Subject: Sun-Symbolic-Math Digest v1n8 Sun-Symbolic-Math Digest July 29, 1988 Volume 1: Number 8 Editor: Steve Christensen, steve@spock.ncsa.uiuc.edu, (217)244-0544 Today's Topics: Administrative Stuff Mathematica Announcements Send contributions to: Sun-Symbolic-Math@spock.ncsa.uiuc.edu Send subscription add/delete requests to: Sun-Symbolic-Math-Request @spock.ncsa.uiuc.edu For BITNET: Sun-Symbolic-Math%spock.ncsa.uiuc.edu@uiucvmd.bitnet Sun-Symbolic-Math-Request%spock.ncsa.uiuc.edu@uiucvmd.bitnet Anonymous ftp to archive: ftp.ncsa.uiuc.edu (128.174.20.) cd to Directory /usr6/ftp/Symbolic See instructions in that directory. --------------------------------------------------------------------------- Date: Friday, July 29, 1988 From: Steve Christensen Subject: Administrative Stuff Activity in this mailing list have been very light over the summer, even though we have added many people to the list. I am hoping this will change when schools start up again in August. To stimulate some discussion, I am attaching the full set of Mathematica announcements that I got at the Mathematica introduction press conference in Santa Clara in June. I am anxious to hear comments on this. I have Mathematica on a Mac II, Sun-3/160, and Sun-4/160. Please contribute to what I hope will be a lively v1n9. ---------------------------------------------------------------------------- Date: Friday, July 29, 1988 From: Steve Christensen Subject: Mathematica Announcements WOLFRAM RESEARCH June 23, 1988 Wolfram Research, Inc. Introduces MATHEMATICA, Announces Relationships with Major Computer Hardware Manufacturers. SANTA CLARA, Calif. - Mathematica, the first software system capable of performing calculations in all areas of mathematics, was introduced today by Wolfram Research, Inc. Designed by well-known scientist and MacArthur prizewinner Stephen Wolfram, 28, Mathematica can be used with many kinds of computers, including personal computers, work-stations and supercomputers. Described by Steven Jobs, president of NeXT, as "mathematics for the rest of us," Mathematica is designed to make algebra, calculus, geometry and other areas of mathematics as straightforward as the electronic calculator has made simple arithmetic. To find the integral of 1/(1+x^6), for example, a Mathematica user simply types Integrate [1/(1+x^6),x]. A few seconds later, Mathematica returns a formula for the result. Working out a complicated formula like this by hand would be time consuming and prone to error. "When calculators came out," says Wolfram, "arithmetic became very easy. We built Mathematica to make all kinds of mathematics that easy." Mathematica operates not only with numbers and algebraic formulae, but also with graphics. Users can produce two-and three-dimensional color pictures that allow them to visualize complex mathematical functions. In addition to having an extensive collection of built-in functions, Mathematica is also a powerful programming language, in which applications can be written. A novel rule-based approach lets users transcribe formulae from textbooks directly into Mathematica. Mathematica is intended for a broad technical market. Says Wolfram, "We wanted to make a tool for anyone who uses mathematics, whether they are math professors, engineers or high school students." Wolfram Research sees four initial markets for Mathematica: * Education at all levels, from elementary school to graduate school; * Research at universities, government and corporate laboratories; * Engineering; * Business analysis, including statistics and modeling. "Mathematica will change forever the way that mathematics is taught," says Jerry Uhl, a professor of mathematics at the University of Illinois. "Students will be able to concentrate on concepts, leaving mechanical details to mathematica. They will learn a lot more mathematics that way." Mathematica is already being tested with graduate, undergraduate, high school and even elementary school students. An undergraduate calculus course based on Mathematica is under development. Some of America's leading mathematicians and scientists have started using pre-release versions of Mathematica in their research. Steve Christensen, a physicist at the National Center for Supercomputing Applications says: "With Mathematica, I repeated in two days a calculation that had taken me two years to do by other means." Dana Scott, Professor of Computer Science at Carnegie-Mellon, describes Mathematica as "a telescope with which I can discover new moons and craters and see details of mathematical landscapes I could never see before." John Gage, Manager of Science Office for Sun Microsystems, sums it up: "Stephens's program will change practical engineering. It will change the way science is done, and the way mathematics is taught. Stephen has created a tool that will bring essentially all of applied mathematics to everyone's desktop." Wolfram Research also announced today that Mathematica has been adopted by a number of major computer hardware manufacturers. "Our goal," says Wolfram, "is to establish Mathematica as a standard throughout the industry." NeXT Inc., the Palo Alto company founded by Steven Jobs, announced today that it plans to bundle Mathematica with every NeXT computer. "Mathematics is at the heart of many of the disciplines that are crucial to America's