Möbius strip or Möbius band (/ˈmɜːrbiəs/ (non-rhotic), US:
/ˈmeɪ-, ˈmoʊ-/; German: [ˈmøːbi̯ʊs]), also spelled Mobius or
Moebius, is a surface with only one side (when embedded in
three-dimensional Euclidean space) and only one boundary. The Möbius
strip has the mathematical property of being unorientable. It can be
realized as a ruled surface. It was discovered independently by the
August Ferdinand Möbius
August Ferdinand Möbius and Johann Benedict
Listing in 1858.
An example of a
Möbius strip can be created by taking a paper strip
and giving it a half-twist, and then joining the ends of the strip to
form a loop. However, the
Möbius strip is not a surface of only one
exact size and shape, such as the half-twisted paper strip depicted in
the illustration. Rather, mathematicians refer to the closed Möbius
band as any surface that is homeomorphic to this strip. Its boundary
is a simple closed curve, i.e., homeomorphic to a circle. This allows
for a very wide variety of geometric versions of the Möbius band as
surfaces each having a definite size and shape. For example, any
rectangle can be glued to itself (by identifying one edge with the
opposite edge after a reversal of orientation) to make a Möbius band.
Some of these can be smoothly modeled in Euclidean space, and others
A half-twist clockwise gives an embedding of the Möbius strip
different from that of a half-twist counterclockwise – that is, as
an embedded object in Euclidean space, the
Möbius strip is a chiral
object with right- or left-handedness. However, the underlying
topological spaces within the
Möbius strip are homeomorphic in each
case. An infinite number of topologically different embeddings of the
same topological space into three-dimensional space exist, as the
Möbius strip can also be formed by twisting the strip an odd number
of times greater than one, or by knotting and twisting the strip,
before joining its ends. The complete open Möbius band is an example
of a topological surface that is closely related to the standard
Möbius strip, but that is not homeomorphic to it.
Finding algebraic equations, the solutions of which have the topology
of a Möbius strip, is straightforward, but, in general, these
equations do not describe the same geometric shape that one gets from
the twisted paper model described above. In particular, the twisted
paper model is a developable surface, having zero Gaussian curvature.
A system of differential-algebraic equations that describes models of
this type was published in 2007 together with its numerical
Euler characteristic of the
Möbius strip is zero.
2 Geometry and topology
2.1 Widest isometric embedding in 3-space
2.3 Computer graphics
2.4 Open Möbius band
2.5 Möbius band with round boundary
3 Related objects
4.1 In stage magic
5 See also
7 External links
August Ferdinand Möbius
Möbius strip has several curious properties. A line drawn
starting from the seam down the middle meets back at the seam, but at
the other side. If continued, the line meets the starting point, and
is double the length of the original strip. This single continuous
curve demonstrates that the
Möbius strip has only one boundary.
Möbius strip along the center line with a pair of scissors
yields one long strip with two full twists in it, rather than two
separate strips; the result is not a Möbius strip. This happens
because the original strip only has one edge that is twice as long as
the original strip. Cutting creates a second independent edge, half of
which was on each side of the scissors. Cutting this new, longer,
strip down the middle creates two strips wound around each other, each
with two full twists.
If the strip is cut along about a third of the way in from the edge,
it creates two strips: One is a thinner
Möbius strip – it is the
center third of the original strip, comprising one-third of the width
and the same length as the original strip. The other is a longer but
thin strip with two full twists in it – this is a neighborhood of
the edge of the original strip, and it comprises one-third of the
width and twice the length of the original strip.
Other analogous strips can be obtained by similarly joining strips
with two or more half-twists in them instead of one. For example, a
strip with three half-twists, when divided lengthwise, becomes a
twisted strip tied in a trefoil knot. (If this knot is unravelled, the
strip has eight half-twists.) A strip with N half-twists, when
bisected, becomes a strip with N + 1 full twists. Giving it extra
twists and reconnecting the ends produces figures called paradromic
Geometry and topology
Möbius strip made with a piece of paper and tape.
