:PMC
7333:doi
7321:107
7261:doi
7229:doi
7119:doi
7074:doi
7062:347
7010:doi
6935:doi
6892:doi
6888:105
6857:doi
6794:doi
6660:doi
6621:doi
6538:doi
6458:doi
6446:114
6395:doi
6383:".
6292:doi
6288:119
6200:doi
6158:doi
6144:177
6101:doi
6059:171
6013:doi
5942:doi
5788:doi
5776:".
5644:doi
5640:143
5595:doi
5510:doi
5421:doi
5407:215
5358:doi
5354:230
5308:doi
5256:doi
5252:119
5201:doi
5197:119
5149:doi
5112:doi
5098:119
5061:doi
5057:312
5012:doi
4965:doi
4916:doi
4868:doi
4822:doi
4770:doi
4709:doi
4657:doi
4433:doi
4339:doi
4302:PMC
4284:doi
4198:doi
4186:111
4002:doi
3926:doi
3854:doi
3790:doi
3742:doi
3402:or
3342:MAY
3338:MOH
3168:by
3075:s "
2885:'s
2774:of
2241:or
2044:sin
2032:sin
2020:sin
2008:cos
1996:cos
1987:sin
1975:cos
1966:cos
1059:sin
1011:sin
990:cos
931:cos
910:cos
371:of
260:in
191:of
42:In
8932::
8705:,
8219:.
8181:.
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8088:.
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