competitiveness," said Jobs. "Mathematica will revolutionize the teaching and learning of mathe by focusing on the prose of mathematics, without getting lost in the grammar." Wolfram added, "NeXT was the first computer company that understood the significance of Mathematica. The name 'Mathematica' actually was suggested by Steve Jobs." Workstation manufacturers Sun Microsystems and Silicon Graphics announced that they will distribute versions of Mathematica for their computers. IBM announced that it will offer a version of Mathematica for the IBM RT PC. Curt Wozniak, Vice President and General Manager of Sun Microsystems' Educational Product Division said, "We are excited about Mathemtica as a tool in the education and research marketplace and about its potential for transforming the way various branches of mathematics will be taught." Supercomputer manufacturers Ardent Computer Corporation and Stellar Computer, Inc. also announced that they will distribute versions of Mathematica for their recently-introduced models. Details of arrangements for Mathematica distribution may be obtained from the computer manufacturers mentioned above. Wolfram says, "We are happy that many leading computer manufacturers share our vision of Mathematica as a general tool that should be distributed like system software or a computer language, rather than as an independent application program." Wolfram Research also announced that it will distribute a version of Mathematica for the Apple Macintosh directly. This version includes a sophisticated user interface written especially for the Macintosh. Wolfram Research announced the immediate availability of versions for the Macintosh and Macintosh II, priced at $495 and $795 respectively. Wolfram Research will offer substantial discounts to educational institutions. Simultaneous with today's announcement of Mathematica, Addison-Wesley Publishing Company is publishing the 768-page book "Mathematica: A System for Doing Mathematics by Computer," written by Stephen Wolfram. The book gives a complete introduction to the use of Mathematica on all types of computers. A number of third parties, including engineering software companies Autodesk and MathSoft, have said that they will develop applications linked to Mathematica. An independent user group for Mathematica is being organized. "Mathematica gives people a new way to use personal computers," says Wolfram. "We expect that with time, Mathematica will define a major new category of software, which will take its place alongside word processors and spreadsheets." Wolfram Research, Inc. was founded in 1987 by Stephen Wolfram and the seven other scientists and mathematicians who built Mathematica. Privately funded by Stephen Wolfram, Wolfram Research is based in Champaign, Illinois, and currently has 25 employees. Mathematica is a trademark of Wolfram Research, Inc. -------------------------------------------------------------------------------- MATHEMATICA: Technical Background Information Mathematica is an interactive software system that incorporates a high-level programming language. Mathematica does three basic kinds of computation: numerical, symbolic and graphical. Unlike an electronic calculator or a standard computer language, Mathematica works with numbers of arbitrary precision. As well as making possible many calculations of mathematical interest, this capability has the practical consequence of giving the user unprecedented control over numerical roundoff errors. Built into Mathematica is a complete set of mathematical functions. Beyond the standard exponential and trigonometric functions, Mathematica includes Bessel functions, hypergeometric functions, elliptic integrals, and much more. Most of these functions were previously accessible only through incomplete books of tables. Mathematica incorporates efficient numerical algorithms for matrix manipulation, numerical integration and numerical equation solving. Beyond numerical computation, Mathematica can also work with algebraic formulae. Built into Mathematica are all the standard operations of algebra and calculus, including polynomial factorization, as well as symbolic integration and indifferentiation. The ability to manipulate and derive exact algebraic formulae makes it possible to automate a wide range of calculations that could hitherto only be done by hand. With Mathematica, the user can get explicit formulas for the solutions to equations, rather than just finding numerical results. This makes a much wider range of calculations possible, and also allows the user to understand the complete structure of a solution. The third major area covered by Mathematica is graphics. Mathematica can plot functions and data in two and three dimensions, in black-and-white or color, allowing the user to visualize results of calculations. Mathematica can also take symbolic descriptions of arbitrary geometrical objects, and produce from them three-dimensional color pictures. Thus, for example, the user could generate a list of the faces in a complex polyhedron using Mathematica's symbolic and numerical capabilities, and then produce a three-dimensional pictuer of the polyhedron using its graphical capabilities. Mathematica uses the PostScript standard for all its graphics output. This resolution-independent form can be rendered on