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Published by:

autoSOCRATIC PRESS

www.rationalsys.com

Copyright 2013 Michael Lee Round

All rights reserved. No part of this book may be reproduced

or utilized in any form or by any means, electronic or

mechanical, including photocopying, recording, or any

information storage retrieval system, without permission in

writing from the publisher.

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BETTER LATE THAN NEVER Cleaning Up Zeno’s Mess 2,500 Years Later

Zeno of Elea

Southern Italy: 490BC – 430BC

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The TORTOISE and the HARE

Zeno of Elea is best known from ancient

times for formulating paradoxes regarding

motion. His most famous paradox is “Achilles

and the Tortoise”. Since I don’t know much

about Achilles but I do know a lot about the

Hare, let’s change this to “The Hare and the

Tortoise”. It’s an odd race, Zeno tells us,

because though the Tortoise has a head start,

the Hare can never catch it, regardless of how

fast the Hare goes! How can this be?

What We Expect

Let’s give the Tortoise a 20 mile head start.

The Hare hops along at 10 mph, while the Tortoise

crawls along at 2 mph.

What happens – in reality?

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In reality, the Hare catches the Tortoise

sometime in the 3rd hour.

What is Zeno talking about?

Zeno says the Hare, in chasing the Tortoise,

must move half the distance to where the

Tortoise is. But in the time it takes the Hare to

move this distance, the Tortoise itself has

moved. When the Hare again tries to overtake

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the Tortoise, it must again move halfway to the

Tortoise. Clearly, every time the Hare moves

halfway, the tortoise has moved, albeit slightly.

Zeno’s conclusion: the slow-moving Tortoise

will never be passed by the fast-moving Hare,

because it has to make infinitely many “half-

distance” moves.

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THE FEYNMAN IMPERATIVE

Richard Feynman, the great physicist,

verbalized this wonderfully in “Surely You’re

Joking, Mr. Feynman!”. While pursuing his

graduate degree at Princeton, Feynman was

talking with some mathematicians. They

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claimed you could cut up an orange into a finite

number of pieces, and, putting it back together,

arrive at something as big as the sun.

“Impossible”, claimed Feynman.

When given the mathematical explanation

about cutting the orange, Feynman interjected:

“But you said an orange! You can’t cut an

orange peel any thinner than the atoms.”

When given further mathematical

justification about being able to cut

continuously, Feynman concluded, “No, you said

an orange, so I assumed that you meant a real

orange.”

THE FEYNMAN IMPERATIVE

always try to think about what is

happening in reality!

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RECONSIDERING ZENO

But now I’m curious, because I can put all this

in a spreadsheet. If I start at ‘0’ and move towards

‘1’, here’s what Zeno said:

This, of course, assumes I’m always heading

towards “ONE”. What happens if I change things

up, and randomly choose “left or right”?

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This new rule (with choice) has me bouncing

back and forth. What happens if I continue the

pattern for 1,000 movements instead of just 10?

I hit every spot between 0 and 1!

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Three Points: The Result

How can this be?

This makes no sense, particularly given the

solid straight line and the filled square earlier. But

this was the result of moving 50,000 times. Let’s

“slow it down”, and capture the results to see how

this took place:

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Random Stepping In Two Dimensions From 25 to 5,000 Steps

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Distance Traveled Changing the Distance from “One-Half”

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Different Number of Points

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Going All the Way to the Next Point And Drawing a Line for the Entire Route

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MICHAEL BARNSLEY

THE CHAOS GAME

The process described here is a recent development

in math. Things like this are now possible – easily

– because of the computer.

Michael Barnsley is the gentleman who discovered

this.

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THE GEOMETRIC MIND

PROBLEMS

The following three problems each have a CHECK

(to make sure you’ve done the problem right).

Once you’ve confirmed you’ve done the problem

right, there’s a KEY. The key is necessary to

unlock the next installment.

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PROBLEM 1

The Tortoise has a 40 mile head start. It now plods

along at 3 mph, while the Hare bounces along at 12

mph. At what mile-marker do the two meet?

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PROBLEM 2

My current position is the red dot. I’ve randomly

chosen to move towards Point #2. What is the

half-way point?

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PROBLEM 3

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THE GEOMETRIC MIND

CONCEPT CARD A prominent concept here is trying to solve a

problem you either don’t know how to do, or “kind

of” remember, but not exactly.

The most important rule: get something on the

table to help. Often time, it’s a “really simple

example”, maybe similar to the one you’re solving,

maybe not, but something!

The goal of this: being able to figure out how to

solve your problem!

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