This presentation describes the team-teaching of a course in mathematics and art. The goal of the class is to show students the interplay between art and math with a focus on having them make physical objects. Most math and art classes we have heard of focus primarily on introducing liberal arts students to mathematics. In a STEM context, we want to instead build on the mathematical knowledge that our students already have. We intend for students to use their existing mathematical (and perhaps artistic) knowledge to reinforce new artistic (and perhaps mathematical) experiences. Ideally, the knowledge and experience gained will increase their appreciation for both the beauty of mathematics and the importance of art.]]>

This presentation describes the team-teaching of a course in mathematics and art. The goal of the class is to show students the interplay between art and math with a focus on having them make physical objects. Most math and art classes we have heard of focus primarily on introducing liberal arts students to mathematics. In a STEM context, we want to instead build on the mathematical knowledge that our students already have. We intend for students to use their existing mathematical (and perhaps artistic) knowledge to reinforce new artistic (and perhaps mathematical) experiences. Ideally, the knowledge and experience gained will increase their appreciation for both the beauty of mathematics and the importance of art.]]>

C.P. Snow famously categorized modern intellectual life as being split between the culture of the sciences and the culture of the humanities, and said that solving the world's problems requires bringing these two cultures together. Math and art classes inherently try to do that. However, most math and art classes we have heard of focus primarily on introducing liberal arts students to mathematics. In the Spring Quarter of 2019 Soully Abas and I team-taught a different sort of course in mathematics and art at the Rose-Hulman Institute of Technology. Rose-Hulman is a private, undergraduate-focused institution, all of whose students major in engineering (predominantly), science, or mathematics. We wanted to build on the mathematical knowledge that our STEM students already have with a focus on having them make physical art objects. Soully is an art professor specializing in oil painting and printmaking. Josh is a math professor interested in the mathematics of fiber arts such as embroidery, knitting, crochet, and weaving. Our goal was for students to use their existing mathematical (and perhaps artistic) knowledge to reinforce new artistic (and perhaps mathematical) experiences. Ideally, the knowledge and experience gained will increase their appreciation for both the beauty of mathematics and the importance of art and help them prepare for productive lives solving the world's problems.]]>

C.P. Snow famously categorized modern intellectual life as being split between the culture of the sciences and the culture of the humanities, and said that solving the world's problems requires bringing these two cultures together. Math and art classes inherently try to do that. However, most math and art classes we have heard of focus primarily on introducing liberal arts students to mathematics. In the Spring Quarter of 2019 Soully Abas and I team-taught a different sort of course in mathematics and art at the Rose-Hulman Institute of Technology. Rose-Hulman is a private, undergraduate-focused institution, all of whose students major in engineering (predominantly), science, or mathematics. We wanted to build on the mathematical knowledge that our STEM students already have with a focus on having them make physical art objects. Soully is an art professor specializing in oil painting and printmaking. Josh is a math professor interested in the mathematics of fiber arts such as embroidery, knitting, crochet, and weaving. Our goal was for students to use their existing mathematical (and perhaps artistic) knowledge to reinforce new artistic (and perhaps mathematical) experiences. Ideally, the knowledge and experience gained will increase their appreciation for both the beauty of mathematics and the importance of art and help them prepare for productive lives solving the world's problems.]]>

ANTS XI, 2014.]]>

ANTS XI, 2014.]]>

One of the first topics often taught in an abstract algebra class is permutations, since they provide good examples of non-commutative finite groups which the students can manipulate and visualize. This visualization is often done through symmetry groups. For students who are less geometrically inclined, however, the use of permutation ciphers provides another good way of motivating permutations. They can easily be used to illustrate composition, non-commutativity, inverses, and the order of group elements, which are fundamental topics in group theory. We will give examples of how this can be done and suggest other courses besides abstract algebra in which this could also prove useful.]]>

One of the first topics often taught in an abstract algebra class is permutations, since they provide good examples of non-commutative finite groups which the students can manipulate and visualize. This visualization is often done through symmetry groups. For students who are less geometrically inclined, however, the use of permutation ciphers provides another good way of motivating permutations. They can easily be used to illustrate composition, non-commutativity, inverses, and the order of group elements, which are fundamental topics in group theory. We will give examples of how this can be done and suggest other courses besides abstract algebra in which this could also prove useful.]]>

