Papers
Preprints:
             Associated Material:
              Associated Material:
    • Gallery of screenshots of images from paper (including input data). These images were produced using the updated Quadratic Rational Map Fractal Generator (see below).
Publications:

Programs
  • Quadratic Rational Map Fractal Generator - Updated Version (last updated by myself in 2010), zipped files
This updated version allows for precise control of the flood-filling parameters used to distinguish between Type II and Type III hyperbolic maps in V_3 and V_4. The version posted here is in source code form and requires a JAVA compiler such as JGrasp to run. For parameter slices other than V1-V4, please use the old version available below as this version may not work correctly for those normal forms.

  • Box Counter (a macro for Excel written by Krystal Bibeau (UMF '10), Dr. Lori Koban, and myself)
This program approximates the box dimension for any finite discrete 2 dimensional data set. This is a work in progress as it still needs automating. Check back in the Fall for an updated version (or update it yourself :)).

Here is a sreenshot of our current  box counter calculating the dimension of Sierpinksi's Triangle.

This applet draws the images of up to 6 families of parametric real plane curves under iteration by up to 4 families of quadratic rational self-maps of the real plane. It allows for orbit tracing, parameter adjustment, and parameter parsing. This is a useful tool for drawing dynamical pictures for quadratic maps of the plane as well as iterated function systems whose functions are quadratic or linear.

Adam and Hunter received a UMF Michael D. Wilson Scholarship for undergraduate research for their work on this program.

Here is a gallery of sample images produced using this applet. The saved files are included.

Here is another gallery made with the Dynamical Grapher.
 
This applet draws parameter space pictures for families of quadratic rational maps of a single     complex variable. The Fixed Point Normal Form, Critical Point Normal Form, and Mixed Normal Form are all included as well as some special cases (e.g. V_n: the parameter slices for maps with a critical n-cycle for n=1,2,3,4).

Dustin won two UMF Wilson Scholarships for undergraduate research for his work on this program.

Other students have used this program for their research as well. For example, see "Fractions and Fractals" below.

Here is a gallery of sample images produced with this applet.


(More) Student Projects

Individual Projects:
  • Properties of Bourbaki's Function. This is a paper written by James McCollum (UMF '11) under my direction. It is currently in preprint form but has be submitted to the Rose-Hulman Undergraduate Mathematics Jounral.
  • The Thinning Ice Sheet. This is a presentation by Jamie Beaulieu (UMF '10) on the fractal analysis of the local glaciated terrain here in Farmington, Maine. The motivation for this work is to help understand the rapid acceleration of the melting ice sheets in Greenland. Jamie won a 2010 Wilson Scholarship for undergraduate research for his work on this project and presented his findings at the 2010 UMF Symposium. This work was co-sponsored by UMF geology professor Dr. Doug Reusch.
  • Fractions and Fractals.  This is a poster by Derek Beaudet ( UMF '10 ) and                          Neil Plummer ( UMF '09 ) written under my direction. The gentlemen presented this poster at UMF Symposium 2008. Professor Mary Rees has a nice tutorial of how to make a poster using LaTex located here.
Class Projects: