Explore the Mandelbrot Set!

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"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth nor does lightning travel in a straight line."- Benoit B. Mandelbrot, 1983

The world around us is full of shapes which the traditional Euclidean geometry (what you've studied or will study in high school geometry) just cannot describe accurately. Even in our manufactured world it is hard to find an example of the perfection of Euclid.

Perhaps others recognized this problem first, but it is Benoit B. Mandelbrot who must be given credit for the solution. Working as a research scientist at I.B.M., he rediscovered the works of Gaston Julia and the "mathematical monsters" of the early twentieth century. When Mandelbrot's ideas had coalesced, he'd forged a new branch of mathematics: fractal geometry.

For his fractal geometry, Mandelbrot introduced a new tool to describe the world: fractals. The most important characteristics of fractals are self-similarity and infinite detail. The world is full of self-similarity (and so, fractals). A mossy pebble at one scale resembles the forested mountain it's a part of at another. A twig grows gnarled as does its tree. Other examples we leave to you to discover.

The advent of computers is what made fractal geometry possible. Though simple formulas may be iterated to produce beautiful, intricate fractals, to do the calculations by hand would be mind numbing. Computers, however, make practical the thousands (or millions or billions) of operations required. Now computer-generated fractals are used to model the world, for pure mathematics, movies, design, and for art.

The Mandelbrot Set is the set of complex numbers which do not cause the function, f(z)=z²+c to diverge when it is iterated. z is originally set to zero, and c is a constant equal to the complex number being tested.

Our explorer works by iterating the function and checking to see if z's absolute value (distance from the origin) is greater than 2. If |z| is greater than 2, then we can prove it will diverge to infinity. Demonstrate this to youself below. Colors are assigned according to how many iterations it takes for the function absolute value to exceed 2. If c passes the test (after 61 iterations), then it is colored black.

Student Mentoring at ISU

posted 5/17/1998
by Patrick Jordan

For the past six months I have been involved in a special project with physics department at the Ames Lab at Iowa State University. I was given this privilige because of a special grant awarded to Rebecca Shivvers linking high school students interested in the area of math and computers with professors working on different research projects ar Iowa State.

Because of this project I have embarked on a new series of subjects that were previously foreign to me. I was quickly exposed to method of approximation called Levin's method. Levin's method is a useful tool in obtaining highly accurate numerical values for slowly converging series. It is done by approximating the correctior term term in a recursion relation by using a polynomial of degree n-2. This gave way to a variety of new studies. One of these, themain emphasis in this project, was to obtain highly accurate values of Fermi-Dirac integrals. Evaluation of these integrals in the semi-classical regime has been a long standing problem. From this I learned of Riemann Zeta functions that were used in the calculation of certain Fermi-Dirac integrals. That then led to confluent hypergeometric functions. From the confluent hypergeometric function I learned of gamma functions.

Though this is an extremely broad range of subjects, Dr. Luban simplified them down to a functional region. This new knowledge will prove to be helpful in deriving mathematical formulas that would otherwise be unsolvable to me.

Now that the research for Dr. Luban's project is complete, I have begun working on Winston Conce for light-collection used in collection photons on the atmosphere for Dr. Krennich in the field of astrophysics.

I feel very fortunate to have been provided the opportunity to participate in the research for both of these projects. The educational experiences afforded by this special grant are immeasurable.

A brain teaser

You have two containers, one 9 gallon and one 4 gallon, and an unlimited amount of water. Neither container has any markings at all.
Your challenge is to fill the larger container with exactly 6 gallons of water.

The answer.