computer screens, laser printers, phototypesetters, or even on huge posters. Mathematica canalso produce encapsulated PostScript, which can be included in desktop publishing systems. In addition to over 700 built-in functions, Mathematica incorporates a powerful programming language, which enables users to add their own functions. The Mathematica language supports several forms of programming, including structured programming in the style of C or Pascal and functional programming in the style of APL. Mathematica also allows a novel rule-based style of programming, in which the user specifies transformation rules for mathematical expressions which match a particular pattern. Such rules can be constructed by almost direct transcription of formulas from textbooks. The high-level nature of the Mathematica language allows users to create complex programs easily. The primitive functions provided by Mathematica correspond directly to primitive operations in mathematics itself. Wolfram Research expects that over time a large number of Mathematica packages will be created, addressing a wide range of specific application needs. Many of these packages will be available through the Mathematica User Group, and can be used on any of the computers that run Mathematica. The Mathematica systems consists of two parts: a kernel that does mathematical computations, and a front end that manages interaction with the user. The kernel is set up to work the same on all types of computers. The front end, on the other hand, is designed to take advantage of the special graphical and other capabilities of each type of computer. The front end and kernel do not necessarily have to be run on the same computer. All versions are compatible, so that, for example, the front end may run on a small personal computer, while the kernel runs on a large supercomputer connected by telephone or network. The most sophisticated front end for Mathematica available at announcement time is the one for the Macintosh. This front end uses the Macintosh's graphical capabilities to allow users to organize their Mathematica calculations. Sophisticated front ends for Mathematica are also planned for the NeXT computer,and under the OpenLook window system. The front end supports "notebooks," which contain text and graphics, mixed with "live" Mathematica input expressions. A user can read the text in a notebook, then activiate embedded Mathematica input expressions to perform particular computations. To give full support for Mathematica notebooks, the front end incorporates an outliner, multi-front text editor with a specialized form of style sheets, and an interpreter for a subset of PostScript. Wolfram Research is publishing the MathTalk communication protocol between the Mathematica kernel and front end, allowing third parties to create their own front ends for Mathematica. Different front ends can be used to support a variety of different styles of Mathematica usage, while retaining the same underlying computational system. Mathematica is designed to interface smoothly with external programs. Mathematica output can be produced in C and Fortran form, suitable for splicing in program source files. Output can also be produced in TeX form, suitable for direct inclusion in TeX documents. On UNIX systems, Mathematica can interact with external programs through pipes. Thus, for example, Mathematica could take data from a laboratory instrument, or could communicate with an existing external numerical program. The internal code for the kernel of Mathematica is written in an object-oriented extension of C developed by Wolfram Research, and is approximatly 150,000 lines long The front end for Mathematica on the Macintosh, written by Theodore Gray, is an additional 50,000 long. Mathematica is a highly portable program. The main limitation is that it requires approximately two megabytes of addressable memory. As a result, Mathematica can run only on Macintoshes which have at least two megabytes of memory. Like all major systems, Mathematica has antecedents. There is, however, no other system with anything like the same breadth and depth of mathematical coverage. Mathematica is much higher level than traditional programming languages such as C and Fortran, as well as being symbolic and fully interactive. Mathematica is immeasurably more powerful than BASIC, yet for simple computations, retains the ease of interactive use pioneered by BASIC. In the area of numerical computation, Mathematica builds on systems like APL and MatLab, but has muc more sophisticated capabilities, including, for example, arbitrary precision arithmetic. Equation solvers, such as TK!Solver and Eureka, are covered by a handful of numerical Mathematica functions. In the area of symbolic computation, Mathematica's closest antecedents are computer algebra systems such as Macsyma, Maple, reduce and SMP. (SMP was designed by Stephen Wolfram in 1980-81). These systems have some of the algebraic capabilities of Mathematica, but lack its generality or programmability. In the area of graphics, Mathematica appears to be unique in providing a full implementation of three-dimensional hidden-surface graphics in resolution-independent PostScript. -------------------------------------------------------------------------------- Addison-Wesley Published Mathematica