Published by: autosocratic PRESS

Transcription

1 1

2 Published by: autosocratic PRESS 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. 2

3 BETTER LATE THAN NEVER Cleaning Up Zeno s Mess 2,500 Years Later Zeno of Elea Southern Italy: 490BC 430BC 3

4 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? 4

5 In reality, the Hare catches the Tortoise sometime in the 3 rd 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 5

6 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 halfdistance moves. 6

7 7

8 8

9 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 9

10 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! 10

11 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? 11

12 12

13 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! 13

14 14

15 15

16 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: 16

17 Random Stepping In Two Dimensions From 25 to 5,000 Steps 17

18 Distance Traveled Changing the Distance from One-Half 18

19 Different Number of Points 19

20 20

21 Going All the Way to the Next Point And Drawing a Line for the Entire Route 21

22 22

23 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. 23

24 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. 24

25 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? 25

26 PROBLEM 2 My current position is the red dot. I ve randomly chosen to move towards Point #2. What is the half-way point? 26

27 PROBLEM 3 27

28 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! 28

29 29

30 30

31 31

32 32