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MÖBIUS STRIP

Surface studied by Listing and Möbius in 1858.
August Ferdinand Möbius (1790-1868): German astronomer and mathematician.
Other name: Möbius (or Moebius) band, ring, belt.

 
Rotoidal representation: see Möbius surface.
Wunderlich representation: surface envelope of the rectifying planes (i.e. the rectifying developable) of the rational curve: with .
Meeks representation, portion of a minimal surface:

Representation on Plücker's conoid.

A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a half-twist, or any topologically equivalent surface.
 
Maple program giving an animation of the opposite construction.
with(plots):a:=1/2:b:=1/3:c:=1/6:d:=2/3:e:=1/3:C:=4/5:
x0:=(1+d^2*t^2+2*d*e*t^4+e^2*t^6)/2:x:=(a*t+b*t^3+c*t^5)/x0:y:=(d*t+e*t^3)/x0:
z:=-C/x0:t:=tan(tt):
a1:=diff(v1,tt):a2:=diff(v2,tt):a3:=diff(v3,tt):
v1:=diff(x,tt):v2:=diff(y,tt):v3:=diff(z,tt):
b1:=v2*a3-a2*v3:b2:=a1*v3-v1*a3:b3:=v1*a2-a1*v2:
n1:=simplify(v2*b3-b2*v3):n2:=simplify(b1*v3-v1*b3):n3:=simplify(v1*b2-b1*v2):
dn1:=diff(n1,tt):dn2:=diff(n2,tt):dn3:=diff(n3,tt):
c1:=n2*dn3-dn2*n3:c2:=dn1*n3-n1*dn3:c3:=n1*dn2-dn1*n2:
facteur:=simplify(sqrt(b1^2+b2^2+b3^2)/(b1*c1+b2*c2+b3*c3)):
c1:=simplify(c1*facteur):c2:=simplify(c2*facteur):c3:=simplify(c3*facteur):
ds:=simplify(sqrt(v1^2+v2^2+v3^2)):
s:=a->evalf(Int(ds,tt=0..a,4))/4:                    d:=a->plot3d([x/s(a)+u*c1/s(a),y/s(a)+u*c2/s(a),(z+2*C)/s(a)+u*c3/s(a)],tt=-a..a,u=-1/3*s(a)..1/3*s(a),grid=[150,2],style=patchnogrid):
n:=40:display([seq(d(k*Pi/2.0001/n,50),k=1..n)],orientation=[-60,80],lightmodel=light2,insequence=true);

Therefore, we obtain a Möbius strip by turning regularly a segment of a line with constant length around a circle with a half-twist or, more generally, an odd number of half-twits; these various strips are homeomorphic, but not isotopic in (it is not possible to pass continuously from one surface to the other), and for each number of half-twists, there exist two isotopy classes, mirror images of one another:
 
1 half-twist 3 half-twists 9 half-twists
right-handed strip
left-handed strip

Moreover, the boundary of the strip with 3 half-twists is a trefoil knot, and, more generally,
the boundary of the strip with 2p + 1 half-twists is a toroidal knot of order (2p + 1, 2).

These rotoidal surfaces are not developable: they cannot be obtained from a piece of paper without tearing; inversely, the strip that is naturally obtained with a rectangular piece of paper does not have a simple parametrization; here is one owed to W. Wunderlich (cf. above), in the right-handed case with one half-twist:
 
The Möbius-Wunderlich strip has the property that it develops into a rectangle and it minimizes at all points the strain energy.

Here is another example of a Möbius strip developable into a rectangle, composed of 3 cylindrical sections joined by planar sections:
 

Left-handed strip with one half-twist: the radius of one of the cylinders is equal to twice that of the other ones and the planar sections are parallel.

Right-handed strip with 3 half-twists.

Here are two other developable Möbius strips obtained by putting strips cut on cones next to one another.
 
The blue and green strips are cut on cones of revolution; the red strip on two cones of revolution; the pattern is unfortunately not a rectangle, but a parallelogram, hence the connection with a right angle.

 
This is a Möbius strip with 3 half-twists instead of 1, obtained thanks to 3 conical strips, the directrix of the cones being Viviani's curve.
The boundary is self-parallel, but the pattern does not have linear edges.

 
This strip with two half-twists (which is, therefore, not a Möbius strip) is developable since it is traced on a cylinder based on a lemniscate. Its pattern does not have linear edges, but our eyes tend to think so, hence our tendency to consider this figure to be impossible!
 

We can also wonder what ratios length/width of the initial rectangle allow to construct a Möbius strip with no self-intersection:
 
Here is already a Möbius strip with one half-twist for which the ratio length/width is equal to 3Ö3 (slightly greater in fact for the sake of clarity).
The view from above is a regular hexagon.
The initial pattern is indicated on the right (the 3 folds are dotted).

But with a rectangle for which the ratio length/width is equal to Ö3, you will construct a Möbius strip for which the view from above is an equilateral triangle.

 
Möbius strip with 3 half-twists 
Strip with 4 half-twists: it is not a Möbius strip, it has two faces!
Möbius strip with 5 half-twists
Strip with 6 half-twists, which is not a Möbius strip

The same one, tightened (almost) as much as possible:
Length of the strip equal to 3Ö3 times the width.
It cannot be tightened more without self-intersections.

