The Math Cafe

Mathwright Microworlds are multi-page documents that you may read either online in your Microsoft Internet Explorer browser, using MathwrightWeb, or off line, using Mathwright32 Reader. These Microworlds tell their stories using publisher-quality HTML mathematical text. But our Microworlds are not static text. All of the books at the Library are live and interactive. They let you ask your own questions as you read them.

Here at the Cafe, we invite you to explore our new technology for bringing Mathematics to life in your browser, before you join the Library. We are certain you will agree that our approach goes well beyond the Java Applet paradigm, even though all of our Microworlds are written in Java! If you like some of the Microworlds below, then we invite you to join the Library so that you may add our Microworlds to your private off line collection on any of your computers, and so that you may view 75+ additional Microworlds that are not in the Cafe.

These Microworlds complement our older collection of 150 live WorkBooks that can only be read offline using Mathwright2000 Reader.

Once you download our free MathwrightWeb, and install it, you may view any of our free Microworlds in the Visualization Studio below. Click the hyperlink to go to the Title page to learn what the Microworld is about. Then, if it interests you, check your browser settings to be sure that it is ready to play ActiveX Controls as explained on the Title page, and press the

button to read in your ActiveX-enabled browser. You may also download and read the free "Introduction to Mathwright32 Microworlds" offline with Mathwight32 Reader. Press

Actually, there are two new Players (MathwrightNET and Mathwright32 NET Reader) but those are written in anticipation of Microsoft's next operating system after XP that should include its new .NET framework. For that, press

For now, it's best to stick with MathwrightWeb and Mathwright32 Reader. If you cannot see the microworld in your browser, just visit our MathwrightWeb page and we will tell you what to do.

Periodic functions

Author(s): jim swift
Topics: trigonometric functions, graphing, sine, cosine, amplitude, frequency (Level: High School)

Description: This workbook aims to help readers visualize the properties of trigonometric functions, beginning with the wrapping functions through the exploration of the sine and cosine functions. Students learn about amplitude, period and frequency, and phase shift.

Introduction to mathwright32 Microworlds

Author(s): margie hale
Topics: this is the mathwright tutorial (Level: High School)

Description: This microworld discusses the mechanics of reading and interacting in mathwright microworlds. It is a good place to start.

Exploring quadratic functions

Author(s): samad mortabit
Topics: college algebra, graphing, relations, quadratic equations, quadratic graphs, acceleration, gravity, growth models, parabolas, inequalities (Level: High School)

Description: This microworld studies quadratic equations and inequalities. It generates problems randomly or lets you make them up. In either case, it draws the solution graphically then explains step-by-step how to solve algebraically using completion of the square.

Stay afloat!

Author(s): james white
Topics: optimization, extrema, archimedes' principle, bouyancy, implicit differentiation (Level: Beginning College)

Description: this playbook explores archimedes' bouyancy principle and optimization problems in a story in which players build a boat by cutting and assembling planks from a board and putting them together. They may view their product in three dimensions.

Discrete mathematics and computational structures, Part I

Author(s): james white
Topics: set theory, logic, propositional calculus, boolean algebra, relations, function composition, permutations, demorgan's laws, order, cardinality, digraphs (Level: Beginning College)

Description: This is the first half of a 14-week course in discrete mathematics. The lectures cover: sets,functions sets and logic,composition of functions,set operations,permutations of sets,boolean algebra,graphs and directed sets,relations,order,cardinality.

Discrete mathematics and computational structures, Part II

Author(s): james white
Topics: set theory, logic, propositional calculus, boolean algebra, relations, iteration, recursion, counting, critical paths, prolog, automatic theorem proving, ruleset, inference engine function composition, permutations, demorgans laws, order, cardinality, (Level: Beginning College)

Description: This is the second half of a 14-week course in discrete mathematics. The lectures cover: iteration and recursion,search,sets defined by propositions,critical path analysis,counting,automatic problem solving,relations and functions,graphs and logic.

Color portraits of complex mappings

Author(s): james white
Topics: complex numbers, complex arithmetic, cubic polynomials, geometry, cauchy integral theorem, lucas theorem, marden's theorem (Level: Intermediate College)

Description: This microworld explores complex analysis in a colorful way. analytic maps also have some deep connections with geometry, but are somewhat more difficult to visualize than real maps. In this book we use colors to represent the properties of these maps.

Dynamical systems primer

Author(s): samad mortabit
Topics: chaos, bifurcation, dynamical systems, logistic, movie, cobweb diagrams, population dynamics, harvesting (Level: Intermediate College)

Description: This interactive microworld is a short course on chaotic discrete dynamical systems. It covers dynamical systems, bifurcation, conjugacy, chaos, and applications. In each chapter, there is a laboratory developing the ideas via interactive explorations.

