Illustrated Mathematics

Visualization of Mathematical Objects with Mathematica

Oliver Gloor
Beatrice Amrhein
Roman E. Maeder

Illustrated Mathematics  is a comprehensive collection of graphics and animations for various topics in mathematics on a CD-ROM. The images can be used without any additional software.

Further, programs for the design of new examples are included. The programs are written in Mathematica's programming language, which means that Mathematica is necessary for their use. Prior knowledge of Mathematica is not required. Examples can be created by modifying parameters in existing examples.

Illustrated Mathematics  is designed for teachers and students at high schools and undergraduate colleges.

The software can be used on Macintosh computers and under Windows (without any additional software), and on other computers (together with Mathematica or MathReader). The CD-ROM contains fully worked out collections for Macintosh and Windows. Versions for other computers (NeXT and Unix) are included in compressed form.

The CD-ROM is formatted in dual-mode, that is, it looks like an ordinary (HFS) volume on the Macintosh. On other computers, the ISO 9660 part can be accessed.

Chapters (and sections, etc.) in this document can be opened by double-clicking the corresponding box at the right margin.

What Is Illustrated Mathematics ?

Visual presentations aid in understanding complicated objects. Understanding abstract constructions frequently encountered in mathematics is improved with visual representations and graphics. Such visualizations are also important for students who have difficulty grasping abstract objects.

If visualization had been our only goal, we could have published this collection as a printed book instead of as a CD-ROM. CD-ROMs can store much more information, however. Besides, we can animate sequences of graphics and the computer allows direct interaction with the material. For example, it is feasible to create additional images during class and respond to students' questions. This capability improves the quality of instruction.

Collection of Visualizations

The graphics and animations on the CD-ROM can be used directly for teaching. No additional software is necessary (on Macintosh and under Windows).

Projection

Examples from this collection can be projected on a screen during class (for example, with an LCD projection panel connected to your computer).

Hardcopy

If no computer-compatible projection equipment is available, the examples can be printed for class notes or to create slides.

Part of Documents

The graphics and animations are part of hierarchically structured documents (Mathematica notebooks). This organization allows fast access to particular examples and enables us to add explanatory text.

For Class Notes

Be it directly on the computer or printed out, the graphics can be included in your documents and used as part of class notes. To help with this use of the graphics, many graphics are stored on the CD-ROM outside of their notebooks, in a graphics standard format.

Color

Because many graphics use color, a color monitor is recommended. However, colors have been chosen in a way that makes presentations with a monochrome monitor possible.

Included: MathReader

The MathReader program included on the CD-ROM allows you to read all documents on this CD-ROM, including animations.

Programs for Your Own Examples

Mathematica Is Necessary

The programs are based on Mathematica. For their use, Mathematica is necessary. In fact, the collection of examples was created with these programs.

Prior Knowledge of Mathematica Is Not Necessary

Some experience with Mathematica is an advantage, but not required for the use of the Illustrated Mathematics  programs. Simple adjustment of parameters in the examples allows their modification. Each program (package) is accompanied by a manual and many examples of its use.

Creation of Teaching Material

The programs allow you to create your own teaching material in the form of Mathematica notebooks, the form also used for this collection.

Directories/Folders and Files on the CD-ROM

This section briefly describes the folders/directories and files, which are grouped by purpose.

Windows version: Filenames and directory names have been truncated after eight characters, for example, COLLECTI instead of Collection. Most notebooks are accompanied by another file with a .MB extension that contains binary information for MathReader.

NeXT/Unix version: Notebooks and programs were compressed into a single tar file. This archive is named NEXT_TAR.Z and UNIX_TAR.Z, respectively. The installation instructions describe how these archives can be unpacked.

Collection of Visualizations

Collection

The Collection directory/folder contains all graphics and animations as notebooks. No further installation is necessary to use them. Windows version: According to the installation instructions two files are to be copied into your WINDOWS directory.