An object that existed in a mobius-strip-shaped universe would be
indistinguishable from its own mirror image - this fiddler crab's
larger claw switches between left to right with every circulation. It
is not impossible that the universe may have this property, see
To turn a rectangle into a Möbius strip, join the edges labelled A so
that the directions of the arrows match.
One way to represent the
Möbius strip as a subset of
Euclidean space is using the parametrization:
displaystyle x(u,v)=left(1+ frac v 2 cos frac u 2
displaystyle y(u,v)=left(1+ frac v 2 cos frac u 2
displaystyle z(u,v)= frac v 2 sin frac u 2
where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This creates a Möbius
strip of width 1 whose center circle has radius 1, lies in the xy
plane and is centered at (0, 0, 0). The parameter u runs around the
strip while v moves from one edge to the other.
In cylindrical polar coordinates (r, θ, z), an unbounded version of
Möbius strip can be represented by the equation:
displaystyle log(r)sin left( frac 1 2 theta right)=zcos left(
frac 1 2 theta right).
Widest isometric embedding in 3-space
If a smooth
Möbius strip in three-space is a rectangular one – that
is, created from identifying two opposite sides of a geometrical
rectangle with bending but not stretching the surface – then such an
embedding is known to be possible if the aspect ratio of the rectangle
is greater than the square root of three. (Note the shorter sides of
the rectangle are identified to obtain the Möbius strip.) For an
aspect ratio less than or equal to the square root of three, however,
a smooth embedding of a rectangular
Möbius strip into three-space may
As the aspect ratio approaches the limiting ratio of √3 from above,
any such rectangular
Möbius strip in three-space seems to approach a
shape that in the limit can be thought of as a strip of three
equilateral triangles, folded on top of one another so that they
occupy just one equilateral triangle in three-space.
Möbius strip in three-space is only once continuously
differentiable (in symbols: C1), however, then the theorem of
Nash-Kuiper shows that no lower bound exists.
A method of making a
Möbius strip from a rectangular strip too wide
to simply twist and join (e.g., a rectangle only one unit long and one
unit wide) is to first fold the wide direction back and forth using an
even number of folds—an "accordion fold"—so that the folded strip
becomes narrow enough that it can be twisted and joined, much as a
single long-enough strip can be joined. With two folds, for
example, a 1 × 1 strip would become a 1 × ⅓ folded strip whose
cross section is in the shape of an 'N' and would remain an 'N' after
a half-twist. This folded strip, three times as long as it is wide,
would be long enough to then join at the ends. This method works in
principle, but becomes impractical after sufficiently many folds, if
paper is used. Using normal paper, this construction can be folded
flat, with all the layers of the paper in a single plane, but
mathematically, whether this is possible without stretching the
surface of the rectangle is not clear.
Möbius strip can be defined as the square [0, 1]
× [0, 1] with its top and bottom sides identified by the relation (x,
0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.
A less used presentation of the
Möbius strip is as the topological
quotient of a torus. A torus can be constructed as the square [0,
1] × [0, 1] with the edges identified as (0, y) ~ (1, y) (glue left
to right) and (x, 0) ~ (x, 1) (glue bottom to top). If one then also
identified (x, y) ~ (y, x), then one obtains the Möbius strip. The
diagonal of the square (the points (x, x) where both coordinates
agree) becomes the boundary of the Möbius strip, and carries an
orbifold structure, which geometrically corresponds to "reflection"
– geodesics (straight lines) in the
Möbius strip reflect off the
edge back into the strip. Notationally, this is written as T2/S2 –
the 2-torus quotiented by the group action of the symmetric group on
two letters (switching coordinates), and it can be thought of as the
configuration space of two unordered points on the circle, possibly
the same (the edge corresponds to the points being the same), with the
torus corresponding to two ordered points on the circle.
Möbius strip is a two-dimensional compact manifold (i.e. a
surface) with boundary. It is a standard example of a surface that is
not orientable. In fact, the
Möbius strip is the epitome of the
topological phenomenon of nonorientability. This is because
two-dimensional shapes (surfaces) are the lowest-dimensional shapes
for which nonorientability is possible and the
Möbius strip is the
only surface that is topologically a subspace of every nonorientable
surface. As a result, any surface is nonorientable if and only if it
contains a Möbius band as a subspace.
