# Topology by yurtgc548

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Topology
The Edible
Lecture
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-
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
• 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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