Imagine that you’ve finally found a parking space after a long and harrowing search, but are now encountering some difficulty in trying to enter this space. Wouldn’t it be great if you knew a formula that allowed you to enter the space without difficulty? Are you annoyed because your soda can doesn’t remain upright during a picnic? Would you like to know why a mirror swaps right and left, but not top and bottom? Are you looking for a mathematical speech to toast your mother-in-law’s 85th birthday? Or do you want to give your heart away mathematically? Dr. Norbert Herrmann provides amusing and entertaining solutions to these and many other problems that we encounter in everyday situations.
“A book for teachers, students of mathematics, and anybody who likes unusual and amusing calculations.”
The Beauty of Everyday Mathematics
Norbert Herrmann
The Beauty of Everyday Mathematics
Copernicus Books
An Imprint of Springer ScienceþBusiness Media
Dr.Dr.h.c. Norbert Herrmann Universität Hannover Institut für Angewandte Mathematik Welfengarten 1 30167 Hannover Germany
[email protected]
Translated from German by Martina Lohmann-Hinner, mlh.communications © Springer-Verlag Berlin Heidelberg 2012 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus Books, an imprint of Springer Science+Business Media. Copernicus Books Springer Science+Business Media 233 Spring Street New York, NY 10013 www.springer.com Library of Congress Control Number: 2011940689
Mathematics Subject Classification (2010): 00A09 Translation from the 3rd German edition Mathematik ist überall by Norbert Herrmann, with kind permission of Oldenbourg Verlag Germany. Copyright © by Oldenbourg Verlag Germany. All rights reserved. Printed on acid-free paper ISBN 978-3-642-22103-3 DOI 10.1007/978-3-642-22104-0
e-ISBN 978-3-642-22104-0
Preface
Once upon a time, there was a group of representatives from the State of Utah in the United States of America around the year 1875. One of them was James A. Garfield. During a break, they were sitting in the congressional cafeteria. To pass the time, one of them, namely Mr. Garfield, suggested that they take a look at the Pythagorean Theorem. Even though this famous theorem had already been studied and proven 2000 years ago, he wanted to come up with a new proof. Together with his colleagues, he worked for a little while, and discovered the following construction: C
A
•
β
B
α • β •
α
D
E
Fig. 0.1 Sketch proving the Pythagorean Theorem.
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Preface
Here, we ave the crosshatched right triangle ABC. We sketch this triangle once more below it, though this time turned slightly so that side AD lies exactly on the extension of side AC. The connecting line EB completes the figure, turning it into a trapezoid because the bottom side is parallel to the top side thanks to the right angles. The two triangles meet, with their angles α and β , at A. Because the triangles are right triangles, the two angles add up to 90◦ , from which we conclude immediately that the remaining angle at A is also a right angle. After all, three angles equal 180◦ when added together. Now, only the little task remains of comparing the area of the trapezoid (central line times height, where the central line equals (base line + top line)/2 with the sum of t