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135-592: In mathematics , a Möbius strip , Möbius band , or Möbius loop is a surface 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 Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE . The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains

270-416: A 1 × 1 3 {\displaystyle 1\times {\tfrac {1}{3}}} 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. The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in

405-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

540-445: 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 x {\displaystyle x} -axis consists of all symmetries that take the axis to itself. Each line ℓ {\displaystyle \ell } corresponds to

675-430: 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 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

810-400: 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 strip. 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. A path along

945-589: A Möbius strip. As an abstract topological space , the Möbius strip can be embedded into three-dimensional Euclidean space 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 knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent . All of these embeddings have only one side, but when embedded in other spaces,

1080-419: A coset, the set of symmetries that map ℓ {\displaystyle \ell } to the x {\displaystyle x} -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. Beyond the already-discussed applications of Möbius strips to

1215-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

1350-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

1485-510: 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. Stereographic projection transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving

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1620-403: 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 . It is also possible to find algebraic surfaces that contain rectangular developable Möbius strips. The edge, or boundary , of a Möbius strip is topologically equivalent to a circle . In common forms of

1755-507: A matching large sculpture of a Möbius strip on display in their building. The Möbius strip has also featured in the artwork for postage stamps from countries including Brazil, Belgium, the Netherlands, and Switzerland. 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

1890-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

2025-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

2160-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

2295-454: A plane . Another family of graphs that can be embedded on the Möbius strip, but not on the plane, are the Möbius ladders , the boundaries of subdivisions of the Möbius strip into rectangles meeting end-to-end. 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

2430-489: A plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction. The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858. 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

2565-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

2700-473: 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 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. The limiting case, a surface obtained from an infinite strip of

2835-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

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2970-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

3105-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

3240-420: A strip of nine equilateral triangles, the result is a trihexaflexagon , which can be flexed to reveal different parts of its surface. 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 1 × 1 {\displaystyle 1\times 1} strip would become

3375-825: A transparent Möbius strip. 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 V {\displaystyle V} , E {\displaystyle E} , and F {\displaystyle F} of vertices, edges, and regions satisfy V − E + F = 0 {\displaystyle V-E+F=0} . For instance, Tietze's graph has 12 {\displaystyle 12} vertices, 18 {\displaystyle 18} edges, and 6 {\displaystyle 6} regions; 12 − 18 + 6 = 0 {\displaystyle 12-18+6=0} . There are many different ways of defining geometric surfaces with

3510-404: A triangular boundary. 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; 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. However, not every abstract triangulation of

3645-505: 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). One of the musical canons by J. S. Bach ,

3780-462: 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 ourobouros or of figure-eight -shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is unclear. Independently of the mathematical tradition, machinists have long known that mechanical belts wear half as quickly when they form Möbius strips, because they use

3915-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

4050-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

4185-423: Is flat " and "a field is always a ring ". Annulus (mathematics) The open annulus is topologically equivalent to both the open cylinder S × (0,1) and the punctured plane . The area of an annulus is the difference in the areas of the larger circle of radius R and the smaller one of radius r : The area of an annulus is determined by the length of the longest line segment within

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4320-413: Is a chiral object with right- or left-handedness. 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. More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine

4455-471: Is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

4590-779: Is a self-crossing minimal surface in the unit hypersphere of 4-dimensional space, the set of points of the form ( cos ⁡ θ cos ⁡ ϕ , sin ⁡ θ cos ⁡ ϕ , cos ⁡ 2 θ sin ⁡ ϕ , sin ⁡ 2 θ sin ⁡ ϕ ) {\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )} for 0 ≤ θ < π , 0 ≤ ϕ < 2 π {\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi } . Half of this Klein bottle,

4725-489: Is called the Meeks Möbius strip, after its 1982 description by William Hamilton Meeks, III . 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. Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to

4860-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

4995-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

5130-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

5265-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

5400-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

5535-611: 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. These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic rings . The Möbius strip can be cut into six mutually-adjacent regions, showing that maps on

Möbius strip - Misplaced Pages Continue

5670-405: 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 orientable surfaces 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

5805-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

5940-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

6075-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

6210-440: Is one of the forms of the open Möbius strip. 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. These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the affine transformations , and the symmetries of hyperbolic lines include

6345-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

6480-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

6615-406: 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, Moebius Chair (2006) by Pedro Reyes is a courting bench whose base and sides have the form of a Möbius strip. As a form of mathematics and fiber arts , scarves have been knit into Möbius strips since the work of Elizabeth Zimmermann in

6750-424: 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. Other polyhedral embeddings of Möbius strips include one with four convex quadrilaterals as faces, another with three non-convex quadrilateral faces, and one using the vertices and center point of a regular octahedron , with

