Advanced Microworlds

(Arranged alphabetically)

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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.

Congruences
Author(s): ravinder kumar
Topics: congruence, systems of congruence equations, fermat theorem, chinese remainder theorem, cayley tables, group suggested use: congruence equation and systems of congruence equations solver (Level: Advanced)

Description: This microworld develops the following: congruences, fermat's theorem, solution of congruence equations, systems of congruence equations, cayley tables. Readers are invited to conjecture fermat's theorem. They may alse explore some simple groups.

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.

Marden's theorem
Author(s): dan kalman
Topics: complex numbers, complex arithmetic, cubic polynomials, geometry, marden's theorem (Level: Advanced)

Description: Marden's theorem concerns polynomials over the complex numbers. Specifically, consider the polynomial p(z) = (z-a)(z-b)(z-c), where a, b, c are fixed complex numbers. The roots of p and the roots of the derivative p' have an interesting relationship.

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.

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