Classes

### Mathematics

### MATH 103: The Nature of Math - Games and Puzzles

Topics such as geometry, computing, algebra, number theory, history of mathematics, logic, probability, statistics, modeling and problem solving intended to give students insight into what mathematics is, what it attempts to accomplish and how mathematicians think.

Each professor in the mathematics department teaches this course differently, with different materials. This Math 103 section is a hands-on discovery course with somewhat odd materials: the card game SET; a series of handouts and worksheets; the Japanese pencil puzzle book PENPAMIX; and some knot-making materials.

### MATH 103: The Nature of Math - Origami

Topics such as geometry, computing, algebra, number theory, history of mathematics, logic, probability, statistics, modeling and problem solving intended to give students insight into what mathematics is, what it attempts to accomplish and how mathematicians think.

In this class we will use origami as the vehicle for exploring mathematical topics such as divisibility, the Euler characteristic, the Mandelbrot set, Platonic solids, semi-regular polyhedra, Buckyballs, Menger sponges, and graph theory.

### MATH 107: Fundamentals of Mathematics

Sets, logic, numeration systems, development of real numbers, number operations, number theory, geometry, measurement, algebra, functions, probability and data analysis. Sequence is required for early childhood, elementary, or middle school teacher licensure.

### MATH 199: Algebra/Precalculus Gateway

Fall 2012, Spring 2013, Fall 2013, Spring 2014

Review of fundamental mathematics required to be successful in Calculus, including graphs of functions, factoring, simplifying, solving equations and inequalities, and exponential/logarithmic/trigonometric functions. Self-paced study with required proctored tests.

### MATH 231: Calculus with Functions I

Fall 2001, Fall 2002, Spring 2003, Fall 2003, Fall 2004, Fall 2005, Fall 2008, Spring 2011, Fall 2013, Fall 2016, Fall 2018

First semester of a sequence that combines first-semester calculus with algebra and trigonometry. The sequence is designed for students whose pre-calculus skills are not strong enough for the standard STEM Calculus sequence. Calculus material includes limits and derivatives of algebraic functions and their applications.

### MATH 232: Calculus with Functions II

Spring 2002, Fall 2002, Spring 2003, Spring 2004, Spring 2009, Fall 2011, Spring 2012, Spring 2014, Spring 2017, Spring 2018, Spring 2019

Second semester of a sequence that combines first-semester calculus with algebra and trigonometry. The sequence is designed for students whose pre-calculus skills are not strong enough for the standard STEM Calculus sequence. Calculus topics include limits and derivatives of transcendental functions, the theory of integration and basic integration techniques.

### MATH 235: Calculus I

Fall 2000, Fall 2010, Fall 2012, Spring 2013

Differential and integral calculus of functions of one variable.

### MATH 236: Calculus II

Spring 2001, Summer 2002, Spring 2011

Differential and integral calculus of functions of one variable. Sequences and infinite series.

### MATH 245: Introduction to Proof and Discrete Mathematics

Fall 2005, Spring 2006, Fall 2009, Fall 2012, Fall 2018

Logic, set theory, relations and functions, mathematical induction and equivalent forms, recurrence relations, counting techniques. Introduction to proof and proof techniques.

### MATH 353: Graph Theory

Graphs and their applications. Possible topics include trees, Euler paths and Hamiltonian circuits, planar graphs, digraphs, adjacency matrices, connectivity and coloring problems.

### MATH 430: Abstract Algebra I

Spring 2004, Fall 2004, Spring 2005

An introduction to groups, rings and fields.

### MATH 434: Advanced Linear Algebra

A proof-based linear algebra course covering such topics as vector spaces, linear transformations and matrices, eigenvalues and eigenvectors, inner product spaces, and canonical forms.

### MATH 435: Algebraic Topology

Metric spaces, limits, continuous maps and homeomorphisms, connectedness, compact topological spaces and applications. This particular class also included topics such as the fundamental group, classification of surfaces, and basic knot theory.

