The
Contents 1 Properties 2 Geometry and topology 2.1 Widest isometric embedding in 3-space 2.2 Topology 2.3 Computer graphics 2.4 Open Möbius band 2.5 Möbius band with round boundary 3 Related objects 4 Applications 4.1 In stage magic 5 See also 6 References 7 External links Properties[edit] August Ferdinand Möbius The
A
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 Non-orientable wormhole 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
x ( u , v ) = ( 1 + v 2 cos u 2 ) cos u displaystyle x(u,v)=left(1+ frac v 2 cos frac u 2 right)cos u y ( u , v ) = ( 1 + v 2 cos u 2 ) sin u displaystyle y(u,v)=left(1+ frac v 2 cos frac u 2 right)sin u z ( u , v ) = v 2 sin 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
the
log ( r ) sin ( 1 2 θ ) = z cos ( 1 2 θ ) . displaystyle log(r)sin left( frac 1 2 theta right)=zcos left( frac 1 2 theta right). Widest isometric embedding in 3-space[edit]
If a smooth
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.
Take a
Open Möbius band[edit]
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
z 1 = sin η e i φ displaystyle z_ 1 =sin eta ,e^ ivarphi z 2 = cos η e i φ / 2 . 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
3-sphere.
To obtain an embedding of the
1 / 2 , i / 2 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
the
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
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.[15]
A
A compact resonator with a resonance frequency that is half that of
identically constructed linear coils[19]
An inductionless resistor[20]
In chemistry/nano-technology as: Molecular knots with special characteristics (Knotane [2], Chirality) Molecular engines[23] Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism[24] 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 backbone. In stage magic[edit]
The
Cross-cap Umbilic torus Ribbon theory References[edit] ^
External links[edit] Look up
Wikimedia Commons has media related to Möbius strip. The Möbius Gear – A functional planetary gear model in which one gear is a Möbius strip Weisstein, Eric W. "Möbius Strip" |