Möbius strip is also a standard example used to illustrate the
mathematical concept of a fiber bundle. Specifically, it is a
nontrivial bundle over the circle S1 with a fiber the unit interval, I
= [0, 1]. Looking only at the edge of the
Möbius strip gives a
nontrivial two point (or Z2) bundle over S1.
A simple construction of the
Möbius strip that can be used to portray
it in computer graphics or modeling packages is:
Take a rectangular strip. Rotate it around a fixed point not in its
plane. At every step, also rotate the strip along a line in its plane
(the line that divides the strip in two) and perpendicular to the main
orbital radius. The surface generated on one complete revolution is
the Möbius strip.
Möbius strip and cut it along the middle of the strip. This
forms a new strip, which is a rectangle joined by rotating one end a
whole turn. By cutting it down the middle again, this forms two
interlocking whole-turn strips.
Open Möbius band
The open Möbius band is formed by deleting the boundary of the
standard Möbius band. It is constructed from the set S = (x, y) ∈
R2 : 0 ≤ x ≤ 1 and 0 < y < 1 by identifying (glueing)
the points (0, y) and (1, 1 − y) for all 0 < y < 1.
It may be constructed as a surface of constant positive, negative, or
zero (Gaussian) curvature. In the cases of negative and zero
curvature, the Möbius band can be constructed as a (geodesically)
complete surface, which means that all geodesics ("straight lines" on
the surface) may be extended indefinitely in either direction.
Constant negative curvature: Like the plane and the open cylinder, the
open Möbius band admits not only a complete metric of constant
curvature 0, but also a complete metric of constant negative
curvature, say −1. One way to see this is to begin with the upper
half plane (Poincaré) model of the hyperbolic plane ℍ, namely ℍ =
(x, y) ∈ ℝ2 y > 0 with the
Riemannian metric given by (dx2
+ dy2) / y2. The orientation-preserving isometries of this metric are
all the maps f : ℍ → ℍ of the form f(z) := (az + b) /
(cz + d), where a, b, c, d are real numbers satisfying ad − bc = 1.
Here z is a complex number with Im(z) > 0, and we have identified
ℍ with z ∈ ℂ Im(z) > 0 endowed with the Riemannian metric
that was mentioned. Then one orientation-reversing isometry g of ℍ
given by g(z) := -conj(z), where conj(z) denotes the complex
conjugate of z. These facts imply that the mapping h : ℍ →
ℍ given by h(z) := −2⋅conj(z) is an orientation-reversing
isometry of ℍ that generates an infinite cyclic group G of
isometries. (Its square is the isometry h(z) := 4⋅z, which can
be expressed as (2z + 0) / (0z + 1/2).) The quotient ℍ / G of the
action of this group can easily be seen to be topologically a Möbius
band. But it is also easy to verify that it is complete and
non-compact, with constant negative curvature −1.
The group of isometries of this Möbius band is 1-dimensional and is
isomorphic to the special orthogonal group SO(2).
(Constant) zero curvature: This may also be constructed as a complete
surface, by starting with portion of the plane R2 defined by 0 ≤ y
≤ 1 and identifying (x, 0) with (−x, 1) for all x in R (the
reals). The resulting metric makes the open Möbius band into a
(geodesically) complete flat surface (i.e., having Gaussian curvature
equal to 0 everywhere). This is the only metric on the Möbius band,
up to uniform scaling, that is both flat and complete.
The group of isometries of this Möbius band is 1-dimensional and is
isomorphic to the orthogonal group SO(2).
Constant positive curvature: A Möbius band 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. The projective plane P2 of constant curvature +1 may
be constructed as the quotient of the unit sphere S2 in R3 by the
antipodal map A: S2 → S2, defined by A(x, y, z) = (−x, −y,
−z). The open Möbius band is homeomorphic to the once-punctured
projective plane, that is, P2 with any one point removed. This may be
thought of as the closest that a Möbius band of constant positive
curvature can get to being a complete surface: just one point away.