### Mentored Mathematical Research

### MATH 497: Advanced Undergrad Research - Knot Theory

Fall 2012, Spring 2013

Students pursue advanced research in a selected area of mathematics and/or statistics. Student must make arrangements with a supervising instructor prior to registration. This course centered on an active research project concerning determinants of spiral knots.

### MATH 499: Honors Thesis - Multiple Gerechte Designs

Spring 2006, Fall 2006, and Spring 2007

Three-semester sequence fulfilling the planning, research, and writing phases of the JMU Undergraduate Honors Thesis program. This particular course centered on new research concerning Shidoku configurations and minimal clue sets.

### MATH 499: Honors Thesis - Spiral Knots

Spring 2014, Fall 2014, Spring 2015

Three-semester sequence fulfilling the planning, research, and writing phases of the JMU Undergraduate Honors Thesis program. This particular course centered on new research concerning determinants of spiral knots.

### JMU M^3 NREUP - Mancala/Tchoukaillon

Summer 2010 Consultant, with Roger Thelwell and Anthony Tongen

The focus of the research project is Mancala, an ancient family of board games popular in Africa and Asia. While there are many possible rule variants, this ‘sowing’ type game is based on moving stone seeds from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn. Surprisingly, there has been little published mathematical research of this very interesting game. John Conway developed his own variant, ‘Sowing,’ which led to some simple mathematical language and structure which could be further developed. The primary research question for this project is “Is there an optimal strategy, against which no other competing strategy can win.” Mancala has been played for more than ten thousand years, suggesting that no obvious optimal strategy exists.

### JMU Internal Math/Stat REU - Knot Determinant Patterns

Summer 2012

Open exploration of determinant patterns for spiral knots, using the Online Encyclopedia of Integer Sequences.

### JMU NSF Mathematics REU - Pretzel Knot Colorings

Two Fox m-colorings of a knot or link K are said to be equivalent if they differ only by a permutation of colors. The set of equivalence classes of m-colorings under this relation is the set Cm(K) of Fox m-coloring classes of K. We develop a combinatorical formula for |Cm(K)| for any knot or link K that depends only on the m-nullity of K. As a practical application, we determine the m-nullity, and therefore the value of |Cm(P(p,q,r))|, for any any (p, q, r) pretzel link P(p,q,r). Resulted in published paper Counting m-coloring classes of knots and links.

### JMU NSF Mathematics REU - Torus Knot Colorings

Summer 2004

We classify by elementary methods the p-colorability of torus knots, and prove that every p-colorable torus knot has exactly one nontrivial p-coloring class. As a consequence, we note that the two-fold branched cyclic cover of a torus knot complement has cyclic first homology group. Resulted in published paper p-Coloring Classes of Torus Knots.

### JMU NSF Mathematics REU - Spiral Knots

Summer 2007, with Len Van Wyk

Spiral knots are a generalization of torus knots we define by a certain periodic closed braid representation. For spiral knots with prime power period, we calculate their genus, bound their crossing number, and bound their m-alternating excess. Resulted in published paper Spiral Knots.

### JMU NSF Mathematics REU - Mancala/Tchoukaillon

Summer 2011, with Anthony Tongen

We introduce a new matrix tool for the sowing game Tchoukaillon that enables us to non-iteratively construct an explicit bijection between board vectors and move vectors. This allows us to provide much simpler proofs than currently appear in the literature for two key theorems, as well as a non-iterative method for constructing move vectors. We also explore extensions of our results to Tchoukaillon variants that involve wrapping and chaining. Resulted in published paper Mancala Matrices.

### JMU NSF Mathematics REU - Singular Knots

Summer 2013, with Len Van Wyk

An open-ended experimental exploration of singular knots and their classification. Specifically, how can we generate a complete enumeration of 1-singular and 2-singular knots?

