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Matthew McCullagh
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LEARNING WITH A TWIST
AN INTRODUCTION TO
MOBIUS STRIPS
I could teach you about the man
Möbius was born in Schulpforta, Saxony-Anhalt, and was descended on his
mother's side from religious reformer Martin Luther.[1] He was home-schooled
until he was 13 when he attended the College in Schulpforta in 1803 and
studied there graduating in 1809. He then enrolled at the University of Leipzig,
where he studied astronomy under the mathematician and astronomer, Karl
Mollweide.[2] In 1813 he began to study astronomy under the mathematically
inclined professor Carl Friedrich Gauss at the University of Göttingen while
Gauss was the director of the Göttingen Observatory. From there he went to
study with Carl Gauss's instructor, Johann Pfaff at the University of Halle,
where he completed his doctoral thesis The occultation of fixed stars in
1815.[3] In 1816 he was appointed as Extraordinary Professor to the "chair of
astronomy and higher mechanics" at the University of Leipzig.[4] Möbius died
in Leipzig in 1868 at the age of 77. His son Theodor was a noted philologist.
I could teach you the maths
Geometry and topology of a mobius strip
One way to represent the Möbius strip as a subset of threedimensional Euclidean space is using the parametrization:
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.
I
I could teach you the
applications
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.[14]
A Möbius resistor is an electronic circuit element that cancels its own
inductive reactance. Nikola Tesla patented similar technology in
1894:[15] "Coil for Electro Magnets" was intended for use with his
system of global transmission of electricity without wires.
I
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(with a twist)
LEARNING WITH A TWIST
LEARNING WITH A TWIST
LEARNING WITH A TWIST
LEARNING WITH A TWIST
LEARNING WITH A TWIST
LEARNING WITH A TWIST
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How many sides does the
mobius strip have?
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How many edges does the
mobius strip have?
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What will happen if we cut along
the line?