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Revisi sejak 20 April 2022 09.35 oleh Dedhert.Jr (bicara | kontrib) (→Asal-usul)
Sebuah strip Möbius yang terbuat dari kertas dan plester.

Dalam matematika, strip Möbius atau pita Möbius adalah suatu permukaan yang dapat dibentuk dengan menempel ujung pita tersebut dengan memutarnya sebagian. Sebagai sebuah objek matematika, pita ini ditemukan oleh Johann Benedict Listing dan August Ferdinand Möbius pada tahun 1858, namun pita ini sudah muncul di mosaik Roman pada abad ketiga masehi. Strip Möbius merupakan permukaan yang tidak dapat diarahkan (atau permukaan takterarahkan), dalam artian bahwa dalam pita tersebut tidak selalu dapat membedakan arah jarum jam dengan arah sebaliknya. Setiap permukaan yang tidak dapat diarahkan memuat sebuah strip Möbius.

Karena berupakan ruang topologis yang abstrak, strip Möbius dapat dibenamkan menjadi ruang Euklides berdimensi tiga dalam berbagai cara: sebuah pita yang diputar setengah dengan arah jarum jam berbeda dengan yang diputar setengah dengan arah yang berlawanan, dan strip Möbius dapat dibenamkan dengan jumlah putaran ganjil yang lebih besar dari satu, atau dengan garis tengah yang dibuhul. Secara topologis dikatakan ekuivalen apabila setiap dua pembenaman dengan buhul dalam garis tengah dan jumlah arah putaran yang sama. Semua pembenaman pada strip Möbius hanya memiliki satu sisi, namun pita dapat mempunyai dua sisi bila dibenamkan dalam ruang lain. Pita ini hanya mempunyai sebuah kurva batas yang tunggal.

Ada beberapa konstruksi geometris strip Möbius yang menyediakannya dengan struktur tambahan. Pita tersebut dapat disapu sebagai permukaan bergaris dengan memutar ruas garis di sebuah bidang yang berputar, dengan atau tanpa menyilang dirinya sendiri. Secarik kertas yang tipis dengan ujungnya yang ditempelkan agar membentuk sebuah strip Möbius dapat dibelokkan dengan lancar sebagai secarik kertas dengan permukaan yang dapat dikembangkan atau dengan rata yang terlipat (contoh mengenai strip Möbius yang diratakan, seperti triheksafleksagon). Strip Möbius Sudan merupakan sebuah permukaan minimal dalam sebuah hiperbola, dan strip Möbius Meeks merupakan permukaan minimal yang memotong diri sendiri dalam ruang Euklides biasa. Strip Möbius Sudan dan strip Möbius yang memotong diri sendiri lainnya (yaitu cross-cap), mempunyai batas yang melingkar. Sebuah strip Möbius tanpa adanya batas (disebut strip Möbius terbuka) dapat membentuk permukaan dari kurva konstanta. Ruang yang sangat simetris dengan titik-titiknya mewakili garis di bidang mempunyai bentuk strip Möbius.

Ada beberapa penerapan terhadap strip Möbius. Penerapan tersebut diantaranya: sabuk dalam mesin yang memakai pada kedua sisi dengan rata, kereta luncur dengan jalur berganda yang mengangkut secara bergantian di antara kedua jalur tersebut, dan peta dunia yang dicetak sehingga antipoda muncul berseberangan. Strip Möbius muncul dalam molekul dengan elektrik yang tidak biasa dan perangkat dengan sifat-sifat elektromekanis, dan pita ini telah dipakai untuk membuktikan hasil kemustahilan dalam teori pemilihan sosial. Dalam budaya populer, strip Möbius muncul dalam hasil karya M. C. Escher, Max Bill, dan tokoh lainnya. Pita ini muncul dalam desain dari simbol daur ulang. Ada banyak konsep tentang arsitek yang terinspirasi oleh strip Möbius strip, seperti desain bangunan NASCAR Hall of Fame. Pemain sandiwara seperti Harry Blackstone Sr. dan Thomas Nelson Downs menggunakan trik sulap yang berasal dari sifat-sifat strip Möbius. Musik kanon milik J. S. Bach telah dianalisis bahwa musiknya menggunakan strip Möbius. Ada banyak karya yang bersifat fiksi dan spekulatif menampilkan strip Möbius; lebih umumnya, struktur alur berdasarkan strip Möbius, atau kejadian yang berulang dengan memutar struktur alur, biasanya terdapat di dalam karya fiksi.