The group of isometries of this Möbius band is also 1-dimensional and
isomorphic to the orthogonal group O(2).
The space of unoriented lines in the plane is diffeomorphic to the
open Möbius band. To see why, let L(θ) denote the line through
the origin at an angle θ to the positive x-axis. For each L(θ) there
is the family P(θ) of all lines in the plane that are perpendicular
to L(θ). Topologically, the family P(θ) is just a line (because each
line in P(θ) intersects the line L(θ) in just one point). In this
way, as θ increases in the range 0° ≤ θ < 180°, the line
L(θ) represents a line's worth of distinct lines in the plane. But
when θ reaches 180°, L(180°) is identical to L(0), and so the
families P(0°) and P(180°) of perpendicular lines are also identical
families. The line L(0°), however, has returned to itself as L(180°)
pointed in the opposite direction. Every line in the plane corresponds
to exactly one line in some family P(θ), for exactly one θ, for 0°
≤ θ < 180°, and P(180°) is identical to P(0°) but returns
pointed in the opposite direction. This ensures that the space of all
lines in the plane – the union of all the L(θ) for 0° ≤ θ ≤
180° – is an open Möbius band.
The group of bijective linear transformations GL(2, R) of the plane to
itself (real 2 × 2 matrices with non-zero determinant) naturally
induces bijections of the space of lines in the plane to itself, which
form a group of self-homeomorphisms of the space of lines. Hence the
same group forms a group of self-homeomorphisms of the Möbius band
described in the previous paragraph. But there is no metric on the
space of lines in the plane that is invariant under the action of this
group of homeomorphisms. In this sense, the space of lines in the
plane has no natural metric on it.
This means that the Möbius band possesses a natural 4-dimensional Lie
group of self-homeomorphisms, given by GL(2, R), but this high degree
of symmetry cannot be exhibited as the group of isometries of any
Möbius band with round boundary
The edge, or boundary, of a
Möbius strip is homeomorphic
(topologically equivalent) to a circle. Under the usual embeddings of
the strip in Euclidean space, as above, the boundary is not a true
circle. However, it is possible to embed a
Möbius strip in three
dimensions so that the boundary is a perfect circle lying in some
plane. For example, see Figures 307, 308, and 309 of "Geometry and the
A much more geometric embedding begins with a minimal Klein bottle
immersed in the 3-sphere, as discovered by Blaine Lawson. We then take
half of this
Klein bottle to get a Möbius band embedded in the
3-sphere (the unit sphere in 4-space). The result is sometimes called
the "Sudanese Möbius Band" , where "sudanese" refers not to the
Sudan but to the names of two topologists, Sue Goodman and
Daniel Asimov. Applying stereographic projection to the Sudanese band
places it in 3-dimensional space, as can be seen below – a version
due to George Francis can be found here.
From Lawson's minimal
Klein bottle we derive an embedding of the band
3-sphere S3, regarded as a subset of C2, which is
geometrically the same as R4. We map angles η, φ to complex numbers
z1, z2 via
displaystyle z_ 1 =sin eta ,e^ ivarphi
displaystyle z_ 2 =cos eta ,e^ ivarphi /2 .
Here the parameter η runs from 0 to π and φ runs from 0 to 2π.
Since z1 2 + z2 2 = 1, the embedded surface lies
entirely in S3. The boundary of the strip is given by z2 = 1
(corresponding to η = 0, π), which is clearly a circle on the
To obtain an embedding of the
Möbius strip in R3 one maps S3 to R3
via a stereographic projection. The projection point can be any point
on S3 that does not lie on the embedded
Möbius strip (this rules out
all the usual projection points). One possible choice is
displaystyle left 1/ sqrt 2 ,i/ sqrt 2 right
. Stereographic projections map circles to circles and preserves the
circular boundary of the strip. The result is a smooth embedding of
Möbius strip into R3 with a circular edge and no
The Sudanese Möbius band in the three-sphere S3 is geometrically a
fibre bundle over a great circle, whose fibres are great semicircles.
