Mobius Strip

thumb|right|335 pxThe Möbius strip or Möbius band, also Mobius or Moebius, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.
== Tossup Questions ==
# The namesake of this object also names a formula that assigns a value of (-1)k to a positive integer n that has k distinct prime factors and also names a related inversion formula. A rectangle consisting of two 2x1 rectangles with a unit square in the middle can be used to tile it, and paradromic rings are created by deforming it. Independently discovered by Listing and its namesake, two of these constructs can be combined to form a Klein bottle. Formed by reattaching the ends of a closed band after cutting the band and giving it a half-twist, name this structure named after a German mathematician, a one-sided, nonorientable surface.
# This structure can be defined by the parametric equations x equals cos t times the quantity A plus b cos one-half t, y equals sin t times the quantity A plus b cos one-half t, and z equals b sin one-half t. This structure can also be formed by cutting a Klein bottle in half; cutting this structure results in paradromic rings. This nonorientable surface has Euler characteristic zero. For 10 points, name this structure often formed by making a half-twist in a strip of paper and connecting the ends, which allows one to draw a single line encompassing both sides of that paper.