Until now, most methods for making a hyperbolic plane from crochet or similar fabrics have fallen into one of two categories. In one type, the work starts from a point or line and expands in a sequence of increasingly long rows, creating a constant negative curvature. In the other, polygonal tiles are created out of a more or less Euclidean fabric and then attached in such a way that the final product approximates a hyperbolic plane on the large scale with an average negative curvature. On the small scale, however, the curvature of the fabric will be closer to zero near the center of the tiles and more negative near the vertices and edges, depending on the amount of stretch in the fabric. The goal of this project is to show how crochet can be used to create polygonal tiles which themselves have constant negative curvature and can therefore be joined into a large region of a hyperbolic plane without significant stretching. Formulas from hyperbolic trigonometry are used to show how, in theory, any regular tiling of the hyperbolic plane can be produced in this way.]]>

Until now, most methods for making a hyperbolic plane from crochet or similar fabrics have fallen into one of two categories. In one type, the work starts from a point or line and expands in a sequence of increasingly long rows, creating a constant negative curvature. In the other, polygonal tiles are created out of a more or less Euclidean fabric and then attached in such a way that the final product approximates a hyperbolic plane on the large scale with an average negative curvature. On the small scale, however, the curvature of the fabric will be closer to zero near the center of the tiles and more negative near the vertices and edges, depending on the amount of stretch in the fabric. The goal of this project is to show how crochet can be used to create polygonal tiles which themselves have constant negative curvature and can therefore be joined into a large region of a hyperbolic plane without significant stretching. Formulas from hyperbolic trigonometry are used to show how, in theory, any regular tiling of the hyperbolic plane can be produced in this way.]]>

An Eulerian walk traverses each edge of a graph exactly once. What happens if you want to traverse each edge of a graph exactly twice? If you want to cover the graph with "double-running stitch", then you need to traverse each edge twice but also put conditions on how many edges you traverse in-between. Then you could add conditions on whether you traverse the edges once in each direction or twice in the same direction. Which graphs can you still traverse? Classical algorithms for solving mazes give us some answers to these questions, but others are still open. ]]>

An Eulerian walk traverses each edge of a graph exactly once. What happens if you want to traverse each edge of a graph exactly twice? If you want to cover the graph with "double-running stitch", then you need to traverse each edge twice but also put conditions on how many edges you traverse in-between. Then you could add conditions on whether you traverse the edges once in each direction or twice in the same direction. Which graphs can you still traverse? Classical algorithms for solving mazes give us some answers to these questions, but others are still open. ]]>

With Nathan Lindle. ANTS VIII, 2008.]]>

With Nathan Lindle. ANTS VIII, 2008.]]>

With Daniel R. Cloutier. ANTS VII, 2006.]]>

With Daniel R. Cloutier. ANTS VII, 2006.]]>

ANTS X, 2012.]]>

ANTS X, 2012.]]>

The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and weaving. Previous work as been shown that rules for cellular automata can be written in order to produce depictions of braids. This talk will extend the previous system into a more flexible one which more realistically captures the behavior of strands in certain media, such as knitting. Some theorems about what can and cannot be represented with these cellular automata will be presented.]]>

The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and weaving. Previous work as been shown that rules for cellular automata can be written in order to produce depictions of braids. This talk will extend the previous system into a more flexible one which more realistically captures the behavior of strands in certain media, such as knitting. Some theorems about what can and cannot be represented with these cellular automata will be presented.]]>

Secure hash functions are the unsung heroes of modern cryptography. Introductory courses in cryptography often leave them out --- since they don't have a secret key, it is difficult to use hash functions by themselves for cryptography. In addition, most theoretical discussions of cryptographic systems can get by without mentioning them. However, for secure practical implementations of public-key ciphers, digital signatures, and many other systems they are indispensable. In this talk I will discuss the requirements for a secure hash function and relate my attempts to come up with a "toy" system which both reasonably secure and also suitable for students to work with by hand in a classroom setting.]]>

Secure hash functions are the unsung heroes of modern cryptography. Introductory courses in cryptography often leave them out --- since they don't have a secret key, it is difficult to use hash functions by themselves for cryptography. In addition, most theoretical discussions of cryptographic systems can get by without mentioning them. However, for secure practical implementations of public-key ciphers, digital signatures, and many other systems they are indispensable. In this talk I will discuss the requirements for a secure hash function and relate my attempts to come up with a "toy" system which both reasonably secure and also suitable for students to work with by hand in a classroom setting.]]>

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The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including knitting and crochet, where braids are called “cables”. We will view some examples of braids and their mathematical representations in these media.]]>

The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including knitting and crochet, where braids are called “cables”. We will view some examples of braids and their mathematical representations in these media.]]>

Integrating the Use of Maple with the Collaborative Use of Wireless Tablet PCs. Workshop on the Impact of Pen-Based Technology on Education (poster session), Purdue University, October 16, 2008. ]]>

Integrating the Use of Maple with the Collaborative Use of Wireless Tablet PCs. Workshop on the Impact of Pen-Based Technology on Education (poster session), Purdue University, October 16, 2008. ]]>

The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macramé, and we will touch on some of these.]]>

The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macramé, and we will touch on some of these.]]>

Many people don't realize that what we now call "algorithm design" actually dates back to the ancient Greeks! Of course, if you think about it, there's always the "Euclidean Algorithm". A more dubious example might be Theseus's use of a ball of string to solve the "Labyrinth Problem". (Google "Theseus, Labyrinth, string".) Solutions to this problem got a lot less dubious after graph theory was invited, since a graph turns out to be a good way of representing a maze mathematically. We will examine the classical solutions to this problem, and then throw in a twist --- a Twisted Painting Machine that puts restrictions on which paths we can take to explore the maze. Applications to sewing may also appear, depending on the presence of audience interest and string. ]]>

Many people don't realize that what we now call "algorithm design" actually dates back to the ancient Greeks! Of course, if you think about it, there's always the "Euclidean Algorithm". A more dubious example might be Theseus's use of a ball of string to solve the "Labyrinth Problem". (Google "Theseus, Labyrinth, string".) Solutions to this problem got a lot less dubious after graph theory was invited, since a graph turns out to be a good way of representing a maze mathematically. We will examine the classical solutions to this problem, and then throw in a twist --- a Twisted Painting Machine that puts restrictions on which paths we can take to explore the maze. Applications to sewing may also appear, depending on the presence of audience interest and string. ]]>

Blackwork embroidery is a needlework technique often associated with Elizabethan England. The patterns used in blackwork generally are traversed in a way that can be described by classical algorithms in graph theory, such as those used in “The Labyrinth Problem”. We investigate which of these maze-traversing algorithms can also be used to traverse blackwork patterns.]]>

Blackwork embroidery is a needlework technique often associated with Elizabethan England. The patterns used in blackwork generally are traversed in a way that can be described by classical algorithms in graph theory, such as those used in “The Labyrinth Problem”. We investigate which of these maze-traversing algorithms can also be used to traverse blackwork patterns.]]>

Blackwork embroidery, also known as "Spanish stitch'' or "Holbein stitch'', is a needlework technique often associated with Elizabethan England. The patterns used in Blackwork generally are strongly geometric, and are traversed in a way that can be easily described by graph theory. We will characterize these graph traversals and present some algorithms that may be of practical use to needleworkers as well as theoretical interest.]]>

Blackwork embroidery, also known as "Spanish stitch'' or "Holbein stitch'', is a needlework technique often associated with Elizabethan England. The patterns used in Blackwork generally are strongly geometric, and are traversed in a way that can be easily described by graph theory. We will characterize these graph traversals and present some algorithms that may be of practical use to needleworkers as well as theoretical interest.]]>

Cryptography and Computer Security for Undergraduates (Panel, SIGCSE 2004) I have two goals in teaching cryptography to computer science students: to use cryptography as a cool way of introducing important areas of mathematics and computer science theory and to educate students in something that may be necessary for them to know in the future. For the past two years I have co-taught a course in cryptography at the Rose-Hulman Institute of Technology with David Mutchler, a colleague from the Computer Science department. The course is cross-listed in both the Computer Science and Mathematics departments, but most of the students are CS majors. The published prerequisites are one quarter of discrete mathematics and two quarters of computer science. ]]>

Cryptography and Computer Security for Undergraduates (Panel, SIGCSE 2004) I have two goals in teaching cryptography to computer science students: to use cryptography as a cool way of introducing important areas of mathematics and computer science theory and to educate students in something that may be necessary for them to know in the future. For the past two years I have co-taught a course in cryptography at the Rose-Hulman Institute of Technology with David Mutchler, a colleague from the Computer Science department. The course is cross-listed in both the Computer Science and Mathematics departments, but most of the students are CS majors. The published prerequisites are one quarter of discrete mathematics and two quarters of computer science. ]]>

Cryptography is a field which has recently attracted a great deal of attention from both students and teachers of mathematics and of computer science. Students of computer science see cryptography as something which is not only "cool" but may be necessary for them to know in their future careers. Many of them do not realize, however, just how much mathematics they need to know in order to understand the algorithms which lie at the heart of modern cryptography.]]>

Cryptography is a field which has recently attracted a great deal of attention from both students and teachers of mathematics and of computer science. Students of computer science see cryptography as something which is not only "cool" but may be necessary for them to know in their future careers. Many of them do not realize, however, just how much mathematics they need to know in order to understand the algorithms which lie at the heart of modern cryptography.]]>

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