Fractals and the Mandelbrot set

Author(s): jim swift
Topics: mandelbrot set, fractals, iteration, fibonacci (Level: Intermediate College)

Description: this microworld is an interactive introduction to fractals and the mandelbrot set. it steps through the construction of that set, developing the notion of complex iterated maps, and provides many exercises that can illustrate the basic ideas.

Special relativity and conic sections

Author(s): james white
Topics: hyperbolic geometry, special relativity, conic sections, geometry, light cone, focus, ellipse, hyperbola (Level: Intermediate College)

Description: An interesting property of conic sections leads to the focus-locus description. We will explore in this microworld a link between the focus locus definition and the plane slicing cone definition, based on the hyperbolic geometry of special relativity.

Cardano

Author(s): james white
Topics: cubic equations, equations, inflection points, graphing, factorization of polynomials, maxima and minima, cubic polynomials, complex numbers (Level: Advanced)

Description: This microworld develops an approach to the study of cardano's method for solving cubic equations that discloses certain new symmetries and points the way to generalization to higher degree equations. Those generalizations are to the quartic case.

Heron's formula

Author(s): james white
Topics: geometry, heron's formula, triangles, optimization, extrema (Level: Advanced)

Description: This microworld explores heron's formula for the area of a triangle in terms of its sides. The formula may be understood by asking which quadrilateral with given side lengths has the largest area. Here the reader varies the shape of the quadrilaterals.

Odds and integrals

Author(s): james white
Topics: probability, integration, approximation, geometry, geometric construction, vectors, permutations, barycentric, topology, simplex, simplicial complex, subdivision (Level: Advanced)

Description: This microworld explores the following question. Given n random numbers chosen from the unit interval [0,1], what is the expected value of the kth shortest segment so determined? We translate this problem from probability to geometry.

Preview the new 10-Chapter Interactive Web Course on College Algebra built on Mathwright Microworld technology. It differs from other textbooks in significant ways. It is a genuine effort to provide students with the right tools and the appropriate level of discussion that are necessary for a successful learning experience. Students can interact with the text, pose their own questions, and are provided the tools to discover the answers to the questions they pose.

The Preview version, Chapter 2 of this book (70 printed pages with 9 embedded explorations) is available here at the MATH Cafe as a demonstration of the new idiom that we are exploring. The entire text is available for purchase at another website. Click the title below to go there.

Mathwright uses some new techniques to help students visualize mathematics. Since it also uses ActiveX controls, which may be unfamiliar at first, we provide a few free demonstration titles that you may use to get familiar with the environment, especially if you need to tweak something. In this way, you can be sure that things work on your system before joining the Library. The books above are designed to support visualization and to encourage your questions. They are arranged in increasing order of academic level.

It is well known how the computer language Java burst suddenly upon the scene in 1995.. As a language for producing browser interactivity (applets) it took the Internet by storm. What is perhaps not so well known is that Java is an extremely powerful programming language that integrates many of the best features of LISP and OOP (Object-oriented programming, a discipline explored early on by LISP and SmallTalk) in a versatile and expressive development environment.

We discuss our free Java ActiveX Control: MathwrightWeb and show you some example WorkBooks in your ActiveX-enabled browser in the MathwrightWeb Room of the Cafe below. MathwrightWeb offers a fresh alternative to Mathematical applets, extending their range, power, and expressiveness. Check it out below.

MathwrightWeb and Mathwright32 are pure LISP in homage to the grace and beauty of that wonderful language. But it is not well known that LISP is an implementation of a theory of mathematical logic, Alonzo Church's lambda calculus. As such, it is a shining contribution of mathematics to computer science, and we say a little about it in our LISP room of the Cafe below. Also, visit the Discussion Room where we take up the topic: Artificial Intelligence in the Classroom in the LISP, Logo, and AI Forum.

One of the most powerful contributions that personal computers can make in the world of educational technology is in the design of compelling microworlds that teach by placing the reader in a virtual context in which the "rules" reflect the properties of an ideal mathematical reality. In such a context, the reader need only surrender to her imagination to come under the spell of that ideal reality. Plato would have been pleased!

Mathematical ideas do, however, have an abstract Platonic reality that makes them difficult and challenging to render in this way. It requires more than imagination and dedication alone to populate a virtual world with mathematical objects and relations that are pedagogically useful. Until recently, it required the resources of a Production Studio! To make those objects dynamic, and responsive to a reader's questions was beyond even Hollywood.

But with the expressive power of realistic 3D graphics, that has changed. 3D graphics is a wonderful new discipline that is based on Solid Geometry and Linear Algebra and that demonstrates in a vivid way the powerful tools that mathematics itself can bring to a new art form. Mathwright32 Author is a Simulation Toolkit. Its microworlds give readers (Players) unique opportunities to visualize, and to participate in, exciting ideas and constructions of mathematics and science, in the interplay between geometry, graphics, and art, and in the underlying logical principles that bind them all together in this vibrant new mind-tool, the three-dimensional desktop.