Acknowledgments
     We live in exciting times and one of the reasons they are exciting is the great electronic resource called the Internet.  It opens up remarkable possibilities involving "publication" in the broadest sense of the word:  dissemination of information in creative, interesting, and even interactive ways.  We have been exploring a few of the dimensions it has to offer in previous works and continue in this one, emphasizing the role of virtual reality in mathematical models.  The Institute of Mathematical Geography (IMaGe) has been a leader in these efforts since 1985.  Over the past 20 years, there have been more changes than we might ever have foreseen.  The IMaGe website archives those efforts.  Here, we note simply one previous work in a similar vein:  a work by these two authors co-authored with the late Frank Harary.  That work was entitled Graph Theory and Geography:  An Interactive View eBook, published by John Wiley and Sons (Wiley-Interscience Series in Optimization and Discrete Mathematics) in April of 2003 as their first eBook publication.  Professor Emeritus, Mitchell J. Rycus, Taubman College of Architecture and Urban Planning, The University of Michigan (Ann Arbor) notes of that one in a request for review sent by Wiley,
"I've had a chance to look at the Arlinghaus et al. book and I think it's just great!  It is what an eBook is supposed to be--interactive, animated, easy to read and easily accessible, and of course, the context has to be as well written and as authoritative as any printed text.  Graph Theory [and Geography] does indeed have all these elements, and more since it is both a math and geography text in one document.  What a treat to have the opportunity to review it and (I hope) to keep it for reference for my work."
The current work draws partially from the success of the Wiley eBook as well as from the continuing experience with IMaGe and its electronic journal, Solstice:  An Electronic Journal of Geography and Mathematics (founded, 1990).  Articles appeared in Solstice as a means of testing book content.  The comments below relate to these tests. We hope that all readers of this document, and of those derivative ones to come, will also enjoy them and learn from them.  This document would not have been possible without the kind help and wise counsel of numerous individuals.  Klaus-Peter Beier, Adjunct Associate Professor of Engineering and Director of the 3D Laboratory of the Duderstadt Center, The University of Michigan, offered a marvelous course in Virtual Reality (Engineering 477) in which the first author had the privilege to participate in the Fall of 2003 and 2004.  She was thus able, through Professor Beier's patience and fine lectures, to add virtual reality scenes to this book.  The Graduate Student Instructors in the course offered clear instruction to help her to work through the laboratory exercises:  Thana Chirapiwat (who had earlier been a student of hers in a GIS course), Bonnie Congxi Bao, and Jamie Cope.  The staff of the 3D Laboratory of the Duderstadt Center was also of great help:  Lars Schumann, Steffen Heise, Brett Lyons, Eric Maslowski, and Scott Hamm.

They also thank Merle Johnson of the City of Ann Arbor ITS Department and Chandra Hurd Gochanour of the City of Ann Arbor Planning Department:  the former for his generosity in sharing aerials and maps from City of Ann Arbor files and the latter for sharing her expertise in analyzing city data and in checking selected files related to material on this site.  Donald T. Uchman, Coordinator of Space Graphics, Space Information and Planning, Plant Extension--AEC, The University of Michigan, kindly shared official University map files on locations and sizes of buildings.

This work has been in progress for over 15 years and is an ongoing venture.  Numerous colleagues have been helpful, often in ways that might be considered indirect, but generally which feed into an important infrastructure in understanding on which we build our work.  At the risk of missing some, we thank particularly, Frank Harary (1921-2005), John D. Nystuen, Kenneth H. Rosen, Matthew Naud, Steve Quigley, Susanne Steitz, Penelope Kaiserlian, Sylvia Hecimovich, Douglas Kelbaugh, Fred Beal, Susan Pollay, Karen Hart, Paul Mohai, Mitchell J. Rycus, Waldo R. Tobler, Michael Batty, Michael J. Woldenberg, John F. Kolars, Marvin MIkesell, Chauncy D. Harris (1914-2003), and Marco Farinelli.

The authors wish to thank members of their respective university communities who were supportive, in various ways, of this project.  S. Arlinghaus thanks the Dean of the School of Natural Resources and Environment of The University of Michigan, Rosina Bierbaum, for her consistent appointment of this author to her annual post.  This author hopes that a strong win-win situation is created when scholars publish materials in association with such appointments.  W. Arlinghaus wrote the chapters on elementary number theory and on binary quadratic forms while on a semester sabbatical from the Mathematics Department at Lawrence Technological University.  He thanks those who made that sabbatical possible, particularly his Department Chair, David Bindschadler.  Also, both authors wish to thank Donald F. Lach (1917-2000), Bernadotte E. Schmitt Professor of History, University of Chicago, and the staff of the Joseph Regenstein Library of the University of Chicago who helped procure German maps and texts from 1941.

Errors that remain are, of course, ours alone.
 

Sandra Lach Arlinghaus
William Charles Arlinghaus

Ann Arbor, Michigan
June 21, 2005.


Software used in the course of making this book was made by the following firms: Hardware used centered on the equipment in the "Usability Lab" of the Duderstadt Center (with two flat screen monitors working together for a wide view) and on a Hewlett-Packard Pavilion f1903, with an Intel Pentium 4 Processor, 512 MB PC3200 DDR SDRAM memory, Intel Graphics Media Accelerator, 200GB hard drive and a 19 inch flat screen monitor set to highest resolution.

Institute of Mathematical Geography.  Copyright, 2005, held by authors.
Spatial Synthesis:  Centrality and Hierarchy, Volume I, Book 1.
Sandra Lach Arlinghaus and William Charles Arlinghaus