![]() Teams of students build a fractal tetrahedron made of toothpicks and marshmallows. Learn about the fractal nature of rivers, understand what a watershed is and create your own fractal design.Ī simple geometric substitution leads to a jagged fractal much like a coastline. Measure the ratios and angles of a tree’s branches to uncover its fractal structure. Make a 3-dimensional fractal with paper and scissors. Learn to make the classic triangle fractal, and join many together into a bigger one. See what people are saying about us on our feedback page! We are sharing these fractivities with teachers for free and request that you provide feedback to us so we can continue to refine and improve our curriculum. This page includes the associated English language arts standards met by the activities. This summary page shows what grades each activity is appropriate for and what common core math standards they meet. Check out our education and outreach page! You can use our curriculum for free or can hire us to come to your school to lead fun, hands-on activities. Pierce Foundation, our fractivities are aligned with common core standards in mathematics and English language arts so you can more easily incorporate teaching fractals in your classroom, combining science, math and art! Students utilize their math skills in real-world applications and also work together as a team to create large fractal designs. Here's a comparison of those two algorithms for $z^2 - 0.77967939051932 + 0.Fractivities are hands-on projects that teach fractal concepts in a fun, artistic way. This is particularly useful for certain hard to generate Julia sets, as described in this answer. One thing that's nice about that version is that (with all the algebraic machinery that Mathematica has to offer), it's not hard to write a program that works for higher order polynomials or rational functions.Īnother technique to get the boundary is called "boundary scanning". That program has been been built into the Mathematica kernel since V10. The basic ideas behind inverse iteration, it's modification appear, and implementation for Mathematica were described in this 1998 paper. Viewing the page source, it's pretty easy to find this Javascript code for the program. Sure! This little web app generates images of Julia sets for functions of the form $f_c(z) = z^2+c$ using the modified inverse iteration algorithm as described in "The Beauty of Fractals" and "The Science of Fractal Images". (My app does not yet support IIM or any other method for plotting Julia sets other than filled Julia sets.)Īre there sample implementations of rendering Julia set boundaries that I could study? The best I can do with my app is to plot the Julia set using Distance Estimates and crank up the boundary I draw: Note how the interior structure is faint and does not connect like the image from the book. The JavaScript app recommended in the answer looks like this: Note the complex, connected interior structure. The image from "The Beauty of Fractals" is a complex number Julia set from the equation Zₓ₊₁ = Zₓ² + C. It talks about using the inverse iteration method, and various other methods that divide the complex plane into a grid of rectangles and keeping track of how many times a rectangle is visited as you iterate the points in a plot in order to identify periodic points.Īre there sample implementations of rendering Julia set boundaries that I could study? (I'm fluent in C and several other C-family languages, and can usually figure out other languages well enough to read them, so the language isn't that important.) EDIT: There is a chapter in "The Beauty of Fractals" titled "Juila Sets and Their Computergraphical Generation", but I find it hard to follow. I use trig, algebra, some linear algebra and matrix math all the time as a developer.) (I took AP calculus in high school almost 40 years ago, so my advanced math is pretty rusty. I own copies of both "The Beauty of Fractals" and "The Science of Fractal Images", but struggle with the heavier mathematics in those texts. It does not, however, render just the boundaries of Julia sets. It renders filled Julia sets using either iteration counts or DE data. (See this article for a writeup on the app: ) It creates 2D and 3D images of Mandelbrot and Julia sets (using complex numbers and the formula Zₓ₊₁ = Zₓ² + C.) It includes support for Distance Estimates (DE) and fractional iteration values, and creates 3D height maps using DE data. I have a commercial application, FractalWorks, for Mac OS.
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