Mobius Band By: Katie Neville. Definitions Mobius strip—a surface with only one side and one...
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Transcript of Mobius Band By: Katie Neville. Definitions Mobius strip—a surface with only one side and one...
Mobius Band
By:Katie Neville
Definitions Mobius strip—a surface with only one
side and one boundary component Boundary component of S—the
maximal connected subsets of any topological space of the boundary of S
In other words, a mobius strip is a one sided surface in the form of a single closed continuous curve with a twist
Properties
Non-orientable Ruled surface Chiral Continuous One boundary component
Non-orientable surfaces Any surface that contains a subset that
is homeomorphic to the Mobius band. No way to consistently define the
notions of "right" and "left“ Anything that is slid around a non-
orientable surface will come back to its starting point as its mirror image
It cannot be mapped one to one in three space.
Non-orientable vs. orientable A torus is
orientable.
A mobius band is non-orientable.
Chirality
The mobius strip has chirality or “handedness” The existence of left/right opposition
The mobius strip is not identical to its mirror image.
Thus, it cannot be mapped to its mirror image by rotations or translations.
Chiral vs. Achiral
Examples of chiral: Right hand and left hand Why? Their reflections are different
from the original objects.
Examples of achiral: a common glass of water
Ruled Surface
A surface S is ruled if through every point of S, there is a straight line that lies on S
Examples of ruled objects: Cone, cylinder, and saddles
Examples of non ruled objects: Ellipsoid and elliptic paraboloid
Real life applications Magic Science Engineering Literature Music Art
Recycling symbol Monumental
sculptures Synthetic
molecules Postage stamps Knitting patterns Skiing acrobatics
Filmstrips, Tape Recorders, and Conveyor Belts
In 1923, Lee De Forest attained a U.S. patent for a mobius filmstrip that records sound on both “sides”.
Tape recorders Twisted tape runs twice as long
Conveyor belts Created to wear evenly on both “sides”
Recycling Symbols The standard
form is a mobius band made with one half-twist and the alternative is a one-sided band with three half-twist.
Making a Mobius Band
Bring the two ends of the rectangular strip together to make a loop.
Give one end of the strip a half twist and bring the ends together again and tape them.
Experiment 1
Demonstrate the mobius band is one sided. Draw a line down the middle, all the
way around the band. You will notice the line is drawn on
the back side and the front side. The back side is the same side as the
front side—one sided!
Experiment 2
Demonstrate the curve is continuous and has only one boundary component Take a crayon and color around the
very edge of the mobius band. Keep going until you get back where you started from.
How many edges are there?
Experiment 3
Take a pair of scissors and cut down middle line.
What shape is created? Band with two full twists and two edges
How long is the band, in terms of the original mobius band, when we cut the it lengthwise down the middle?
Experiment 4 Create another mobius band. Cut the band
lengthwise, so that the scissors are always 1/3 inch away from the right edge. Creates two strips
A mobius band with a third of the width of the original A long strip with two full twists
This strip is a neighborhood of the edge of the original strip!
This occurs since the original mobius strip had one edge that is twice as long as the strip of paper. The cut created a second independent edge
Developing Ideas
Imagine a mobius band thickened so the edge is as thick as the side. What shape is it? How many edges does it have? How many faces? produces a three-dimensional object
with a square cross section (a twisted prism)
The resulting form has two edges and two faces .
Why did the chicken cross the mobius band? To get to the same side!
References Burger, E and Starbird, M. (2005) The Heart of
Mathematics: An invitation to effective thinking.
http://www.daviddarling.info/encyclopedia/N/non-orientable_surface.html
http://www.sciencenews.org/articles/20000902/mathtrek.asp
http://chirality.ouvaton.org/homepage.htm
Peterson, Ivars. (2002) Mobius and his Band Peterson, Ivars (2003) Recycling Topology www.wikipedia.com