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Topology

The Edible Lecture

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

• 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 + Y2 = 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