Intermediate College Level Microworlds

(Arranged alphabetically)

 

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3D graphics
Author(s): james white
Topics: lorenz attractor, conic sections, platonic solids,differential equations, chaos, tutorial (Level: Intermediate College)

Description: This microworld illustrates various uses for 3d graphics. On the first page is a simulation on conic sections in three dimensions. Next is a differential equation solver for ordinary differential equations in R3 starting with the lorenz equations.

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.

Cycloids
Author(s): samuel masih
Topics: cycloids, parametric curves, hypocycloids, epicycloids, polar coordinates, trigonometric functions, trigonometric identities (Level: Intermediate College)

Description: This microworld explores a number of parametric curves with animations and opportunity for readers to practice graphing curves. The curves of interest are various types cycloids, including hypocyloids, epicycloids, prolate cycloids, and curate cycloids.

Deformable templates
Author(s): kate johnson and j.t.ratnanather
Topics: metric mapping,diffeomorphism,template,orbit of a diffeomorphism group,groups of diffeomorphisms (Level: Intermediate College)

Description: Pattern theory proposes the idea that the world can be understood in terms of patterns. The goal of pattern theory is to build a mathematical framework to describe these patterns.

De Moivre's theorem
Author(s): samuel masih
Topics: complex numbers, euler number, de moivre theorem, roots of complex numbers, polar coordinates, polar graphs (Level: Intermediate College)

Description: This microworld discusses complex roots of complex numbers. The exercises introduce: the representation of complex numbers in the plane in polar and cartesian form, euler's form of complex numbers in exponential form,the calculation of powers and roots.

Dynamic programming
Author(s): priyan weerappuli and j.t.ratnanather
Topics: dynamic programming,shortest path,network optimization,minimizing expenditure,computational anatomy (Level: Intermediate College)

Description: dynamic programming is an important tool of computational anatomy. Imagine the cortex of the brain as a folded sheet. One would think that due to the functional significance of these structures, they would appear uniformly in all human brains.

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.

Evolutes
Author(s): samuel masih
Topics: curvature, radius of curvature, osculating circles, evolutes, differential geometry, geometry, plane curves (Level: Intermediate College)

Description: This microworld provides a way to visualize the construction of evolutes of the graphs of functions, and of parametric curves. This construction shows how these evolutes are the locus of the centers of the osculating circles at points along the curve.

Group actions and orbits
Author(s): kate johnson and j.t.ratnanather
Topics: lie group,manifold,group,continuous group,differentiable manifold,group action,orbit (Level: Intermediate College)

Description: Applying an action to a point results in a transformed point. Similarly, applying an action to a group of points results in a transformed group. In the two examples, the point and the group of points are part of the same space, the 2-D real space.

Implicit functions and level sets
Author(s): james white
Topics: implicit functions, implicit curves, pencils of tangent lines, implicit equations, implicit differentiation (Level: Intermediate College)

Description: An equation of the form f(x,y) = 0 often defines a curve in the plane. We say that the curve is defined implicitly but we may not be able to solve for y in terms of x. Implicit differentiation often allows us to solve for dy/dx as function of x and y.

Introduction to group theory
Author(s): mark mattson, kate johnson and j.t.ratnanather
Topics: group, group action, orbit,binary operations,number groups,cycles,transpositions,orbits,stabilizers,symmetry (Level: Intermediate College)

Description: Here, you can play with cycles and transpositions. For the cycle section of the page, click new cycle to have the computer give you a random permutation that can be expressed as 1 or more cycles.

Lie groups
Author(s): kate johnson and j.t.ratnanather
Topics: lie group,manifold,group,continuous group,differentiable manifold (Level: Intermediate College)

Description: A sphere and a torus are just two examples of smooth (infinitely differentiable) manifolds. The earth, as it is a sphere, is locally flat and so it appears flat.Here we study manifolds that also have the structure of groups.

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.

Mapping geometry
Author(s): dan kalman
Topics: linear algebra, matrices, geometry, linear transformations (Level: Intermediate College)

Description: This microworld explores the effect of linear transformations on points, polygons, circles and other objects (drawn freehand) in the plane. One may define any linear transformation and see the effect of the map on geometric objects.

Matrix groups
Author(s): kate johnson and j.t.ratnanather
Topics: group, orthogonal group, symmetry (Level: Intermediate College)

Description: A matrix group is a collection of square matrices that satisfies the group properties. The group composition is matrix multiplication, and the group inverse is the matrix inverse. These properties are in addition to those that define all groups.

Metric mapping
Author(s): kate johnson and j.t.ratnanather
Topics: metric mapping,gradient descent,geodesic (Level: Intermediate College)

Description: the goal of this book is to explain how an algorithm for metric mapping in 1-dimension works. A map is a function. It is used here as a transformation function between one coordinate system and another.

Nonlinear equations
Author(s): ravinder kumar
Topics: nonlinear equations, newton's method, bisection method, secant method (Level: Intermediate College)

Description: When f(x) is a nonlinear function, there is often no procedure for finding exact solutions to the equation: f(x) = 0. The author illustrates various methods for finding approximate solutions: the bisection method, the secant method, and newton's method.

Parametric curves 13 easy pieces
Author(s): mike pepe
Topics: plane curves, curvature (Level: Intermediate College)

Description: This microworld begins with a graphical introduction to parametric curves. The user then can work through a series of dynamic explorations that consider topics such as: speed, curvature, cycloids, evolutes, involutes, bezier curves and more.

Parametric curves all-in-one
Author(s): mike pepe
Topics: plane curves, curvature (Level: Intermediate College)

Description: This microworld begins with a graphical introduction to parametric curves. The user then can work through a series of dynamic explorations that consider topics such as: speed, curvature, cycloids, evolutes, involutes, bezier curves and more.

Periodic points of certain dynamical systems
Author(s): samad mortabit and juan estrada
Topics: dynamical systems, cobweb diagrams, transients, orbits (Level: Intermediate College)

Description: One of the manifestations of chaotic behavior in discrete dynamical systems is the existence of a dense set of periodic points. In this short paper and in the context of some specific classes of one-dimensional discrete dynamical systems.

Space curves
Author(s): mike pepe
Topics: space curves, space graphs, differential geometry (Level: Intermediate College)

Description: This microworld explores properties of curves in three dimensions. It builds frenet frames, normal and binormal ribbons, osculating circles, and tube plots.

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.

Symmetry and groups
Author(s): ravinder kumar
Topics: symmetric groups, rigid motions, cayley tables, group theory (Level: Intermediate College)

Description: This microworld offers visualization of the symmetric group s3 of degree 3. Rotations and reflections of an equilateral triangle are shown using animations. Calculator for the symmetries helps compute various combinations of the symmetries.

Taylor polynomials and their graphs
Author(s): james white
Topics: polynomial approximations to functions, taylor series, maclaurin series, infinite series, series, graphing polynomial approximations (Level: Intermediate College)

Description: This microworld studies taylor-maclaurin series approximations, for which one needs only the value of the function and of its derivatives at a single specified point. These polynomials lead to infinite series representations of the function.

Vectors in the plane
Author(s): mike pepe
Topics: vectors, linear algebra (Level: Intermediate College)

Description: This workbook is an introduction to the basic algebra and geometry associated with vectors in the plane. It includes vector representation of a line and vector projections.

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