Möbius strip - Knowledge

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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 2676: 2226: 331: 233: 35: 484: 449: 437: 2900: 821: 8201: 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 855: 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

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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

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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

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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

1897: 7882: 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 1267:

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

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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}}} 394:

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

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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

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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 844:

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 1885: 3581:

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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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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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

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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

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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, 1263: 675: 116:

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

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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

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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. 2421:

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

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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

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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 2249:

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 7315: 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.

3982: 3906: 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

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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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transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its

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This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the

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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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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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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.). 5479: 3053:

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

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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. 3844: 1788: 1500:

in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the

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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 7945: 2069: 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 7404: 8523: 8238: 7585: 6697: 6112: 6066: 6024: 5978: 5887: 5685: 5521: 5160: 4668: 4647: 4479: 3644: 3614: 3113: 7689: 2340:

The open Möbius strip also admits complete metrics of constant negative curvature. One way to see this is to begin with the

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Yoon, Zin Seok; Osuka, Atsuhiro; Kim, Dongho (May 2009). "Möbius aromaticity and antiaromaticity in expanded porphyrins".

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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: 8693: 4232: 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

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even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this

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centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are

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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

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These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called

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To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the

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However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral

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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: 8631: 8600: 6654:

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

5585: 2749: 175:. Many architectural concepts have been inspired by the Möbius strip, including the building design for the 8950: 8779: 8546: 7644: 5675: 3260: 2928: 8230: 6960: 6432: 3712:; González, Diego L. (2016). "Möbius strips before Möbius: topological hints in ancient representations". 2341: 1546:

can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its

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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. 6325: 6136:

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 2124: 1831: 1505: 7533: 3980:

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 1657: 8829: 3019: 7665: 6494: 6331: 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: 3486: 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: 2743:

A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in

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A single off-center cut produces a Möbius strip (purple) linked with a double-length two-sided strip

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copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously

8864: 8800: 8570: 8427: 8116: 6792:. Encyclopaedia of Mathematical Sciences. Vol. 20. Springer-Verlag, Berlin. pp. 164–166. 3390: 3126: 2871: 2522: 2170: 1847: 1458: 560: 6571:. Aportaciones Mat. Notas Investigación. Vol. 8. Soc. Mat. Mexicana, México. pp. 67–79. 2485: 2284:

of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a

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In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an

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after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a

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With an even number of twists, however, one obtains a different topological surface, called the

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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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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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swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains

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Ticket To Ride: The Essential Guide to the World's Greatest Roller Coasters and Thrill Rides

6360: 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 2697: 2257:

The two parts of the surface formed by the two glued pairs of edges cross each other with a

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exist other topological spaces in which the Möbius strip can be embedded so that it has two

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The flat-folded Möbius strip formed from three equilateral triangles does not come from an

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paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space?

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These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the

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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

2567: 2345: 2285: 1921:. In common forms of the Möbius strip, it has a different shape from a circle, but it is 1910: 1611: 196: 184: 176: 117: 106: 8069: 8048: 7328: 7069: 7028: 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

4279: 4193: 3737: 3682: 2422: 1472: 1206: 700: 680: 8879: 8702: 8677: 8354: 8097: 8081: 7950: 7804: 7464: 7456: 7439: 7347: 7310: 7268: 7089: 6864: 6838: 6636: 6591: 6307: 6173: 6147: 5873: 5795: 5610: 5535: 5458: 5436: 5410: 5367: 5315: 5273: 5216: 5127: 5101: 5019: 4982: 4923: 4885: 4785: 4643: 4442: 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. 4068: 4017: 3991: 3941: 3915: 3861: 3757: 3723: 3686: 3669: 3395: 3211: 2771: 2641: 2578: 2453: 2351: 2289: 1185: 1164: 845: 720: 622: 602: 582: 516: 351: 98: 59: 7782: 7578:

Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture

4503: 4306: 4261: 3215: 2712: 2680: 8945: 8935: 8786: 8657: 8333: 8213: 8121: 8089: 8056: 7808: 7762: 7616: 7615:. MAA Spectrum. Mathematical Association of America, Washington, DC. pp. 31–35. 7581: 7489: 7468: 7170: 7143: 7122: 7109: 7093: 7081: 7056: 6942: 6899: 6801: 6758: 6731: 6693: 6667: 6640: 6473: 6311: 6252: 6242: 6207: 6177: 6108: 6096:

An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity

6062: 6020: 5974: 5883: 5681: 5614: 5539: 5517: 5440: 5220: 5156: 5131: 4789: 4716: 4664: 4610: 4604: 4580: 4543: 4515: 4475: 4381: 4358: 4311: 4151: 4113: 4021: 3945: 3884: 3797: 3761: 3709: 3640: 3610: 3187: 2705: 2562:

These symmetries also provide another way to construct the Möbius strip itself, as a

2506: 2393:

projective plane, the surface obtained by removing any one point from the projective

2281: 1865: 532: 490: 417: 360: 303: 212: 144: 90: 6868: 5553: 5277: 4986: 4889: 3242: 2853: 2296:("straight lines" on the surface) may be extended indefinitely in either direction. 8899: 8859: 8824: 8755: 8727: 8712: 8595: 8315: 8101: 8073: 7796: 7514: 7448: 7373: 7342: 7332: 7264: 7260: 7228: 7118: 7073: 7009: 6934: 6891: 6856: 6793: 6789:

Lie groups and Lie algebras I: Foundations of Lie Theory; Lie Transformation Groups

6659: 6620: 6541: 6537: 6461: 6457: 6394: 6291: 6199: 6157: 6100: 6012: 5941: 5787: 5746: 5643: 5594: 5509: 5504:. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge. p. 5420: 5357: 5311: 5307: 5255: 5200: 5148: 5111: 5060: 5011: 4964: 4915: 4903: 4867: 4821: 4769: 4708: 4656: 4432: 4414: 4338: 4301: 4283: 4211: 4197: 4180: 4141: 4001: 3925: 3853: 3789: 3741: 3634: 3356: 3294: 3288: 3248: 3222: 3195: 3183: 3143: 3106: 3086: 2986: 2949: 2913: 2905: 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: 8839: 8667: 8652: 8417: 8142: 8044: 8014: 7735: 7626: 7518: 7276: 7246: 7216: 7190: 6965: 6811: 6768: 6721: 6703: 6628: 6572: 6545: 6465: 6433:"Instability of a Möbius strip minimal surface and a link with systolic geometry" 6402: 6299: 6262: 6238: 6217: 6165: 6118: 6072: 6030: 5984: 5951: 5803: 5653: 5602: 5565: 5527: 5428: 5375: 5323: 5265: 5208: 5166: 5119: 5070: 5027: 4974: 4931: 4877: 4831: 4803: 4777: 4748: 4726: 4674: 4553: 4485: 4450: 4391: 4346: 4293: 4228: 4123: 4076: 4009: 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: 8884: 8874: 8854: 8849: 8732: 8565: 8515: 8174: 7604: 7306: 7232: 6191: 6052: 6048: 5701: 5424: 5065: 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".

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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).

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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

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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: 7487: 7486: 7478: 7470: 7466: 7462: 7458: 7454: 7450: 7446: 7442: 7441: 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: 7120: 7116: 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: 6944: 6940: 6936: 6932: 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: 6539: 6535: 6531: 6527: 6523: 6505: 6487: 6479: 6475: 6471: 6467: 6463: 6459: 6455: 6451: 6447: 6443: 6442: 6434: 6430: 6426: 6422: 6416: 6408: 6404: 6400: 6396: 6392: 6388: 6387: 6370: 6367: 6364: 6342: 6327: 6321: 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: 6171: 6167: 6163: 6159: 6154: 6149: 6145: 6141: 6140: 6132: 6124: 6120: 6116: 6110: 6106: 6102: 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: 5108: 5103: 5099: 5095: 5087: 5085: 5076: 5072: 5067: 5062: 5058: 5054: 5053: 5048: 5041: 5033: 5029: 5025: 5021: 5017: 5013: 5009: 5005: 5004: 4996: 4988: 4984: 4980: 4976: 4971: 4966: 4962: 4958: 4957: 4952: 4945: 4937: 4933: 4929: 4925: 4921: 4917: 4913: 4909: 4905: 4899: 4891: 4887: 4883: 4879: 4874: 4869: 4865: 4861: 4860: 4855: 4851: 4845: 4837: 4833: 4828: 4823: 4819: 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: 3705: 3703: 3701: 3692: 3688: 3684: 3680: 3676: 3672: 3671: 3663: 3661: 3659: 3657: 3648: 3642: 3638: 3637: 3632: 3626: 3618: 3612: 3608: 3604: 3598: 3594: 3578: 3571: 3567: 3549: 3545: 3534: 3525: 3498: 3488: 3470: 3466: 3457: 3451: 3444: 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: 2742: 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: 7198: 7185: 7165: 7158: 7138: 7131: 7114: 7108: 7102: 7061: 7055: 7048: 7039: 7035: 7022: 6997: 6993: 6987: 6975: 6964: 6955: 6922: 6918: 6912: 6887: 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: 5548: 5501: 5483: 5473: 5449: 5406: 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. 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