The most symmetrical image of a stereographic projection of this band
into R3 is obtained by using a projection point that lies on that
great circle that runs through the midpoint of each of the
semicircles. Each choice of such a projection point results in an
image that is congruent to any other. But because such a projection
point lies on the Möbius band itself, two aspects of the image are
significantly different from the case (illustrated above) where the
point is not on the band: 1) the image in R3 is not the full Möbius
band, but rather the band with one point removed (from its
centerline); and 2) the image is unbounded – and as it gets
increasingly far from the origin of R3, it increasingly approximates a
plane. Yet this version of the stereographic image has a group of 4
symmetries in R3 (it is isomorphic to the Klein 4-group), as compared
with the bounded version illustrated above having its group of
symmetries the unique group of order 2. (If all symmetries and not
just orientation-preserving isometries of R3 are allowed, the numbers
of symmetries in each case doubles.)
But the most geometrically symmetrical version of all is the original
Sudanese Möbius band in the three-sphere S3, where its full group of
symmetries is isomorphic to the
Lie group O(2). Having an infinite
cardinality (that of the continuum), this is far larger than the
symmetry group of any possible embedding of the Möbius band in R3.
A closely related 'strange' geometrical object is the Klein bottle. A
Klein bottle can be produced by gluing two Möbius strips together
along their edges; this cannot be done in ordinary three-dimensional
Euclidean space without creating self-intersections.
Another closely related manifold is the real projective plane. If a
circular disk is cut out of the real projective plane, what is left is
a Möbius strip. Going in the other direction, if one glues a disk
Möbius strip by identifying their boundaries, the result is the
projective plane. To visualize this, it is helpful to deform the
Möbius strip so that its boundary is an ordinary circle (see above).
The real projective plane, like the Klein bottle, cannot be embedded
in three-dimensions without self-intersections.
In graph theory, the
Möbius ladder is a cubic graph closely related
to the Möbius strip.
In 1968, Gonzalo Vélez Jahn (UCV, Caracas, Venezuela) discovered
three dimensional bodies with Möbian characteristics; these were
later described by
Martin Gardner as prismatic rings that became
toroidal polyhedrons in his August 1978
Mathematical Games column in
Mathematical art: a scarf designed as a Möbius strip
There have been several technical applications for the Möbius strip.
Giant Möbius strips have been used as conveyor belts that last longer
because the entire surface area of the belt gets the same amount of
wear, and as continuous-loop recording tapes (to double the playing
time). Möbius strips are common in the manufacture of fabric computer
printer and typewriter ribbons, as they let the ribbon be twice as
wide as the print head while using both halves evenly.
Möbius resistor is an electronic circuit element that cancels its
own inductive reactance.
Nikola Tesla patented similar technology in
1894: "Coil for Electro Magnets" was intended for use with his
system of global transmission of electricity without wires.
Möbius strip is the configuration space of two unordered points
on a circle. Consequently, in music theory, the space of all two-note
chords, known as dyads, takes the shape of a Möbius strip; this and
generalizations to more points is a significant application of
orbifolds to music theory.
In physics/electro-technology as:
A compact resonator with a resonance frequency that is half that of
identically constructed linear coils
An inductionless resistor
Superconductors with high transition temperature
Möbius resonator 
In chemistry/nano-technology as:
Molecular knots with special characteristics (Knotane , Chirality)
Graphene volume (nano-graphite) with new electronic characteristics,
like helical magnetism
A special type of aromaticity: Möbius aromaticity
Charged particles caught in the magnetic field of the earth that can
move on a Möbius band
The cyclotide (cyclic protein) kalata B1, active substance of the
plant Oldenlandia affinis, contains Möbius topology for the peptide
In stage magic
Möbius strip principle has been used as a method of creating the
illusion of magic. The trick, known as the Afghan bands, 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
Harry Blackstone Sr.
Harry Blackstone Sr. and Thomas Nelson Downs.
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Möbius strip in Wiktionary, the free dictionary.
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Weisstein, Eric W. "Möbius Strip". Math