Robert G. Niemeyer

Assistant Professor

About Me

Photograph of Robert G. Niemeyer

I am an assistant professor in the Department of Mathematics & Statistics at the University of the Incarnate Word in San Antonio, TX. My research interests mostly reside in the intersection of fractal geometry and mathematical billiards. Recent projects have allowed me to branch out into number theory and physically-inspired billiard problems. If you would like a copy of my research statement, please send me an e-mail.

After receiving my PhD from the University of California, Riverside under the supervision of M. L. Lapidus, I went to the University of New Mexico as an NSF MCTP Postdoctoral Fellow (2012-2015). Afterwards, I spent two years at the University of Maine as a term assistant professor. Luckily, I missed seeing a true Maine winter! Now at the University of the Incarnate Word, I can say I've certainly seen everything in between, from the desert to the green.

In the fall of 2018, I am teaching Precalculus and Calculus II. I am also currently putting together a seminar series on "mathematical curiosities." Email me if you are interested.

Curriculum Vitae Teaching Statement Diversity Statement

Robert G. Niemeyer
University of the Incarnate Word
Dept. of Mathematics & Statistics
234B Bonilla Science Hall
San Antonio, TX 78239
robert.niemeyer "at"


I am a dynamicist with specific research interests in topological dynamical systems, both pure and applied. Below I give brief details about my own work and projects with collaborators. The central theme of my research in dynamical systems is computer experiment as a means of supporting and proving conjectures.

C. Johnson and I are currently investigating the flow on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, but in specific instances can be formally defined.

My joint work with Joe P. Chen on a project that extends the work of Jeremy Tyson and Estibalitz Durand-Cartagena [Du-CaTy] has recently appeared in the Journal of Mathematical Analysis & Applications. The project involves describing periodic orbits of a self-similar Sierpinski carpet billiard table.

I have worked with M. L. Lapidus on various papers on the topic of fractal billiards. Specifically, a recent article with M. L. Lapidus and R. L. Miller consists of determining periodic orbits of the T-fractal billiard table. An eventual goal is understanding whether or not there are equidistributed orbits on the T-fractal billiard table. An initial step in this direction is to understand whether or not every orbit with an irrational direction is recurrent in the T-fractal billiard.

Mathematical Andreev reflection is retro-reflection with parity. By this, we mean that when a pointmass intersects a portion of a billiard table not obeying the standard Law of Reflection, the pointmass returns upon the incoming path.

In addition to this, the ball is assigned a parity and such parity changes with each intersection with said side. The side for which the billiard ball experiences Andreev reflection is call the Andreev subset of the billiard table boundary. We begin building a mathematical foundation for dynamics in an Andreev billiard table, as it is discussed within the context of physics. To such end, we have begun simulating billiard trajectories in a 2D rectangular billiard table with the base of the billiard table being the Andreev subset. Andreev reflection, as it is discussed above, is meant to ideally model the dynamics of an electron in a nanowire lying on a superconductor plate. Elementary results in mathematical billiards are quickly applied to classify dynamics in such a model. We also perturb the boundary of the 2D model so as to introduce explanations for observed behavior not fitting existing models for electron-hole dynamics in a nanowire.

In collaboration with R. E. Niemeyer, we examine the effect of particular parameter changes in a system of ordinary differential equations modeling the interaction between people of a nation-state, the amount of internal strife and the amount of resources within the system.

Peter Turchin et al, construct a system of ordinary differential equations that accurately models the rise and fall of particular empires. However, they do not fully examine the robustness of their model, nor do they fully justify the simplification of a 3D system to a 2D system. While their work is empirically sound, we argue that one does not need to reduce the complexity of the system by collapsing the state resources variable S and the internal strife variable W into a new variable I, called the instability index. If one maintains the conjecture that all three variables are important in describing the rise and fall of an empire, then other dynamical systems may provide better explanatory power and insight into how to predict the effect of diminishing resources in a city-state and an increase in internal strife.

Additional work has focused on computational modeling of number-theoretic problems so as to gain insight into the behavior of particular systems. Joint work with D. Bradley, A. Khalil and E. Ossanna has focused on finding a rigorous connection between a matrix representation of Pascal's triangle modulo a square-free integer and a planar representation of such a matrix, with the immediate goal being to calculate the box-counting of said planar representation. For the sake of brevity, further discussion on these topics will not be given here.