Book June 23, 1988 REDWOOD CITY, Calif. -- Addison-Wesley Publishing Company today announced the publication of "Mathematica: A System for Doing Mathematics by Computer," by Stephen Wolfram. This announcement cam e at the same time as the announcement by Wolfram Research, Inc. of the Mathematica software system. Mathematica, said Steven Jobs, president of NeXT, Inc., "will revolutionize the teaching and learning of mathe by focusing on the prose of mathematics, without getting lost in the grammar." A revolutionary software system for doing mathematical calculations, Mathematica is available on a full range of computers, from personal systems to workstations and supercomputers, Mathematica is expected to be widely used in education, research , engineering and business. The Addison-Wesley book is the primary documentation for the Mathematica system. Written by Stephen Wolfram, principal designer of Mathematica, a well-known scientist and MacArthur prizewinner, the 768-page book includes both a tutorial introduction and a definitive reference guide to Mathematica. Fully illustrated with graphics produced by Mathematica, the book is available immediately in both softcover and hardcover. The softcover version (ISBN 0-201-19334-5) retails for $29.95; the hardcover version (ISBN 0-201-19330-2) retails for $44.25. According to Bill Press, professor of Astrophysics at Harvard Unviersity, and author of Numerical Recipes," "Mathematica has a real chance of repplacing pencil and paper as a standard desktop environment for scientists and engineers, and the book, by itself, provides insight into a powerful new mode of thinking at the boundary between mathematics and algorithmics." "Publication of the Mathematica book reinforces our leadership position as the premier publisher of computer science and technology products," says John Connoly, corporate vice-president and director of Higher Education at Addison-Wesley. Over the past several years, Addison-Wesley has formed strategic alliances with many of the world's leading computer hardware and software companies, including Apple Computer, Inc., International Business Machines Corporation, Lotus Development Corporation, Adobe Systems, Inc., Minitab, Inc., and MathSoft, Inc. Addison-Wesley publishes the complete series of Macintosh technical books from Apple Computer, Inc., the official Postscript books from Adobe Systems, Inc., and the IBM programming language series volumes. "Addison-Wesley was the obvious choice of a publisher for the Mathematica book," said Wolfram. "Their leadership both in computer book and textbook publishing made Addison-Wesley uniquely suitable." "Addison-Wesley intends to maintain its lead in Mathematica-related publishing," said Vice President Allan Wylde, director of the Advanced Book Program which spearheaded the Mathematica book project. "We will publish both educational and professional bookds based on Mathematica." Addison-Wesley plans a series of books based on Mathematica. Already under contract are books on Mathematica programming techniques, and a calculus course based on Mathematica. The Mathematica software will be offered by a number of major computer manufacturers, including IBM, Sun Microsystems, and Silicon Graphics. NeXT plans to bundle Mathematica with each one of its eagerly-awaited new computers. Mathematica is also available for the Macintosh directly from Wolfram Research, Inc. of Champaign, Illinois. -------------------------------------------------------------------------------- Calculus Course Based on Mathematica to be Published by Addison-Wesley. June 23, 1988 REDWOOD CITY, Calif. -- Mathematica will be used as the basis for a major new initiative in undergraduate education. A calculus course based on Mathematica is now under development at the University of Illinois at Urbana-Champaign. The course, which uses a combination of printed material and on-line Mathematica notebooks will be published by Addison-Wesley Publishing Company, the leading information-disseminator of computer books, software, and related materials, and the world's largest publisher of mathematics books. Calculus, with a nationwide enrollment of over 900,000 students, has one of the larges enrollments of any single course in American universities today. Calculus is taken by all students who intend to go on in science, engineering or business. Robert White, president of the National Academy of Engineering calls calculus "a critical waystation for the technical manpower this country needs." Yet calculus instruction in America is widely perceived to be troubled. Two national conferences on calculus reform have been held in the past two years, including one sponsored by the National Research Council (NRC). Lynn Arthur Steen, past president of the Mathematical Association of America said, "Think, as many did at the NRC Colloquium on Calculus for a New Century, about the contrast between the five thousand exercises in typical calculus books that mostly ask students to imitate calculator buttons and the discovery potential in symbolic computer systems." "Mathematica will revolutionize calculus teaching the way the calculator has revolutionized arithmetic," says J. Jerry Uhl, a professor of Mathematics at the University of Illinois and one of the authors of the calculus course based on Mathematica. Uhl, and