6885-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

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7020-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

7155-428: 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. 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. Lawson's Klein bottle

7290-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

7425-524: The Möbius transformations . The affine transformations and Möbius transformations both form 6-dimensional Lie groups , topological spaces having a compatible algebraic structure describing the composition of symmetries. 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

7560-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

7695-506: The Björling problem , which defines a minimal surface uniquely from its boundary curve and tangent planes along this curve. 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. One way to see this is to extend the Euclidean plane to

7830-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

7965-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

8100-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

8235-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

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8370-413: The orthogonal group O ( 2 ) {\displaystyle \mathrm {O} (2)} , the group of symmetries of a circle. 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

8505-405: 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. Therefore, the space of Euclidean lines is a punctured projective plane, which

8640-415: The third century CE. 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 untwisted rings . 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

8775-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

8910-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

9045-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

9180-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

9315-527: The Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture. 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. One notable building incorporating a Möbius strip is the NASCAR Hall of Fame , which

9450-479: The Möbius strip can be represented geometrically, as a polyhedral surface. To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the triangulation. 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 3 ≈ 1.73 {\displaystyle {\sqrt {3}}\approx 1.73} ,

9585-439: 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 R n {\displaystyle \mathbb {R} ^{n}} . 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

9720-411: 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. Cutting

9855-532: The Möbius strip include an untitled 1947 painting by Corrado Cagli (memorialized in a poem by Charles Olson ), 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. It is also a popular subject of mathematical sculpture , including works by Max Bill ( Endless Ribbon , 1953), José de Rivera ( Infinity , 1967), and Sebastián . A trefoil-knotted Möbius strip

9990-413: The Möbius strip may have two sides. It has only a single boundary curve . Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a ruled surface 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 developable surface or be folded flat ;

10125-497: 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. 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

10260-404: 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 subset. Relatedly, when embedded into Euclidean space , the Möbius strip has only one side. A three-dimensional object that slides one time around the surface of the strip

10395-424: The annulus up into an infinite number of annuli of infinitesimal width dρ and area 2π ρ dρ and then integrating from ρ = r to ρ = R : The area of an annulus sector of angle θ , with θ measured in radians, is given by In complex analysis an annulus ann( a ; r , R ) in the complex plane is an open region defined as If r = 0 {\displaystyle r=0} ,

10530-424: The annulus, which is the chord tangent to the inner circle, 2 d in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so d and r are sides of a right-angled triangle with hypotenuse R , and the area of the annulus is given by The area can also be obtained via calculus by dividing

10665-442: The aspect ratio must be at least 2 3 3 + 2 3 ≈ 1.695. {\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.} For aspect ratios between this bound and 3 {\displaystyle {\sqrt {3}}} , it has been an open problem whether smooth embeddings, without self-intersection, exist. In 2023, Richard Schwartz announced

10800-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

10935-421: The circularity of its boundary. 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. Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to

11070-503: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

11205-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

11340-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

11475-485: The design of stage magic . One such trick, known as the Afghan bands, uses the fact that the Möbius strip remains in one piece as 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 . Mathematics Mathematics

11610-483: 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: Scientists have also studied the energetics of soap films shaped as Möbius strips, the chemical synthesis of molecules with a Möbius strip shape, and the formation of larger nanoscale Möbius strips using DNA origami . Two-dimensional artworks featuring

11745-449: The design of the recycling symbol . Many architectural concepts have been inspired by the Möbius strip, including the building design for the NASCAR Hall of Fame . Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. The canons of J. S. Bach have been analyzed using Möbius strips. Many works of speculative fiction feature Möbius strips; more generally,

11880-645: The early 1980s. In food styling , Möbius strips have been used for slicing bagels , making loops out of bacon , and creating new shapes for pasta . 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

12015-496: 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. A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it

12150-411: 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 to its width – is 3 ≈ 1.73 {\displaystyle {\sqrt {3}}\approx 1.73} , and the same folding method works for any larger aspect ratio. For

12285-473: 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 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. Another use of this surface

12420-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

12555-556: 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. In music theory , tones that differ by an octave are generally considered to be equivalent notes, and

12690-404: 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. Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with

12825-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

12960-511: The flattened Möbius strips include the trihexaflexagon . The Sudanese Möbius strip is a minimal surface in a hypersphere , 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 surfaces of constant curvature . Certain highly symmetric spaces whose points represent lines in

13095-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

13230-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

13365-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

13500-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

13635-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

13770-445: 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. It has applications in the design of gears . A strip of paper can form a flattened Möbius strip in the plane by folding it at 60 ∘ {\displaystyle 60^{\circ }} angles so that its center line lies along an equilateral triangle , and attaching

13905-479: The other. However, this is a property of its embedding into space rather than an intrinsic property of the Möbius strip itself: there exist other topological spaces in which the Möbius strip can be embedded so that it has two sides. 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 Cartesian product of

14040-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

14175-415: The plane between two parallel lines, glued with the opposite orientation to each other, is called the unbounded Möbius strip or the real tautological line bundle . Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean space. The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have

14310-577: The plane have the shape of a Möbius strip. The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes 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 social choice theory . In popular culture, Möbius strips appear in artworks by M. C. Escher , Max Bill , and others, and in

14445-412: The plane of this circle, 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. The two parts of the surface formed by

14580-1123: The plane's rotation. This can be described as a parametric surface defined by equations for the Cartesian coordinates of its points, x ( u , v ) = ( 1 + v 2 cos ⁡ u 2 ) cos ⁡ u y ( u , v ) = ( 1 + v 2 cos ⁡ u 2 ) sin ⁡ u z ( u , v ) = v 2 sin ⁡ u 2 {\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}}} for 0 ≤ u < 2 π {\displaystyle 0\leq u<2\pi } and − 1 ≤ v ≤ 1 {\displaystyle -1\leq v\leq 1} , where one parameter u {\displaystyle u} describes

14715-422: 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 . 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

14850-605: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

14985-410: The region is known as the punctured disk (a disk with a point hole in the center) of radius R around the point a . As a subset of the complex plane , an annulus can be considered as a Riemann surface . The complex structure of an annulus depends only on the ratio ⁠ r / R ⁠ . Each annulus ann( a ; r , R ) can be holomorphically mapped to a standard one centered at

15120-531: The rotation angle of the plane around its central axis and the other parameter v {\displaystyle v} 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 x y {\displaystyle xy} -plane and is centered at ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} . The same method can produce Möbius strips with any odd number of half-twists, by rotating

15255-410: The same knot and they have the same number of twists as each other. With an even number of twists, however, one obtains a different topological surface, called the annulus . 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

15390-569: The same ratio as for the flat-folded equilateral-triangle version of the Möbius strip. 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 planes. Mathematically, a smoothly embedded sheet of paper can be modeled as a developable surface , that can bend but cannot stretch. As its aspect ratio decreases toward 3 {\displaystyle {\sqrt {3}}} , all smooth embeddings seem to approach

15525-690: The same triangular form. 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. 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 π / 2 ≈ 1.57 {\displaystyle \pi /2\approx 1.57} , even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this bound. Without self-intersections,

15660-400: 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. A line or line segment swept in a different motion, rotating in a horizontal plane around

15795-566: 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 . Modern musical groups taking their name from the Möbius strip include American electronic rock trio Mobius Band and Norwegian progressive rock band Ring Van Möbius . Möbius strips and their properties have been used in

15930-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

16065-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

16200-451: The subset with 0 ≤ ϕ < π {\displaystyle 0\leq \phi <\pi } , gives a Möbius strip embedded in the hypersphere as a minimal surface with a great circle as its boundary. This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the 1970s. Geometrically Lawson's Klein bottle can be constructed by sweeping

16335-507: The surface of the Möbius strip can sometimes require six colors, in contrast to the four color theorem for the plane. Six colors are always enough. This result is part of the Ringel–Youngs theorem , which states how many colors each topological surface needs. 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

16470-431: The surface) may be extended indefinitely in either direction. 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

16605-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

16740-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

16875-419: The topology of the Möbius strip, yielding realizations with additional geometric properties. 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. 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

17010-563: 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, the same topological structure seen in Plücker's conoid. 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

17145-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

17280-413: Was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not match up. Another mosaic from the town of Sentinum (depicted) shows the zodiac , held by the god Aion , 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

17415-457: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

17550-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

17685-550: 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. The Möbius strip has several curious properties. It is a non-orientable surface : 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

17820-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

17955-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

18090-475: Was the logo for the environmentally-themed Expo '74 . Some variations of the recycling symbol use a different embedding with three half-twists instead of one, and the original version of the Google Drive logo used a flat-folded three-twist Möbius strip, as have other similar designs. The Brazilian Instituto Nacional de Matemática Pura e Aplicada (IMPA) uses a stylized smooth Möbius strip as their logo, and has

18225-461: Was used in John Robinson ' s Immortality (1982). Charles O. Perry 's Continuum (1976) is one of several pieces by Perry exploring variations of the Möbius strip. Because of their easily recognized form, Möbius strips are a common element of graphic design . The familiar three-arrow logo for recycling , designed in 1970, is based on the smooth triangular form of the Möbius strip, as

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