Mark's Scholarly activities

My scholarly interests include real analysis, fractal geometry, real and complex dynamics, and computer graphics illustrating these ideas. My images have been featured on the covers of The Mathematica Journal, Mathematica in Education and Research, The Mathematical Intelligencer, and Math Horizons.

My publication rate has slowed in recent few years as my interests shifted to some other things. First, I worked heavily as a consultant for Wolfram Research; mostly, I developed mathematical content for Wolfram|Alpha but I also developed some functionality in Mathematica V10. More recently, I've been involved developing open source tools for mathematical visualization.

In the Summers of 2015 and 2016 I participated in a teacher training workshops organized by FCR-STEM and funded by the US DOE Math-Science Partnership. The long term purpose is to study the impact of teacher content knowledge on student learning outcomes. It was loads of fun to develop dynamic illustrations of basic mathematics to support this project; I've archived my tools on this web page.

Recent presentations

• Computational Adventures in Complex Dynamics
  • The code
• Classroom Discourse

Publications

Fractals in real and abstract analysis

• Fractal Measures on Infinite Dimensional Sets
  My 1994 OSU dissertation
• "Entropy dimensions of the hyperspace of compact sets"
  The Real Analysis Exchange, V21 (1995/96), #2, pp 194-202
• "The Hausdorff dimension of the hyperspace of compact sets"
  The Real Analysis Exchage, V22 (1996/97), #2, pp 611-625
• "The prevalent dimension of graphs"
  The Real Analysis Exchage, V23 (1997/98), #1, pp 241-246
• "The Hausdorff dimension of Hilbert's coordinate functions"
  The Real Analysis Exchange V24 (1998/99), #2, pp 875-883
• "The Borel structure of the collections sub-self-similar sets and super-self-similar sets"
  Acta Mathematica UC, V69, #2, (2000) pp 145-150
  with Bob Vallin
• "Self-similar structure in Hilbert's space filling curve"
  Mathematics Magazine, V76, #1, (2003) pp 40-47

Fractals from a computational perspective

• "Inverse iteration algorithms for Julia sets"
  Mathematica in Education and Research, V7 (1998), #2, pp 22-28
  Mathematica V10 compatible notebook
• "Directed-graph iterated function systems"
  Mathematica in Education and Research, V9, #2, (2000)
• "Digraph self-similar sets and aperiodic tilings"
  The Mathematical Intelligencer, V24, #2 (2002), pp 33-42
• "Self-similar intersections"
  Fractals, V16 #2 (2008), pp 187-197
• "Images of a vibrating Koch drum"
  Computers and Graphics, V32, #6, (2008), pp 711-715
• "Generating self-affine tiles and their boundaries"
  The Mathematica Journal, V11, #1, (2009).
• "Parameterized dynamics for the Weierstrass elliptic function over square period lattices"
  Intl. Jour. of Bifurcation and Chaos., V21, #1, (2011), pp 1-11
  with Jane Hawkins
• "Through the Looking-Glass, and What the Quadratic Camera Found There"
  The Mathematical Intelligencer., Volume 34, Issue 3, (2011) pp 30-34.
  with Stan Wagon, Bart de Smit, Willem Jan Palenstijn, and Isaac Sparling

Computational discrete mathematics

• "A stochastic cellular automaton for three-coloring Penrose tiles"
  Computers and Graphics, V26, #3, (2002) pp 519-524
• "Decremental tag systems and random trees"
  Complex Systems, V18, #2 (2009)
  With Patrick Bahls and Josh Knox.
  
• "Four-coloring the US counties"
  Math Horizons April, 2009 pp 20-21
  with Stan Wagon

Columns from Mathematica in Education and research, plus one invited review

• A review of Stan Wagon's Mathematica in Action, 2nd Ed
  V8, issue 3/4 (1999).
• "The connected locus for complex cubic iteration"
  V10, #2 (2005)
• "Parametric L-systems and borderline fractals"
  V10, #3 (2005)
• "Steffen's flexible polyhedron"
  V10, #4 (2005)
• "Bifurcation sets and critical curves"
  V11, #1 (2006)
• "Newton's method for complex polynomials"
  V11, #2 (2006)
• "The Read-Bajraktarevic operator"
  V11, #3 (2006)
• "The kings problem"
  V11, #4 (2006)
• "Boundary scanning and complex dynamics"
  V12, #1 (2006)
• "Vibration of the Koch drum"
  V12, #2 (2007)
• "Generating Google maps"
  V12, #3 (2007)