Geometric Etudes in Combinatorial Mathematics
The etudes presented here are not simply those of Czerny, but are better compared to the etudes of Chopin, not only technically demanding and addressed to a variety of specific skills, but at the same time possessing an exceptional beauty that characteriz
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Alexander Soifer
Geometric Etudes in Combinatorial Mathematics Second Edition With over 350 Illustrations Forewords by Philip L. Engel Paul Erd˝os Branko Gr¨unbaum Peter D. Johnson, Jr. and Cecil Rousseau
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Alexander Soifer College of Letters, Arts and Sciences University of Colorado 1420 Austin Bluffs Parkway Colorado Springs, CO 80918, USA [email protected]
ISBN: 978-0-387-75469-7 e-ISBN: 978-0-387-75470-3 DOI: 10.1007/978-0-387-75470-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010928053 Mathematics Subject Classifications (2010): 52-01, 52-02, 52A05, 52A10, 52A15, 52A35, 52A37, 05C99, 00A07, 00A08 c Alexander Soifer, Center for Excellence in Mathematical Education, Colorado Springs, 1st edition: CO, 1990 c Alexander Soifer 2010 2nd edition: All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design: Mary Burgess Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
In memory of ISAAC YAGLOM, the great expositor of the world of mathematics.
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Frontispiece reproduces the front cover of the original edition. It was designed by my late father Yuri Soifer, who was a great artist. Will Robinson, who produced a documentary about him for the Colorado Springs affiliate of ABC Broadcasting Company, called him “the artist of the heart.” For his first American one-man show at the University of Colorado in June–July 1981, Yuri sketched his autobiography: I was born in 1907 in the little village Strizhevka in the Ukraine. From the age of three, I was taught at the Cheder (elementary school by a synagogue), and since that time I have been painting. At the age of ten, I entered Feinstein’s Jewish High School in the city of Vinniza. The art teacher, Abram Markovich Cherkassky, a graduate of the Academy of Fine Arts at St. Petersburg, looked at my book of sketches of praying Jews, and consequently taught me for six years, until his departure for Kiev. Cherkassky was my first and most important teacher. He not only critiqued my work and explained various techniques, but used to sit down in my place and correct mistakes in my work until it was nearly unrecognizable. I couldn’t then touch my work and continue—this was unforgettable. In 1924, when I was 17, my relative, the American biologist, who later won the Nobel Prize in (1952) Selman A. Waksman, offered to t
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