Scholarship

I received my Ph.D. in 1994 from Ohio State under the direction of Gerald Edgar. After that, I held visiting positions at Washington and Lee University and St. Olaf College before coming to UNCA in the Fall of 1997. I took a sabbatical at Cornell University during academic year 2003-04.

My scholarly interests include real analysis, fractal geometry, and computer graphics illustrating these ideas. Some of my research is very abstract. For example, I've written several papers applying ideas from fractal geometry to hyperspaces of compact sets. I'm also interested in graphs of fractal functions and tilings using sets with fractal boundary.

I use Mathematica in my research more and more. Originally, I saw my fractal research and my interest in Mathematica as fairly separate. I have long used Mathematica in my teaching and for generating mathematical images. However, I have come to appreciate computational experimentation as a serious research tool as well. In particular, I have been able to develop some novel ways to think about fractal functions and self-similar tilings. I don't believe my recent paper on self-similar intersections would have been possible without the use of some computational tool. My digraph fractals package has been used by a number of other mathematicians to illustrate their papers.

From 2005 through 2008, I wrote a regular column on mathematical graphics for Mathematica in Education and Research. MiER was a journal independent of Wolfram Research, the makers of Mathematica, that published a broad range of articles on the use of Mathematica. Sadly, it folded in 2008. Since then, I've joined the editorial board of Math Horizons. My work has been on the cover of both these journal, as well as The Mathematica Journal.

Preprints

Most of my publications are available in preprint form as PDFs below.

• Fractals in real and abstract analysis
  0. Fractal Measures on Infinite Dimensional Sets
     My 1994 OSU dissertation.
     [PDF]
  1. "Entropy dimensions of the hyperspace of compact sets"
     The Real Analysis Exchange, Volume 21 (1995/96), issue 2, pages 194-202.
     [PDF]
  2. "The Hausdorff dimension of the hyperspace of compact sets"
     The Real Analysis Exchage, Volume 22 (1996/97), issue 2, pages 611-625.
     [PDF]
  3. "The prevalent dimension of graphs"
     The Real Analysis Exchage, Volume 23 (1997/98), issue 1, pages 241-246.
     [PDF]
  4. "The Hausdorff dimension of Hilbert's coordinate functions"
     The Real Analysis Exchange Volume 24 (1998/99), issue 2, pages 875-883.
     [PDF]
  5. "The Borel structure of the collections sub-self-similar sets and super-self-similar sets".
     with Bob Vallin
     Acta Mathematica UC, Volume 69, issue 2, (2000) pages 145-150.
     [PDF]
  6. "Self-similar structure in Hilbert's space filling curve".
     Mathematics Magazine, Volume 76, issue 1, (2003) pages 40-47.
     [PDF]
• Fractals from a computational perspective
  7. "Inverse iteration algorithms for Julia sets".
     Mathematica in Education and Research, Volume 7 (1998), issue 2, pages 22-28.
     [HTML, notebook, package]
  8. "Directed-graph iterated function systems".
     Mathematica in Education and Research, Volume 9, issue 2, (2000).
     [HTML, notebook, package]
  9. "Digraph self-similar sets and aperiodic tilings".
     The Mathematical Intelligencer, Volume 24, issue 2 (2002), pages 33-42.
     [PDF]
  10. "Self-similar intersections".
      Fractals, Volume 16 issue 2 (2008), pages 187-197.
      [PDF, Computational supplement]
  11. "Images of a vibrating Koch drum".
      Computers and Graphics, Volume 32, Issue 6, (2008), Pages 711-715.
      [PDF]
  12. "Generating self-affine tiles and their boundaries".
      The Mathematica Journal, Volume 11, Issue 1, (2009).
      [PDF, notebook]
• Computational discrete mathematics
  13. "A stochastic cellular automaton for three-coloring Penrose tiles".
      Computers and Graphics, Volume 26, issue 3, (2002) pages 519-524.
      [PDF]
  14. "Decremental tag systems and random trees".
      With Patrick Bahls and Josh Knox.
      Complex Systems, Volume 18, Issue 2 (2009)
      [PDF, Computational supplement]
  15. "Four-coloring the US counties".
      With Stan Wagon.
      Math Horizons April, 2009 pp 20-21.
      [ PDF, Computational supplement]
• Columns from Mathematica in Education and research, plus one invited review
  16. A review of Stan Wagon's Mathematica in Action, 2nd Edition.
      Mathematica in Education and Research, Volume 8, issue 3/4 (1999).
      [HTML, notebook]
  17. "The connected locus for complex cubic iteration".
      Mathematica in Education and Research, Volume 10, issue 2 (2005).
      [PDF, gzipped notebook]
  18. "Parametric L-systems and borderline fractals".
      Mathematica in Education and Research, Volume 10, issue 3 (2005).
      [PDF, gzipped notebook]
  19. "Steffen's flexible polyhedron" -
      Mathematica in Education and Research, Volume 10, issue 4 (2005).
      [PDF, notebook]
  20. "Bifurcation sets and critical curves".
      Mathematica in Education and Research, Volume 11, issue 1 (2006).
      [PDF, notebook]
  21. "Newton's method for complex polynomials".
      Mathematica in Education and Research, Volume 11, issue 2 (2006).
      [PDF, notebook]
  22. "The Read-Bajraktarevic operator".
      Mathematica in Education and Research, Volume 11, issue 3 (2006).
      [PDF, notebook]
  23. "The kings problem".
      Mathematica in Education and Research, Volume 11, issue 4 (2006).
      [PDF, notebook]
  24. "Boundary scanning and complex dynamics".
      Mathematica in Education and Research, Volume 11, issue 4 (2006).
      [PDF, gzipped notebook]
  25. "Boundary scanning and complex dynamics".
      Mathematica in Education and Research, Volume 12, issue 1 (2007).
      [PDF, gzipped notebook]
  26. "Vibration of the Koch drum".
      Mathematica in Education and Research, Volume 12, issue 2 (2007).
      [PDF, notebook]
  27. "Generating Google maps".
      Mathematica in Education and Research, Volume 12, issue 4 (2007).
      [PDF, notebook]