Asal-usul

 
Mosaik dari Sentinum kuno yang menggambarkan Aion sedang memegang sebuah pita Möbius
 
Lukisan oleh Ismail al-Jazari (1206), yang menggambarkan pompa rantai dengan rantai penggerak Möbius.

Penemuan strip Möbius sebagai objek matematika dihubungkan dengan matematikawan Jerman Johann Benedict Listing dan August Ferdinand Möbius secara terpisah pada tahun 1858.[1] Akan tetapi, pita öbiusi sudah dtkenal sejak lama sebagai benda fisik dan gambaran artistik. Strip Möbius khususnya dapat dilihat dalam berbagai mosaik Roma pada abad ketiga masehi.[2][3] Pada umumnya, mosaik tersebut hanya menggambarkan pita yang bergelung sebagai batasnya. Ketika jumlah gelungnya adalah ganjil, pita-pita tersebut merupakan strip Möbius, namun bila jumlahnya adalah genap, pita-pita tersebut secara topologis ekuivalen dengan gelanggang yang tidak diputar. Karena itu pita yang merupakan strip Möbius hanyalah kebetulan, bukan dipilih dengan sengaja. Pada setidaknya satu kasus, sebuah pita dengan warna yang berbeda pada sisi yang berbeda digambar dengan putaran gelung yang berjumlahkan ganjil, forcing its artist to make a clumsy fix at the point where the colors did not match up.[2] Mosaik lain yang berasal dari kota Sentinum memperlihatkan gambar seorang dewa Aion sedang memegang zodiak sebagai pita yang hanya dengan setengah putaran. Tidak ada bukti jelas yang memihak representasi visual mengenai celestial time was intentional; melainkan hanya dapat dipilih sebagai cara untuk membuat semua lambang zodiak muncul pada sisi pita yang terlihat. Ada juga yang mengatakan bahwa beberapa gambaran kuno seperti gambar ouroboros atau hiasan berbentuk angka delapan menggambarkan strip Möbius, namun belum jelas apakah strip Möbius bermaksud untuk menggambar segala jenis pita yang rata atau tidak.[3]

Independently of the mathematical tradition, machinists have long known that mechanical belts 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.[2] 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 after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a chain pump in a work of Ismail al-Jazari from 1206 depicts a Möbius strip configuration for its drive chain.[3] 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 garment.[2]

Sifat-sifat

 
Sebuah objek dua dimensi bergerak melintang sekali disekitar strip Möbius dan kembali ke posisi semula dalam bentuk yang tercermin.

Strip Möbius mempunyai beberapa sifat yang aneh. Strip Möbius merupakan permukaan takterarahkan, yang berarti bahwa jika sebuah benda dimensi dua asimetris yang meluncur sekali di sekitar pita tersebut, maka benda tersebut kembali ke posisi semula dengan bentuk yang tercermin. Khususnya, sebuah kurva dengan panah yang mengarah ke jarum jam (↻) akan kembali ketika sebuah panah yang mengarah ke arah jarum jam yang berlawanan (↺). Hal ini menyiratkan bahwa dalam strip Möbius mustahil untuk selalu menentukan apakah benda mengarah ke jarum jam atau sebaliknya. Strip Möbius merupakan permukaan takterarahkan yang sederhana, yang mengatakan bahwa setiap permukaan lain adalah takterarahkan jika dan hanya jika permukaan tersebut mempunyai strip Möbius sebagai subhimpunan.[4] Relatedly, ketika pembenamannya menjadi ruang Euklides, strip Möbius hanya mempunyai satu sisi. Sebuah benda tiga dimensi yang berjalan sekali di sekitar permukaan strip tersebut tidak tercermin, melainkan kembali ke titik yang sama yang muncul di sisi lain, memperlihatkan bahwa kedua posisinya hanya merupakan bagian dari satu sisi pada strip tersebut. Perilaku pada strip ini berbeda dengan permukaan terarahkan yang terkenal dalam tiga dimensi seperti strip yang dimodelkan dengan lembaran kertas yang rata, sedotan minuman yang berbentuk tabung, atau bola berongga, yang suatu sisi permukaannya tidak terhubung dengan yang lain.[5] Namun perilaku tersebut merupakan sifat pembenaman strip tersebut yang menjadi ruang, bukan sebuah sifat intristik dari strip Möbius sendiri: terdapat ruang topologis lain yang strip Möbius dapat dibenamkan sehingga mempunyai dua sisi.[6] Sebagai contoh, jika wajah kubus di depan dan di belakang ditempelkan ke satu sama lain dengan mencerminkan sebelah kiri dan sebelah kanan, maka hasilnya berupakan sebuah ruang topologis tiga dimensi (yaitu, darab Cartesius suatu strip Möbius dengan sebuah interval) yang bagian atas dan bawah kubus dapat dipisah dari satu sama lain dengan dua sisi pada strip Möbius.[a] In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an uncountable set of disjoint copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously embedded.[8][9][10]

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 this sense, the Möbius strip is different from an untwisted ring and like a circular disk in having only one boundary.[5] A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a chiral object with right- or left-handedness.[11] 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 surfaces.[12] 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 other.[13] With an even number of twists, however, one obtains a different topological surface, called the annulus.[14]

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 deformation retraction, 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 fundamental group is the same as the fundamental group of a circle, an infinite cyclic group. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up to homotopy) only by the number of times they loop around the strip.[15]

 
Cutting the centerline produces a two-sided (non-Möbius) strip
 
A single off-center cut separates a Möbius strip (purple) from a two-sided strip

Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with two half-twists in it, rather than two separate strips. 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 half-twists.[5] These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic rings.[16][17]

 
Subdivision into six mutually-adjacent regions, bounded by Tietze's graph
 
Solution to the three utilities problem on a Möbius strip

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 four color theorem for the plane.[18] Six colors are always enough. This result is part of the Ringel–Youngs theorem, which states how many colors each topological surface needs.[19] The edges and vertices of these six regions form Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not n the plane, are the Möbius ladders, the boundaries of subdivisions of the Möbius strip into rectangles meeting end-to-end.[20] These include the utility graph, a six-vertex complete bipartite graph whose embedding into the Möbius strip shows that, unlike in the plane, the three utilities problem can be solved on a transparent Möbius strip.[21] The Euler characteristic of the Möbius strip is zero, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers  ,  , and   of vertices, edges, and regions satisfy  . For instance, Tietze's graph has   vertices,   edges, and   regions;  .[18]

Konstruksi

Ada berbagai cara dalam mendefinisikan permukaan geometris melalui strip Möbius dalam topologi, yielding realizations with additional geometric properties.

Menyapu sebuah ruas garis

 
Sebuah strip Möbius disapu dengan memutar ruas garis dalam sebuah bidang putaran.
 
Konoid Plücker disapu dengan gerakan suatu ruas garis yang berbeda.

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 lines.[22] 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 parametric surface defined by equations for the Cartesian coordinates of its points, for   and  , where one parameter   describes the rotation angle of the plane around its central axis and the other parameter   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  -plane and is centered at  .[23] 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 solid torus swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains connected.[24]

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 Plücker's conoid or cylindroid, an algebraic ruled surface in the form of a self-crossing Möbius strip.[25] It has applications in the design of gears.[26]

Permukaan polihedral dan lipatan rata

 
Trihexaflexagon being flexed

A strip of paper can form a flattened Möbius strip in the plane by folding it at   angles so that its center line lies along an equilateral triangle, 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 aspect ratio – the ratio of the strip's length[b] to its width – is  , and the same folding method works for any larger aspect ratio.[27][28] For a strip of nine equilateral triangles, the result is a trihexaflexagon, which can be flexed to reveal different parts of its surface.[29] 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   strip would become a   folded strip whose cross section 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 would be.[27][28]

 
 
Five-vertex polyhedral and flat-folded Möbius strips

The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the cylinder, which requires six triangles and six vertices, even when represented more abstractly as a simplicial complex.[30][c] A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a four-dimensional regular simplex. This four-dimensional polyhedral Möbius strip is the only tight Möbius strip, one that is fully four-dimensional and for which all cuts by hyperplanes separate it into two parts that are topologically equivalent to disks or circles.[31]

Other polyhedral embeddings of Möbius strips include one with four convex quadrilaterals as faces, another with three non-convex quadrilateral faces,[32] and one using the vertices and center point of a regular octahedron, with a triangular boundary.[33] Every abstract triangulation of the projective plane can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its faces;[34] 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 edges.[30] However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral surface.[35] To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the triangulation.[36]

Pita berbentuk persegi panjang yang dibenamkan dengan lancar

A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than  , the same ratio as for the flat-folded equilateral-triangle version of the Möbius strip.[37] This flat triangular embedding can lift to a smooth[d] embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the planes.[37] Mathematically, a smoothly embedded sheet of paper can be modeled as a developable surface, that can bend but cannot stretch.[38][39] As its aspect ratio decreases toward  , all smooth embeddings seem to approach the same triangular form.[40]

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 folds.[28] Instead, unlike in the flat-folded case, there is a lower limit to the aspect ratio of smooth rectangular Möbius strips. Their aspect ratio cannot be less than  , even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this bound.[28][41] Without self-intersections, the aspect ratio must be at least[42] 

Masalah yang belum terpecahkan dalam matematika:

Dapatkah secarik kertas berbentuk persegi panjang dengan ukuran   ditempelkan dari ujung ke ujung agar membentuk sebuah pita Möbius mulus yang dibenamkan di sebuah ruang? [e]

(lebih banyak masalah yang belum terpecahkan dalam matematika)

For aspect ratios between this bound and  , it is unknown whether smooth embeddings, without self-intersection, exist.[28][41][42] If the requirement of smoothness is relaxed to allow continuously differentiable surfaces, the Nash–Kuiper theorem implies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratio becomes.[f] The limiting case, a surface obtained from an infinite strip of the plane between two parallel lines, glued with the opposite orientation to each other, is called the unbounded Möbius strip or the real tautological line bundle.[43] Although it has no smooth embedding into three-dimensional space, it can be embedded smoothly into four-dimensional Euclidean space.[44]

The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have a known analytic description, but can be calculated numerically, and has been the subject of much study in plate theory since the initial work on this subject in 1930 by Michael Sadowsky.[38][39] It is also possible to find algebraic surfaces that contain rectangular developable Möbius strips.[45][46]

Membuat lingkar batas

 
Gluing two Möbius strips to form a Klein bottle
 
A projection of the Sudanese Möbius strip

The edge, or boundary, of a Möbius strip is topologically equivalent to a circle. In common forms of the Möbius strip, it has a different shape from a circle, but it is unknotted, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly circular.[47] One such example is based on the topology of the Klein bottle, a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can be immersed (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 strips.[48] 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 edges.[49]

Lawson's Klein bottle is a self-crossing minimal surface in the unit hypersphere of 4-dimensional space, the set of points of the form for  .[50] Half of this Klein bottle, the subset with  , gives a Möbius strip embedded in the hypersphere as a minimal surface with a great circle as its boundary.[51] This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the 1970s.[52] 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 to all of the swept circles.[49][53] Stereographic projection transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its boundary.[49] 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 centerline.[51] Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the orthogonal group  , the group of symmetries of a circle.[50]

 
Schematic depiction of a cross-cap with an open bottom, showing its level sets. This surface crosses itself along the vertical line segment.

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 cross-cap or crosscap, also has a circular boundary, but otherwise stays on only one side of the plane of this circle,[54] 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 quadrilateral 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 orientation.[55] The two parts of the surface formed by the two glued pairs of edges cross each other with a pinch point like that of a Whitney umbrella at each end of the crossing segment,[56] the same topological structure seen in Plücker's conoid.[25]

Permukaan kelengkungan konstan

The open Möbius strip is the relative interior of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a Riemannian geometry of constant positive, negative, or zero Gaussian curvature. The cases of negative and zero curvature form geodesically complete surfaces, which means that all geodesics ("straight lines" on the surface) may be extended indefinitely in either direction.

Kelengkungan nol
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 bundle.[43] 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 quotient space of a plane by a glide reflection, and (together with the plane, cylinder, torus, and Klein bottle) is one of only five two-dimensional complete flat manifolds.[57]
Kelengkungan negatif
The open Möbius strip also admits complete metric of constant negative curvature. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane, a geometry of constant curvature whose lines are represented in the model by semicircles that meet the  -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 half-plane (actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard surfaces.[58] Again, this can be understood as the quotient of the hyperbolic plane by a glide reflection.[59]
Kelengkungan positif
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 projective plane.[57] However, in a sense it is only one point away from being a complete surface, as the open Möbius strip is homeomorphic to the once-punctured projective plane, the surface obtained by removing any one point from the projective plane.[60]

The minimal surfaces are described as having constant zero mean curvature 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 the Meeks Möbius strip,[61] after its 1982 description by William Hamilton Meeks, III.[62] Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimal surfaces.[63] 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 Björling problem, which defines a minimal surface uniquely from its boundary curve and tangent planes along this curve.[64]

Ruang garis

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 topologically equivalent to the open Möbius strip.[65] One way to see this is to extend the Euclidean plane to the real projective plane by adding one more line, the line at infinity. By projective duality 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 lines.[66] Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius strip.[60] The space of lines in the hyperbolic plane can be parameterized by unordered pairs 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 strip.[67]

These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the affine transformations, and the symmetries of hyperbolic lines include the Möbius transformations.[68] The affine transformations and Möbius transformations both form 6-dimensional Lie groups, topological spaces having a compatible algebraic structure describing the composition of symmetries.[69][70] Because every line in the plane is symmetric to every other line, the open Möbius strip is a homogeneous space, a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called solvmanifolds, and the Möbius strip can be used as a counterexample, showing that not every solvmanifold is a nilmanifold, and that not every solvmanifold can be factored into a direct product of a compact solvmanifold with  . These symmetries also provide another way to construct the Möbius strip itself, as a group model of these Lie groups. A group model consists of a Lie group and a stabilizer subgroup of its action; contracting the cosets 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  -axis consists of all symmetries that take the axis to itself. Each line   corresponds to a coset, the set of symmetries that map   to the  -axis. Therefore, the quotient space, a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbius strip.[71]

Penerapan

 
Electrical flow in a Möbius resistor

Adapun penerapan strip Möbius yang telah dibahas di atas, seperti desain sabuk mekanis yang memakai di seluruh permukaannya dengan rata, dan konoid Plücker dalam desain gerigi. Selain itu, ada penerapan lain mengenai strip Möbius, seperti:

  • Graphene ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism[72]
  • Möbius aromaticity, a property of organic chemicals whose molecular structure forms a cycle, with molecular orbitals aligned along the cycle in the pattern of a Möbius strip[73][74]
  • The Möbius resistor, a strip of conductive material covering the single side of a dielectric Möbius strip, in a way that cancels its own self-inductance[75][76]
  • Resonators with a compact design and a resonant frequency that is half that of identically constructed linear coils[77][78]
  • Polarization patterns in light emerging from a q-plate[79]
  • A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in social choice theory[80]
  • Möbius loop roller coasters, 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[81][82]
  • World maps projected onto a Möbius strip with the convenient properties that there are no east–west boundaries, and that the antipode 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[83][84]

Scientists have also studied the energetics of soap films shaped as Möbius strips,[85][86] the chemical synthesis of molecules with a Möbius strip shape,[87][88] and the formation of larger nanoscale Möbius strips using DNA origami.[89]

Dalam budaya populer

 
Endless Twist, Max Bill, 1956, from the Middelheim Open Air Sculpture Museum

Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting by Corrado Cagli (memorialized in a poem by Charles Olson),[90][91] and two prints by M. C. Escher: Möbius Band I (1961), depicting three folded flatfish biting each others' tails; and Möbius Band II (1963), depicting ants crawling around a lemniscate-shaped Möbius strip.[92][93] It is also a popular subject of mathematical sculpture, including works by Max Bill (Endless Ribbon, 1953), José de Rivera (Infinity, 1967), and Sebastián.[90] A trefoil-knotted Möbius strip was used in John Robinson's Immortality (1982).[94] Charles O. Perry's Continuum (1976) is one of several pieces by Perry exploring variations of the Möbius strip.[95] As a form of mathematics and fiber arts, scarves have been knit into Möbius strips since the work of Elizabeth Zimmermann in the early 1980s.[96]

 
Recycling symbol
 
Google Drive logo (2012–2014)
 
IMPA logo on stamp

Because of their easily recognized form, Möbius strips are a common element of graphic design.[94] The familiar three-arrow logo for recycling, designed in 1970, is based on the smooth triangular form of the Möbius strip,[97] as was the logo for the environmentally-themed Expo '74.[98] Some variations of the recycling symbol use a different embedding with three half-twists instead of one,[97] and the original version of the Google Drive logo used a flat-folded three-twist Möbius strip, as have other similar designs.[99] The Brazilian Instituto Nacional de Matemática Pura e Aplicada (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.[100] The Möbius strip has also featured in the artwork for postage stamps from countries including Brazil, Belgium, the Netherlands, and Switzerland.[101][102]

 
NASCAR Hall of Fame entrance

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 functional part of the architecture.[103][104] An example is the National Library of Kazakhstan, 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.[105] One notable building incorporating a Möbius strip is the NASCAR Hall of Fame, 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.[106] On a smaller scale, Moebius Chair (2006) by Pedro Reyes is a courting bench whose base and sides have the form of a Möbius strip.[107] In food styling, Möbius strips have been used for slicing bagels,[108] making loops out of bacon,[109] and creating new shapes for pasta.[110]

Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in speculative fiction as the basis for a time loop into which unwary victims may become trapped. Examples of this trope include Martin Gardner's "No-Sided Professor" (1946), Armin Joseph Deutsch's "A Subway Named Mobius" (1950) and the film Moebius (1996) based on it. An entire world shaped like a Möbius strip is the setting of Arthur C. Clarke's "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever inventions in multiple stories of William Hazlett Upson from the 1940s.[111] 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 Marcel Proust's In Search of Lost Time (1913–1927), Luigi Pirandello's Six Characters in Search of an Author (1921), Frank Capra's It's a Wonderful Life (1946), John Barth's Lost in the Funhouse (1968), Samuel R. Delany's Dhalgren (1975) and the film Donnie Darko (2001).[112]

One of the musical canons by J. S. Bach, the fifth of 14 canons (BWV 1087) discovered in 1974 in Bach's copy of the Goldberg Variations, features a glide-reflect symmetry in which each voice in the canon repeats, with inverted notes, 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.[113][g] In music theory, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, the chromatic circle. Because the Möbius strip is the configuration space of two unordered points on a circle, the space of all two-note chords takes the shape of a Möbius strip. This conception, and generalizations to more points, is a significant application of orbifolds to music theory.[114][115] Modern musical groups taking their name from the Möbius strip include American electronic rock trio Mobius Band[116] and Norwegian progressive rock band Ring Van Möbius.[117]

Möbius strips and their properties have been used in the design of stage magic. 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 Harry Blackstone Sr. and Thomas Nelson Downs.[118][119]

Lihat pula

  • Möbius counter, a shift register whose output bit is complemented before being fed back into the input bit
  • Penrose triangle, an impossible figure whose boundary appears to wrap around it in a Möbius strip
  • Ribbon theory, the mathematical theory of infinitesimally thin strips that follow knotted space curves
  • Smale–Williams attractor, a fractal formed by repeatedly thickening a space curve to a Möbius strip and then replacing it with the boundary edge
  • Umbilic torus, a three-dimensional shape with its boundary formed by a Möbius strip, glued to itself along its single edge

Notes

  1. ^ Essentially this example, but for a Klein bottle rather than a Möbius strip, is given by (Blackett 1982).[7]
  2. ^ 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
  3. ^ The flat-folded Möbius strip formed from three equilateral triangles does not come from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle to have a different set of vertices.
  4. ^ This piecewise planar and cylindrical embedding has smoothness class  , and can be approximated arbitrarily accurately by infinitely differentiable (class  ) embeddings.[38]
  5. ^ Di antara 1,695 dan 1,73, 127 merupakan bilangan rasional paling sederhana dalam sejumlah rasio aspek yang keberadaan suatu pembenaman mulus belum diketahui.
  6. ^ These surfaces have smoothness class  . For a more fine-grained analysis of the smoothness assumptions that force an embedding to be developable versus the assumptions under which the Nash–Kuiper theorem allows arbitrarily flexible embeddings, see remarks by (Bartels & Hornung 2015), p. 116, following Theorem 2.2.[38]
  7. ^ 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.[113]

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