Since MathReader does not allow users to copy and paste graphics into other applications, many graphics have been stored as individual files in the Images subdirectory. More information about them can be found in the Hints.ma document in the Images directory.

Macintosh: Some notebooks are rather large. Additional copies of these have been created, with their graphics converted into a more compact format. These versions should be used on machines with less main memory. The PICT folder contains those notebooks whose graphics were converted into PICT format. Those notebooks that were still over 2MB after this conversion have been put into the Bitmap folder, using bitmap graphics.

Table of Contents

The Contents.ma file contains an overview of the visualization and program topics. It is reprinted in the booklet accompanying the CD-ROM.

Using Notebooks and MathReader

Macintosh and Windows version: The MathReader directory/folder contains the MathReader program from Wolfram Research, Inc. If Mathematica is not installed on your computer, you can use this program to read notebooks.

Versions of MathReader for other computers may be available on MathSource (see Installation Instructions).

The Notebooks.ma file in the Introduction directory/folder contains an introduction to the use of MathReader. It is reprinted in the booklet accompanying the CD-ROM.

About Illustrated Mathematics

A description of Illustrated Mathematics  is in the About_IlluMath.ma (ABOUT_IM.MA) file and reprinted in the booklet accompanying the CD-ROM.

Programs for Your Own Examples

The following directories/folders and files pertain to the use of the programs.

Programs and Their Descriptions

The Visual directory contains the programs. They should be copied to a hard disk according to the installation instructions. Alternatively, you can tell Mathematica the pathname where they can be found.

Each program, or more precisely, each package, is accompanied by a detailed description in the Manuals directory, which includes many examples.

Installation Instructions

For best use, the packages should be copied to a hard disk. The Installation.ma file contains instructions for installing the software on various computers. Installation is necessary only if you want to create your own examples. These installation instructions are also printed in the accompanying booklet.

Computer-dependent Settings

The Settings.ma notebook contains settings for configuring Illustrated Mathematics  for various computers (for example, color or monochrome monitors). It is needed only if you want to create your own examples.

Introduction into Notebooks and Mathematica

The Introduction directory/folder contains introductory files. Notebooks.ma describes how to look at notebooks. Notebooks can be displayed either with MathReader or with Mathematica. Programs.ma describes the use of the programs on the CD-ROM. Mathematica.ma explains the basic principles of Mathematica and describes how to work with Mathematica graphics. It contains also a summary of the built-in graphics capabilities of Mathematica.

Versions

The CD-ROM is formatted in dual-mode, HFS/ISO 9660. If read on a Macintosh, only the files for the Macintosh will be visible. If read on another computer (Windows or Unix), the documents for Windows and the two files NEXT_TAR.Z and UNIX_TAR.Z will be visible. How to install these versions is described in Installation.ma.

Hardware and Software Prerequisites

To read the CD-ROM, a CD drive with any necessary operating system extensions is required. The CD-ROM is formatted in dual mode, HFS/ISO 9660, and can be read on most computers.

Collection of Visualizations

Macintosh / Windows

All graphics and animations can be read or printed without any additional software on a Macintosh (System 7) or under Windows (Version 3.0 or later), using the enclosed program MathReader. At least 4MB of RAM are required.

NeXT

No notebook reader is included on the CD-ROM; therefore, Mathematica is required even to read or print the notebooks. All files have been compressed into a single archive (see installation instructions).

Unix/X11

No notebook reader is included on the CD-ROM. Such versions of MathReader may be available on MathSource (see Installation Instructions). Otherwise, a version of Mathematica with the notebook front end is required even to read or print the notebooks. For computers supporting X11, Mathematica Version 2.2 provides such a front end. It can be ordered from your Mathematica dealer, if necessary.

You can open the notebooks of the Windows version with any notebook reader. However, we recommend you use the proper version for Unix. All files have been compressed into a single archive (see installation instructions).

