Agenda Motivation

Limit problems in grade school !

Alvin Moon

Math Circle on Oct. 17th 2015 [email protected] Agenda Motivation What we’re doing today

• Motivation for talking about limits

• Classical example: Zeno’s paradox

• Interactive example:

• Modern example: Clothing store window Agenda Motivation Motivation

Limits are found everywhere in math.

They’re used to describe events that happen: • ”infinitely many times...” or • ”very close to...” Agenda Motivation Motivation

In this lecture, we’ll learn about limits through some cool examples found in the wild. Agenda Motivation Classical example: Zeno’s paradox

Zeno of Elea, ancient Greek philosopher who studied motion and physics. Agenda Motivation 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. Agenda Motivation Zeno’s Paradox

Step 2: Then he must run half the distance from C to B. Call this point D Agenda Motivation Zeno’s Paradox

Zeno’s paradox: ”If Hercules must run half the distance from each midpoint at each step, how can he ever reach point B?” Agenda Motivation Zeno’s Paradox

Suppose the distance from A to B is 1 mile Discussion: With a partner, describe the remaining distance from Hercules to point B at each step n. Agenda Motivation Zeno’s Paradox

1 n At the nth step, the distance remaining is ( 2 ) miles. Agenda Motivation Zeno’s Paradox

Questions:

• Does the remaining distance get smaller after each step?

• Does the remaining distance ever reach 0 miles? Agenda Motivation Zeno’s Paradox

• ”Does the remaining distance get smaller after each step?” Agenda Motivation Zeno’s Paradox

• ”Does the remaining distance get smaller after each step?”

1 1 1 1 > > > > . . . 2 4 8 16 In general, if n < m, then 1 1 < 2m 2n Agenda Motivation Zeno’s Paradox

How small do the distances get?

After 30 steps, the distance between Hercules and point B is smaller than the average distance between water molecules in liquid water. Agenda Motivation Zeno’s Paradox

• “Does the remaining distance ever reach 0 miles?” Agenda Motivation Zeno’s Paradox

• “Does the remaining distance ever reach 0 miles?”

1 No! The distance remaining, 2n miles, at step n is always positive. But we say as the number of steps increases, the remaining distance approaches zero. Agenda Motivation Zeno’s Paradox

In Zeno’s paradox, we are using a limit to describe how close Hercules gets to the end of the line.

“The limit of the remaining distances, as the number of steps approaches infinity, is zero.” Agenda Motivation Interactive example: Midpoint Polygons

Activity: (1) On a piece of paper, draw a big closed .

(2) Find the midpoint of each side of your polygon. (3) Connect the to form a new polygon, called the midpoint polygon. Agenda Motivation 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 first midpoint polygon?

• How many midpoint polygons can you make, one inside the other? Agenda Motivation Midpoint Polygons Agenda Motivation Midpoint Polygons Agenda Motivation Midpoint Polygons

• What happens to the vertices of your midpoint polygons after each step? • What happens to the areas? Agenda Motivation Midpoint Polygons

• “What happens to the vertices of your midpoint polygons after each step?” Surprisingly, this is a hard problem to solve. Agenda Motivation Midpoint Polygons

Theorem: (Schoenberg) Consider a closed polygon. The vertices of the midpoint polygons approach the center of mass of the original vertices. Proof. Can be found in a paper by Schoenberg. Uses an advanced method called the finite Fourier transform! Agenda Motivation Midpoint Polygons

Example: Consider the with vertices: (0, 1), (3, −9), (5, 8). 8 Then the midpoint shrink down to the point: ( 3 , 0). Agenda Motivation Midpoint Polygons Agenda Motivation Midpoint Polygons

• “What happens to the areas of the midpoint polygons?” Agenda Motivation Midpoint Polygons

Activity Let’s get into big groups and try prove the following statement:

“Consider a triangle T with vertices z1, z2, z3. Then the area of the midpoint triangles of T get smaller and smaller - in fact, the areas approach zero.” After everyone has had enough time to think, each group will present their thoughts to us! Agenda Motivation Limits and geometry in real life Agenda Motivation Limits and geometry in real life Agenda Motivation 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