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 http://chirality.ouvaton.org/homepage.htm 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