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					   Topology
      The Edible
       Lecture
(Help yourself, but please
 don’t eat the lecture just
            yet.)
              Topology
• A branch of mathematics that deals with
  the basic structure of objects
• Concerned with shape, symmetry,
  transformation, classification
• Does *NOT* involve the concepts of
  size, distance, or measurement
A couple of the concepts that
   we’ve looked at in the
geometry chapter have been
        topological.

       Any guesses?
• Whether or not a curve is closed?
• Whether or not a curve is simple?
• Whether or not a figure is convex?
           Classification
• Figures are classified according to
  genus – the maximum number of cuts
  that can be made in a figure without
  cutting it into two pieces
• This corresponds to the number of
  holes in a figure
Genus 0
Genus 1
Genus ???
Genus ???
Genus ???
Genus ???
   Topological Oddity:
The MÖBIUS (MOEBIUS)
         STRIP
(by M.C. Escher)
          Topological Oddity:
          The KLEIN BOTTLE
The Klein bottle is another unorientable surface. It can be
     constructed by gluing together the two ends of a
  cylindrical tube with a twist. Unfortunately this can't be
 realized physically in 3-dimensional space. The best we
 can do is to pass one of the ends into the interior of the
 tube at the other end (while simultaneously inflating the
   tube at this second end) before gluing the ends. The
        resulting picture looks something like this:
        (from http://www.math.ohio-state.edu/~fiedorow/math655/Klein2.html)
           Topological Oddity:
           The KLEIN BOTTLE
The result is not a true picture of the Klein bottle, since it
   depicts a self-intersection which isn't really there. The
Klein bottle can be realized in 4-dimensional space: one
 lifts up the narrow part of the tube in the direction of the
4-th coordinate axis just as it is about to pass through the
    thick part of the tube, then drops it back down into 3-
     dimensional space inside the thick part of the tube.

         (from http://www.math.ohio-state.edu/~fiedorow/math655/Klein2.html)
            Deformation
• Suppose your object was malleable
  (could be squished, stretched, twisted,
  etc – suppose it were made of Play-
  Doh). If you start with an object in a
  given genus, you can transform it into
  *ANY* other object in that genus without
  tearing it.
          Deformation
• The topological properties of an object
  are the ones that are invariant under
  deformations such as stretching and
  twisting (but not tearing, breaking, or
  puncturing)
• Which explains why length, angle, and
  measurement are not topological
  properties
• Whether or not a curve is closed?



          Topological
• Whether or not a curve is simple?



          Topological
• Whether or not a figure is convex?



        Not Topological

         (Can be altered by
            deforming)
When mathematicians get
  hold of topology …

        torus_paper.pdf
      It’s not really that bad
• The point of mathematics is to describe
  precisely just what it is you are
  observing
• This requires inventing an entire
  vocabulary and notation to describe a
  concept
Something you’re familiar with
            …

Equations that describe geometric shapes
Y = x2




Parabola
x2 + Y 2 = 1




   Circle
Sphere (genus 0)




  X2 + Y2 + Z2 = 1
Torus (genus 1)




Z2+((x2+y2)1/2 -2)2 =1
      Why is this useful?
• Manufacturing
• Aerodynamics
• Hydrodynamics
 Other Topological Concepts
         Coming Up
• Transformations and Symmetry
• Networks
• Non-Euclidean Geometry

				
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posted:6/26/2012
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