### JMU NSF Mathematics REU - Best Choice Games

Summer 2018, with Brant Jones

The game of best choice, also known as the secretary problem, is a model for sequential decision making with many variations in the literature. Notably, the classical setup assumes that the sequence of candidate rankings is uniformly distributed over time and that there is no expense associated with the candidate interviews. Here, we weight each ranking permutation according to the position of the best candidate in order to model costs incurred from conducting interviews with candidates that are ultimately not hired. We compare our weighted model with the classical (uniform) model via a limiting process. It turns out that imposing even infinitesimal costs on the interviews results in a probability of success that is about 28%, as opposed to 1/e (about 37%) in the classical case. Resulted in the published paper Opportunity costs in the game of best choice.

### 3D Printing and Design

### MATH 103: The Nature of Math -- 3D Printing

Topics such as geometry, computing, algebra, number theory, history of mathematics, logic, probability, statistics, modeling and problem solving intended to give students insight into what mathematics is, what it attempts to accomplish and how mathematicians think.

Our explorations of mathematics in this course will be supported by 3D printing and design in the JMU 3SPACE Classroom. This will be a highly digital course and we will be using the internet as our primary textbook. We will also make use of online collaboration and publication tools, and use 3D design software and coding to create and explore mathematical objects. Main topics will include fractals and the fourth dimension, but we may also investigate puzzles, knots, polyhedra, graph theory, computational geometry, topology, squircles, splines, or whatever we find interesting.

### ISCI 104: Introduction to 3D Printing

Fall 2013, Spring 2014, Fall 2017, Spring 2019

A study of topics selected to allow students to participate in mathematical and scientific problem solving approaches to knowledge.

Our explorations of problem solving and iteration in this course will be supported by 3D printing and design in the JMU 3SPACE Classroom. This will be a highly digital course and we will be using the internet as our primary textbook. We will use both online and locally installed 3D design software and coding programs, and make use of online collaboration and publication tools. The main objective of this class is to put you in a hands-on environment where you can apply modern problem solving approaches such as design thinking, project iteration, and independent learning.

### MATH 167: Independent Study - Mathematical 3D Fabrication

Topics or projects in mathematics which are of interest to the lower-division student. In this particular course, students will explore new 3D printing technologies at IVS, Engineering, and the new Math MakerLab, and learn how to create 3D-printable mathematical forms using 3D design software.

### UNST 390: Special Studies - Representing the World in 3D

Designed to give students an opportunity to complete independent study and/or research under faculty supervision in university studies. This semester’s research projects were chosen by students and were electoral maps, GIS data, facial recognition, and tactile floorplans, designed and 3D printed in the JMU 3SPACE classroom.

### CIT 393: Faculty Sandbox - Introduction to 3D Printing

A multi-week exploratory workshop for faculty interested in getting started with the 3D printing technologies available through JMU 3SPACE and the Library Makeries.

### CIT 393: Faculty Sandbox - Introduction to 3D Scanning

A multi-session exploration of current 3D scanning technologies including scanners, photogrammetry, apps, and mesh manipulation software.

### MSSE 570M: Mathematics Methods - Desktop Fabrication

Research findings about teaching in the content area will be used to identify the most effective instructional strategies for teaching that content to students in grades 9-12. Emphasis will be on developing plans for employing the strategies and making appropriate instructional decisions based on instructional goals, the learner, and available resources. This particular class will also introduce students to Desktop Fabrication tools such as 3D printing, 3D design, and Lasercutting.

### STAR Student Program - Multimaterial 3D Printing

Fall 2013, Spring 2014

Explorations of cutting-edge technology and beta-level hardware for multimaterial 3D printing.

### ISFW: Sketchup and Virtual Reality

Explore using Sketchup in JMU 3SPACE to design 3D models based on historical blueprints, then experience your structures in Virtual Reality at the Carrier Makery.

### ISFW: Carvey and ShopBot

Did you know that desktop fabrication is more than 3D printing? In the JMU Library Makeries we have cababilities to use CNC machines like the Carvey and ShopBot to design and carve into wood and other materials. Come learn with us how to design and create projects using these tools!

### ISFW: 3D Printing Beyond Plastic

Have you wondered if it was possible to 3D print objects in a material other than plastic? In this session we will explore 3D printing using silicone, food, and resin. Join us to explore the benefits of being able to print objects in alternative materials.

