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Agenda Motivation
Limit problems in grade school geometry!
Alvin Moon
Math Circle on Oct. 17th [email protected]
Agenda Motivation
What we’re doing today
• Motivation for talking about limits
• Classical example: Zeno’s paradox
• Interactive example: Midpoint polygons
• Modern example: Clothing store window
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Motivation
Limits are found everywhere in math.
They’re used to describe events that happen:
• ”infinitely many times...”
or
• ”very close to...”
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Motivation
In this lecture, we’ll learn about limits through some cool examplesfound in the wild.
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Classical example: Zeno’s paradox
Zeno of Elea, ancient Greek philosopher who studied motion andphysics.
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Zeno’s Paradox
Hercules runs from point A to point B.
Step 1: He must run half of the distance from A to B.
Call this midpoint: C.
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Zeno’s Paradox
Step 2: Then he must run half the distance from C to B.
Call this point D
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Zeno’s Paradox
Zeno’s paradox: ”If Hercules must run half the distance from eachmidpoint at each step, how can he ever reach point B?”
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Zeno’s Paradox
Suppose the distance from A to B is 1 mile
Discussion: With a partner, describe the remaining distance fromHercules to point B at each step n.
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Zeno’s Paradox
At the nth step, the distance remaining is (12)n miles.
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Zeno’s Paradox
Questions:
• Does the remaining distance get smaller after each step?
• Does the remaining distance ever reach 0 miles?
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Zeno’s Paradox
• ”Does the remaining distance get smaller after each step?”
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Zeno’s Paradox
• ”Does the remaining distance get smaller after each step?”
1
2>
1
4>
1
8>
1
16> . . .
In general, if n < m, then
1
2m<
1
2n
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Zeno’s Paradox
How small do the distances get?
After 30 steps, the distance between Hercules and point B issmaller than the average distance between water molecules
in liquid water.
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Zeno’s Paradox
• “Does the remaining distance ever reach 0 miles?”
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Zeno’s Paradox
• “Does the remaining distance ever reach 0 miles?”
No! The distance remaining, 12n miles, at step n is always positive.
But we say as the number of steps increases, the remainingdistance approaches zero.
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Zeno’s Paradox
In Zeno’s paradox, we are using a limit to describe how closeHercules gets to the end of the line.
“The limit of the remaining distances, as the number of stepsapproaches infinity, is zero.”
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Interactive example: Midpoint Polygons
Activity:
(1) On a piece of paper, draw a big closed polygon.
(2) Find the midpoint of each side of your polygon.
(3) Connect the midpoints to form a new polygon, called themidpoint polygon.
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Midpoint Polygons
Discussion: With your partner, discuss the following questions:
• How many sides does a midpoint polygon have?
• Can you make another midpoint polygon inside the firstmidpoint polygon?
• How many midpoint polygons can you make, one inside theother?
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Midpoint Polygons
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Midpoint Polygons
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Midpoint Polygons
• What happens to the vertices of your midpoint polygonsafter each step?
• What happens to the areas?
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Midpoint Polygons
• “What happens to the vertices of your midpoint polygonsafter each step?”
Surprisingly, this is a hard problem to solve.
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Midpoint Polygons
Theorem: (Schoenberg) Consider a closed polygon. The verticesof the midpoint polygons approach the center of mass of theoriginal vertices.
Proof. Can be found in a paper by Schoenberg. Uses an advancedmethod called the finite Fourier transform!
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Midpoint Polygons
Example: Consider the triangle with vertices: (0, 1), (3,−9), (5, 8).Then the midpoint triangles shrink down to the point: (83 , 0).
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Midpoint Polygons
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Midpoint Polygons
• “What happens to the areas of the midpoint polygons?”
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Midpoint Polygons
Activity Let’s get into big groups and try prove the followingstatement:
“Consider a triangle T with vertices z1, z2, z3. Then the area ofthe midpoint triangles of T get smaller and smaller - in fact, theareas approach zero.”
After everyone has had enough time to think, each group willpresent their thoughts to us!
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Limits and geometry in real life
Agenda Motivation
Limits and geometry in real life
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Limits and geometry in real life
What’s the equation of the resulting curve when you draw “every”line in the store window corner?
Agenda Motivation
Limits and geometry in real life