A recent project with J. Collins focuses on using linear algebra to determine all possible combinations of positive integers summing to a particular value, but without repeats and increasing so as to determine an optimal solution to a digital stegonoraphy problem. As described by my collaborator J. Collins, this research explores the concept of multilevel bit plane expansion of digital multimedia formats for data hiding. Using select numerical analysis techniques, we will explore alternatives representations to the normal binary based bit-plane decomposition of digital multimedia files to determine an optimal solution for high capacity data hiding.

Publications · Papers · Notes


  • Towards the Koch snowflake fractal billiard--Computer experiments and mathematical conjectures (with M. L. Lapidus), in "Gems in Experimental Mathematics", T. Amdeberhan, L. A. Medina and V. H. Moll (Eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, R. I., 517 (2010), 231-263.
  • Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards (with M. L. Lapidus), Discrete and Continuous Dynamical Systems Ser. A Vol 33, No 8 (2013), 3719-3740.
  • The current state of fractal billiards (with M. L. Lapidus), in: ``Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics'' (D. Carfi, M. L. Lapidus, M. van Frankenhuijsen, E. P. J. Pearse, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 601 (2013), pp. 251-288.
  • Periodic billiard orbits of self-similar Sierpinski carpets (with J. P. Chen), J. Math. Anal. Appl. 416 (2014), 969--994.
  • Nontrivial paths and periodic orbits of the T-fractal billiard (with M. L. Lapidus and R. L. Miller), Nonlinearity 29 (2016), 2145--2172.
  • The Box-Counting Dimension of Pascal's Triangle r mod p (with D. M. Bradley, A. Khalil, E. Ossanna), Fractals, doi: 10.1142/S0218348X18500718.

Papers Submitted for Publication

  • The Wild, Elusive Singularities of the T-fractal Surface (with C. C. Johnson), 2018.

Papers in Preparation/Progress

  • A fractal flat surface for the Koch snowflake fractal billiard (with C. C. Johnson), 2016.
  • Generating upward sweeps in population using the Turchin--Korotayev model (with R. E. Niemeyer), 2016.

Other Manuscripts not Intended for Publication

  • Families of periodic orbits of the Koch snowflake billiard (with M. L. Lapidus), preprint (2011), 63 typed pages. [E-print: arXiv:math.DS:1105.0737v1, 2011.] (Pieces of this were taken out and put elsewhere.)

Notes and Videos

My slides used in conjunction with R. Larson's "Precalculus: Real Mathematics, Real People 7e"
Topics pertaining to Set Theory and Mathematical Foundations
  • A brief review of functions and associated terminology (using Explain Everything) MP4 Format
  • A brief review of induction (using Explain Everything) MP4 Format
  • A brief review of logical quantifiers and their use (using Explain Everything) MP4 Format
  • A brief review of relations and equivalence relations (using Explain Everything) MP4 Format
  • A brief review of truth tables (using Explain Everything) MP4 Format
Topics pertaining to Calculus I
These were written in my own voice and constitute either supplemental material for a Calculus I course or specific notes for such a course.
Topics pertaining to Calculus II (coming soon)

Courses & Mentoring

Simply put, I believe in challenging my students and making it known that I care about their success and academic well-being. My interest in mentoring is rooted in a strong belief that it is not enough to teach, but one must help others along their path.

In the fall of 2018, I will be teaching Calculus II and Precalculus.

Calculus II

Integration techniques, applications of integration, improper integrals, infinite series and calculus using polar and parametric curves.


Exponential and logarithmic functions, trigonometric functions, principles of trigonometry and geometry, polar coordinates/equations and parametric equations


While an MCTP Postdoctoral Fellow at the University of New Mexico, I organized various discussions for graduate students. Slides from two of this discussions are below. I continue to organize seminars and information sessions for students and recent hires in an attempt to pass on my own knowledge.

Texts that I've found useful: suggested texts

Student Projects

I encourage interested students to contact me about doing either an undergraduate honors thesis or an independent research project. Below are some topics that can be modified or expanded upon to suit a student's interest and abilities.

The Koch snowflake fractal billiard table is a fractal billiard table with nowhere differentiable boundary. A priori, reflection at any of the points is not well-defined. Building on previous works, you will be asked to simulate orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

The T-fractal billiard table is a somewhat nicer fractal billiard table in that a fair amount of the boundary of the billiard table yields a well-defined tangent. You will be asked to simulate the orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

For those more interested in the broader subject of dynamics, one can also compare and contrast the notions of ergodic, weak mixing and strong mixing. This project is best suited for an undergraduate student finished with advanced calculus and interested in learning about measure theory. In preparing the student for the senior thesis, one will first be asked to read and understand the proof of the Poincare Recurrence Theorem and the definitions involved.

© 2018 Jennifer Kwan