his coauthors Horacio Porta and Don Brown, believe that with Mathematica students will be able to understand the underlying principles of calculus more clearly than has been possible before. By having the computer take care of routine calculations, the students will be able to concentrate on concepts. A key difficulty in present-day calculus courses is that the problem must be simple enough to allow for hand solution. As a result, the problems that students are given have little to do withthose that they would encounter in practical situations. Mathematica provides a resolution to this dilemma. Once students have understood the principles involved in solving a problem, they can use Mathematica to work through details. "With Mathematica, students will be able to solve calculus problems far beyond what even professors can solve today," says Horacio Porta, also a professor of Mathematics at the University of Illinois. "Mathematica is not only great at calculations, it is also a powerful tool for exposition," says J. Jerry Uhl. "With Mathematica's graphics, students can visualize the formulas they are working with." In a typical case, a student might work out the algebraic formula for a derivative using Mathematica, then plot the formula and confirm that it represents the slope of the original curve. With Mathematica, students can also make use of a recently-popular research technique known as "experimental mathematics." Students then can infer from these examples the underlying principles at work. Field testing of the Mathematica-based course is scheduled to begin inthe fall of 1988 at the University of Illinois and the University of Missouri, together with other universities, colleges, and highl schools. -------------------------------------------------------------------------------- Corporate Speakers at the MATHEMATICA Product Announcement - June 23, 1988 * Forest Baskett, Vice President, Research & Development, Silicon Graphics Computer Systems * Gordon Bell, Vice President, Research Development, and Engineering, Ardent Computer * Steven Jobs, President NeXT, Inc. * William Joy, vice President, Research & Development, Sun Microsystems * Vicky Markstein, Research Staff Member IBM * Eric Lyons, Director of Technology, Autodesk, Inc. * Larry Tesler, Vice President, Advanced Technology, Apple Computer * Stephen Wolfram, President, Wolfram Research, Inc., and Professor of Physics, Mathematics and Computer Science, University of Illinois -------------------------------------------------------------------------------- IBM OFFERS MATHEMATICA ON ITS RT SYSTEM June 23, 1988 SANTA CLARA, CA, . . . IBM announced today that it will offer IBM AIX/RT Mathematica symbolic mathematics software for use with the company's reduced instruction set computing (RISC) technology workstation -- the IBM AIX/RT system. AIX/RT Mathematica will run on IBM's Advanced Interactive Executive (AIX) operating system, an industry standard that was recently selected as the UNIX operating system for the Open Software Foundation. "There is every reason to believe that AIX/RT Mathematica will radically change the teaching of engineering," said IBM Fellow John Cocke, the inventor of RISC technology and winner of the Association for Computing Machinery's A.M. Turing Award, the association's highest award for technical contributions in computing. IBM said it expects that AIX/RT Mathematica, together withthe RT workstation, will initially have its greatest impact on the scientific and engineering community. But the company also views it as a tool for the development of new applications in such largely untapped areas as finance and education. "The combination of AIX/RT Mathematica and the RT begin to fulfill a long-time dream of computer scientists," said Cocke. "We now have a tool that can automatically process long, complex algebraic formulations that we previously had to do by hand. Mathematica is a huge step forward in interactive computing, and will complement a lot of other software available for the RT, expecially CATIA and CAD/CAM packages. AIX, Advanced Interactive Executive and RT are trademarks of IBM Corporation. Mathematica is a trademark of Wolfram Research Inc. -------------------------------------------------------------------------------- IBM AIX/RT MATHEMATICA PRODUCT SPECIFICATIONS June 23, 1988 The IBM AIX/RT Mathematica is designed to execute on an IBM RT 6150 or 6151 system configured withthe following software and hardware: Software Prerequisites * IBM AIX RT Operating System Version 2.2 (5601-061) * IBM/AIX RT X-Windows Version 2.1 (5601-125) Hardware Prerequisites An IBM RT 6150 or 6151 configured with the following IBM components: * At least 8MB memory * At least a 70MB file capacity * A 1.2MB diskette drive * An IBM supported graphics display and associated adapter for the RT * An IBM RT mouse or OBM 5083 Tablet * An IBM RT Floating Point Accelerator Adapter or an IBM RT Advanced Floating Point Accelerator Adapter * Contour and density plots and 3D surface with color, shading, and lighting requires an IBM supported color graphics display and associated adapter for the RT. Performance Considerations For generally acceptable response times, these minimum requirements may vary depending upon RT system activities, and the IBM AIX/RT Mathematica function and applications being accessed and executed. In some cases, additional fixed disk storage, memory, and/or the IBM RT Advanced Floating Point Accelerator may be required for generally acceptable resonse times. DESCRIPTION IBM AIX/RT MATHEMATICA provides the capability to: 1) Evaluate many mathematical functions with numbers of any size and precision (subject to limites of the machine configuration). Work with vectors and matrices and perform all standard operations of linear algebra. 