### ISFW: Design Beyond Tinkercad

Summer 2018 and Fall 2018

Already know some 3D design basics? Expand your toolbox with Scribble, Fusion 360, and OpenSCAD and learn three ways to create 3D-printable spiralized vase designs.

### ISFW: Design It, Print It, Share It

Spring 2018 and Fall 2018

Join us in JMU’s 3D printing classroom 3SPACE to take your 3D design skills to the next level. We will use Fusion 360 to design a ring from scratch that we will have 3D printed in Metal at Shapeways. 3D design and printing experience not required.

### ISFW: 3D Design in Color

Summer 2018 and Fall 2018

Join us to learn how to color your 3D prints to print on the da Vinci Full Color 3D Printer! This desktop 3D printer uses CMKY inkjet technology and color absorbing PLA to create full color 3D prints. We will explore how to design in color utilizing TinkerCad, how to blend colors using Meshmixer as well as how to design in XYZ Maker. We will schedule times for everyone to print their designs in full color.

### Mentored Design Research

### MATH 297: Intro Research - Knot Theory and 3D Printing

Students pursue research in a selected area of mathematics and/or statistics. Topics include knot theory, in particular knot classification and special knot types, and 3D modeling skills required to design and print physical models of various knot types.

### MATH 297: Intro Research - Homotopy and 3D Printing

Spring 2017

Students pursue research in a selected area of mathematics and/or statistics. Topics include knot classification, homotopies, and graph theory, as well as 3D modeling skills using OpenSCAD and Fusion 360 to create relevant 3D-printable models.

### MATH 297: Intro Research - Spatial Graphs and 3D Printing

Fall 2017

Students pursue research in a selected area of mathematics and/or statistics. Topics in graph theory include generalized Petersen graphs, intrinsically linked graphs, intrinsically knotted graphs, and representations of non-planar graphs. Students will also learn 3D modeling skills in order to create new physical models of spatial graphs that exhibit symmetry or particular efficiency.

### JMU Internal Math/Stat REU - 3D Printing Laboratory

Two students will lead the construction and administration of the new Math/Stat MakerLab, and along the way explore new software and hardware solutions to support this new technology.

### MakerLab Research Project - 3D Modelling Rolling Knots

By changing parameters in equation for tritangentless trefoils we get different tritangentless conformations. Some of these conformations roll better than others; which one is optimal? And can we prove that one is optimal? We will experiment with 3D printed conformations of tritangentless trefoils, and also try to use calculus, linear algebra, and geometry to arrive at an optimal solution.

### MakerLab Research Project - 3D Graph Conformations

The model shown in the image is a 3D spatial representation of the complete bipartite graph K4,4. When drawn in the plane, this graph necessarily intersects itself, and it has quite a few overlapping edges. When working in three dimensions, we can arrange the vertices so that the edges do not intersect. Determining the “best” way to do this is an aesthetic and artistic challenge.

### MakerLab Research Project - Printed Rocket Exhaust Trench

One of the ways in which engineers reduce rocket launch noise is through the use of a trench which channels the rocket exhaust away. The shape of this trench is very important. In particular, Dr. Lubert has been working with Orbital ATK (now Northrop Grumman) – one of the two commercial resuppliers of the International Space Station, together with SpaceX – who launch their Antares rocket from NASA’s Wallops Flight Facility on the Eastern Shore of Virginia. We will use engineering drawings of the Wallops trench to create a 3D printed version of it, so that we can more easily visualise and understand the flow processes that take place inside it.

### Ars Geometrica Seminar - Pushing 3D Printing Boundaries

Explorations of advanced 3D printing topics such as mid-print error detection, mathematical visualization, and algorithmic infill generation.

### Ars Geometrica Seminar - 3D Graphs and Discrete Geometry

Explorations of advanced 3D printing topics, including representations of generalized Petersen graphs, large-scale scanning and printing, and the creation of an inversive geometry library.