The same one, tightened (almost) as much as possible:
Length of the strip equal to 4 times the width.


The same one, tightened (almost) as much as possible:
Length of the strip equal to times the width.

The same one, tightened (almost) as much as possible:
Length of the strip equal to  times the width.


 
Therefore, except in the case n = 3, the Möbius strip with n half-twists tightened as much as possible, with circumference a regular n-gon, needs a rectangle  (alternating folds hollow/bump).

 
When a Möbius strip is cut at its center, we get only one strip but with two edges, and 4 half-twists (and not 2 half-twists like it could be believed); if it is cut at one third of its length, we get a Möbius strip (at the center) and a strip with 2 entangled edges:
 
 
 

 

Strip with two half-twists, hence homeomorphic to the cylinder, that provides a two-sheeted covering of the Möbius strip:
  Animation: Daniel Audet

Just as the Klein bottle cannot be represented in  without self-intersections, the Möbius strip cannot be represented in the plane without self-intersections.
 
The Möbius strip is a surface that can be characterized by the fact that it has only one face (in other words, it is one-sided, hence non-orientable), a unique boundary and its genus is equal to 1 (i.e. a closed curve traced inside it can leave it connected, but not two curves).

 
 
We also get a topological Möbius strip by identification of the opposite sides of a rectangle with an inversion of direction.
The oriented segment line [A1 B1] is identified to [A2 B2], so that there is only one boundary [A1 B2] linked to [A2 B1].
In the real strip, A1 = A2 =A and B1 = B2 = B.
If the strip is cut along the dotted lines opposite...
... and the pieces 1 and 2 are moved like opposite...
...the we get a less classic triangular representation of the Möbius strip.
Here, the oriented segment line [J1 J2] is identified to [J3 J1], so that the 3 points J1, J2, J3 are identified.
Here, it is easier to see the unique boundary.
Another way of obtaining the Möbius strip: take a circular ring and identify the points that are symmetric about the center (a half-ring is then equivalent to a rectangle where two widths are identified, in opposite directions, which indeed gives a Möbius strip)
Consider the sphere where the antipodal points are identified, i.e. a real projective plane. Then, a strip located between to tropics is a Möbius strip. The complement is an open disk.
 

The Möbius strip can therefore be obtained by puncturing an open hole into a projective plane.
 

This is why the punctured cross-cap is a Möbius strip (or rather, an immersion of a Möbius strip, because of the self-intersection), see opposite:

Inversely, when a Möbius strip and a disk are sewed together by their boundary, we get a real projective plane that can be immerged in  as: the cross-cap, the Roman surface, or Boy's surface.

Le ruban de Möbius en bonnet croisé trouéor

The chromatic number of the Möbius strip is therefore equal to that of the projective plane, i.e. 6.


In this map with 6 countries traced on a Möbius strip, each country meets the 5 others; 6 is the maximum number possible, and any map can be colored with no more than 6 colors.

When two Möbius strips are sewed together by their boundaries, we get the connected sum of two real projective planes, i.e. a Klein bottle.
 
   +
    =

The Möbius strip without its boundary is called the open Möbius strip; it is homeomorphic to the projective plane minus one point.
 
 
The Möbius strip can be represented by a union of three non coplanar polygons, necessarily non convex, as opposite (see Brehm's polyhedron):

See also the boundary of the Möbius strip with 3 half-twists, which is a trefoil knot and the fake Möbius strip on the torus.

The engraver M.C. Escher worked a lot on the Möbius strip:
 

Escher's famous Möbius strip.

Another Möbius strip by Escher, left-handed with three half-twists, cut at its center; 
this way he gets only one strip, knotted like a trefoil knot, with 6 half-twists.

 
Escher's famous horsemen are represented on a strip with two half-twists, hence with two faces (colored in grey and beige) and two boundaries, but the central fusion that simulates the identification between two segment lines of each boundary (indicated in green) makes it a topological Möbius strip with only one face, and only one boundary, indicated in red (cf. Möbius shorts) Same principle for these swans with black-and-white faces.

 
Möbius strip braided with only one braid by Juan Pablo Baudry  Möbius strip with 3 half-twists knotted as a trefoil knot.
University of Flensburg.
Ring 3, by Markus Raetz
sculpture en noeud de trèfle, université de Flensburg

Like the Borromean rings, the Möbius strip is a strong symbol, hence the many logos:

Logo of recycled products:
1 half-twist, cf. the strip with 3 cylinders above

Logo of Commerzbank 
right-handed strip with 3 half-twists

Logo of the Swiss institute of intellectual property, the title of which is written in the four national languages.


Logo of a German university
left-handed strip with 3 half-twists

Canadian Mathematical Society



The logo of Renault is a strip with two half-twists, and therefore is not a Möbius strip
Also look at the foot of this Apple monitor!

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© Robert FERRÉOL, Jacques MANDONNET, Samuel BOUREAU, Alain ESCULIER, Christoph SOLAND 2017