2) Deal with symbolic formulas; do algebra with polynomial, rational expressions, evaluate integrals and derivatives and find power series expansions. 3) Perform an extensive repertoire of 2D and 3D graphical representation. -------------------------------------------------------------------------------- ARDENT COMPUTER ADDS WOLFRAM RESEARCH'S MATHEMATICA TO SOFTWARE PORTFOLIO FOR TITAN GRAPHICS SUPERCOMPUTER. June 23, 1988 SANTA CLARA, Calif., -- Ardent Computer Corp., maker of the Titan graphics supercomputer, has signed a licensing agreement with Wolfram Reserch, Inc., of Champaign, Ill., to offer Wolfram's Mathematica software on its Titan graphics supercomputer. Mathematica, a new software package that performs numerical symbolic and graphical mathematics, will be sold by Ardent under its Application Alliance program, which offers a comprehensive portfolio of third-party software for the Titan. The Titan port of Mathematica has been implemented by Wolfram and will be available from Ardent for $8,000 beginning in the third quarter of 1988. James R. Newcomb, Ardent's vice president of strategic software business development, said Mathematica "represents a new era of computer-based mathematics whose large computational requirements are an excellent match for the Titan's high-performance architecture. This powerful hardware-software combination will be especially attractive to our customers in academic and research institutions. It fits in with our charter to assemble the best available set of math software to run on the Titan." Titan, introduced March 1, represents a new class of computer -- the graphics supercomputer -- that couples the computational speed of a supercomputer with high-performance interactive graphics. Designed using a 64-bit parallel vector architecture typical of supercomputers such as the Cray XMP, Titan achieves a peak processing rate of 64 million floating-point operatiosn per second (MFLOPS). Formed in 1985, Ardent Computer Corp. is headquartered in Sunnyvale, Calif. -------------------------------------------------------------------------------- About the Developers of MATHEMATICA Mathematica was developed by a team of eight scientists and mathematicians, led by Stephen Wolfram. Wolfram, 28, is best known for his pioneering research on cellular automata and complex physical systems. Educated at Eton, Oxford and Caltech, Wolfram spent time at Princeton before becoming Professor of Physics, Mathematics and Computer Science at the University of Illinois. Wolfram's research has spanned many areas of science. His early resarch was in high-energy physics and cosmology. Wolfram's recent work has included developing new models for physical and biological pattern formation, and inventing new schemes for cryptography and for computational fluid dynamics. In 1980-81, Wolfram developed the SMP system for computer algebra. Wolfram was the main designer of Mathematica. He also wrote much of the original core code of Mathematica, and its primary documentation. Wolfram is now President and Cheif Executive Officer of Wolfram Research, Inc. Daniel Grayson implemented many of the mathematical algorithms in Mathematica. Grayson is Professor of Mathematics at the University of Illinois. His research has concentrated on applying computers to solve fundamental mathematical problems, particularly in number theory and algebraic K theory. Roman Maeder wrote the code for integration and polynomial factorization in Mathematica. After receiving his Ph.D. in Zurich, Maeder came to the U.S. to organize "mathematical laboratories" in which computer algebra is used to teach graduate mathematics. Theodore Gray wrote the front end for Mathematica on the Macintosh and the NeXT computer. Before working on Mathematica, Gray did graduate work in theoretical chemistry at Berkeley. Henry Cejtin was responsible for the final implementations of many central parts of Mathematica. Cejtin worked on operating system software, and earned a Ph.D. in pure mathematics from Northwestern University. He was largely responsible for the decision to separate the kernel and front end for Mathematica. Stephen Omohundro wrote the three-dimensional graphics code in Mathematica. Omohundro is presently ASsistant Professor of Computer Science at the University of Illinois. Previously, he did mathematical physics at Berkeley, and algorithm development at Thinking Machines Corporation. Omohundro's main research interests are in geometrical approaches to machine learning. David Ballman was responsible for the external system interface of Mathematica. Ballman is now Vice President of Wolfram Research. Jerry Keiper implemented many of the numerical algorithms in Mathematica. Before working on Mathematica, Keiper earned two masters' degrees in mathematics. ***************************************************************************