Programs for Your Own Examples

Mathematica Version 2.0 and enough memory (at least 5MB) for its use are required to run the programs of Illustrated Mathematics. No CD drive is necessary to run the programs.

The programs are machine-independent.

All the examples were computed with Mathematica Version 2.2 on Macintosh, NeXT, Windows, or Sun/SPARC systems.

Macintosh / Windows / NeXT

Mathematica allows you to read the notebooks. Programs are easiest to use by modifying an example in one of the notebooks from either the Collection or Manual directories/folders.

Other Computers

The programs can be used with Mathematica on any computer. To read the notebooks, however, you need the notebook front end. Such a front end is available for the X11 window system starting with Version 2.2 of Mathematica.

Using the front end you can modify an example in one of the notebooks of Illustrated Mathematics  from either the Collection or Manual directories.

Color Monitor

A color monitor is recommended, but not required.

Colophon

Author addresses:

Oliver Gloor
Universität Tübingen
WSI Sand 13
D-72076 Tübingen
gloor@informatik.uni-tuebingen.de

Beatrice Amrhein
Universität Tübingen
WSI Sand 13
D-72076 Tübingen
amrhein@informatik.uni-tuebingen.de

Roman Maeder
ETH Zürich
ETH Zentrum IFW
CH-8092 Zürich
maeder@inf.ethz.ch

In one form or another, the following persons have helped us with this project: Urs Kirchgraber, Werner Hartmann, Hermann-Josef Biner, Stéphane Collart, Markus Furter, and Albrecht Plümicke. We thank all of them.
We thank Allan Wylde, Paul Wellin, and the staff at TELOS for the cooperation in publishing Illustrated Mathematics.

Zurich, March 1995               Oliver Gloor,  Beatrice Amrhein,  Roman E. Maeder

Further Development of Illustrated Mathematics

We welcome comments, hints, and bug reports. Please direct your correspondence to the publisher or by email to the authors.

Copyright

© 1995 TELOS/Springer-Verlag
Published by TELOS, The Electronic Library Of Science, Santa Clara, California.
TELOS is an imprint of Springer-Verlag New York, Inc.

ISBN  0-387-14222-3

This work consists of a CD-ROM disc packaged with a booklet, and is protected by federal copyright law and international treaty.  This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA) except for brief exerpts 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 other than those expressly granted in the disc copyright and disclaimer information is forbidden.

Springer-Verlag or the authors makes no warranty of representation, either expressed or implied, with respect to this work, including its quality, merchantability, or fitness for a particular purpose.  In no event will Springer-Verlag or the authors be liable for direct, indirect, special, incidental, or consequential damages arising out of the use or inability to use the disc or booklet, even if Springer-Verlag or the authors has been advised of the possibility of such damages.

MathReader contained on this CD-ROM is proprietary software created and owned by Wolfram Research, Inc.  Wolfram Research retains such copyright ownership, but does hereby grant the Publisher worldwide rights to use and distribute this software as it pertains to this work.  Permission to use and reproduce copies of any part of this software must be obtained from the creators by contacting the Publisher in writing at TELOS, The Electronic Library of Science, 3600 Pruneridge Avenue, Suite 200, Santa Clara, CA 95051.

The use of general descriptive names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used by anyone.  Where those designations appear in this work and Springer-Verlag was aware of a trademark claim, the designations follow the capitalization style used by the manufacturer.   
MathReader  is a trademark and Mathematica is a registered trademark of Wolfram Research, Inc., Macintosh is a registered trademark of Apple Computer, Inc., NeXT and NeXTSTEP are trademarks of NeXT Computer, Inc., Windows is a trademark of Microsoft Corporation, Unix is a registered trademark of AT&T. SPARC and SunOS are registered trademarks of Sun Microsystems, Inc.

Selected Publications about Mathematica

Introductory

Blachman, Nancy. Mathematica: A Practical Approach,
	Prentice Hall, 1992. ISBN 0-13-563826-7

Gray, Theodore, Glynn, Jerry. The Beginner's Guide to Mathematica Version 2,
	Addison-Wesley, 1992. ISBN 0-201-58221-X

Shaw, WilliamT., Tigg, Jason. Applied Mathematica: Getting Started, Getting it Done,
	Addison-Wesley, 1994. ISBN 0-201-54217-X

Wagon, Stan. Mathematica in Action,
	W. H. Freeman, 1991. ISBN 0-7167-2229-1

Handbooks/Reference

Wolfram, Stephen. Mathematica: A System for Doing Mathematics by Computer,
	Second Edition, Addison-Wesley, 1991. ISBN 0-201-51502-4

Wolfram, Stephen. The Mathematica Reference Guide,
	Addison-Wesley, 1992.

Guide to Standard Mathematica Packages,
	Wolfram Research Technical Report, 1991. ISBN 1-880083-09-4

Wolfram, Stephen, with Beck, George. Mathematica: the Student Book,
	Addison-Wesley, 1994.

Trott, Michael. The Mathematica Guidebook,
	TELOS/Springer-Verlag, 1995. Includes CD-ROM. ISBN 0-387-94282-3

Wickham-Jones, Tom. Mathematica Graphics: Techniques and Applications, 
	TELOS/Springer-Verlag, 1994. ISBN 0-387-94047-2

Programming

Gaylord, Richard J., Kamin, Samuel N., Wellin, Paul R.. Introduction to Programming with 	Mathematica. TELOS/Springer-Verlag, 1993. ISBN 0-387-94048-0

Gray, John W. Mastering Mathematica: Programming Methods and Applications,
	AP Professional, 1994. ISBN 0-12-296040-8

Maeder, Roman E. Programming in Mathematica,
	Second Edition, Addison-Wesley, 1991. ISBN 0-201-54877-1

Maeder, Roman E. The Mathematica Programmer,
	AP Professional, 1994. ISBN 0-12-464990-4

Applications

Gaylord, Richard J., Wellin, Paul R. Computer Simulations with Mathematica,
	TELOS/Springer-Verlag, 1994. with CD-ROM. ISBN 0-387-94274-2

Gray, Theodore, and Glynn, Jerry. Exploring Mathematics with Mathematica,
	Addison-Wesley, 1991. ISBN 0-201-52809-6

Skeel, R., and Keiper, Jerry. Elementary Numerical Computing with Mathematica,
	McGraw-Hill,1993  ISBN 0-07-057820-6

Skiena, Steven S. Implementing Discrete Mathematics:
	Combinatorics and Graph Theory with Mathematica,
	Addison-Wesley, 1990.  ISBN 0-201-50943-1

Varian, Hal R. (ed.). Economic and Financial Modeling with Mathematica,
	TELOS/Springer-Verlag, 1993. ISBN 0-387-97882-8

Vvedensky, Dimitri. Partial Differential Equations with Mathematica,
	Addison-Wesley, 1993. ISBN 0-201-54409-1

Teaching

Davis, Bill, Porta, Horacio, and Uhl, Jerry. Calculus&Mathematica,
	Addison-Wesley,  1994.
	ISBN 0-201-58461-1 (Windows), 0-201-58153-1 (Macintosh/NeXT).

Crooke, Philip, and Ratcliffe, John. A Guidebook to Calculus with Mathematica,
	Wadsworth Publishing Company, 1991.

Braden, Bart. Discovering Calculus with Mathematica,
	John Wiley&Sons, 1992.

Finch, James K., and Lehmann, Millanne. Exploring Calculus with Mathematica,
	Addison-Wesley, 1992. ISBN 0-201-55572-7

Periodicals

The Mathematica Journal, Miller Freeman, Inc. ISSN 1047-5974

Mathematica in Education, Paul Wellin, editor, (1994), TELOS/Springer-Verlag. ISSN 1065-2965

Mathematica World (floppy disk magazine), Stephen M. Hunt, editor.
	(also available on CD-ROM)