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.
Braids, Cables, and Cells I: An Interesting Intersection of Mathematics, Comp...Joshua Holden
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.
Wagner online have all your PC Parts covered for PC Gamers/Enthusiast to PC and Computer repairs and provide an extensive range of computer maintenance parts and accessories such as PC SATA, SAS, eSATA HDD cables, HDD power adapters and leads. High Quality and great feeling soft touch keyboards. Connect to your broadband internet with our ADSL routers and fibre optic cables. Buy branded computer and laptop accessories at significantly low price with a world class service from us.
http://www.wagneronline.com.au/data-computer/90709/ct/
Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, ...Joshua Holden
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.
Stitching Graphs and Painting Mazes: Problems in Generalizations of Eulerian ...Joshua Holden
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.
I made this slide about fractals for one of my math course's presentation. I chose fractal as it has this beautiful pattern and different kinds of variation.
ps- There might be some mistake. Your corrections will be appreciated.
Braids, Cables, and Cells I: An Interesting Intersection of Mathematics, Comp...Joshua Holden
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.
Wagner online have all your PC Parts covered for PC Gamers/Enthusiast to PC and Computer repairs and provide an extensive range of computer maintenance parts and accessories such as PC SATA, SAS, eSATA HDD cables, HDD power adapters and leads. High Quality and great feeling soft touch keyboards. Connect to your broadband internet with our ADSL routers and fibre optic cables. Buy branded computer and laptop accessories at significantly low price with a world class service from us.
http://www.wagneronline.com.au/data-computer/90709/ct/
Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, ...Joshua Holden
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.
Stitching Graphs and Painting Mazes: Problems in Generalizations of Eulerian ...Joshua Holden
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.
I made this slide about fractals for one of my math course's presentation. I chose fractal as it has this beautiful pattern and different kinds of variation.
ps- There might be some mistake. Your corrections will be appreciated.
The Interplay Between Art and Math: Lessons from a STEM-based Art and Math co...Joshua Holden
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.
Between the Two Cultures: Teaching Math and Art to Engineers (and Scientists ...Joshua Holden
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.
Teaching the Group Theory of Permutation CiphersJoshua Holden
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.
Granny’s Not So Square, After All: Hyperbolic Tilings with Truly Hyperbolic C...Joshua Holden
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.
A Good Hash Function is Hard to Find, and Vice VersaJoshua Holden
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.
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.
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 and algorithms for maze traversalsJoshua Holden
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.
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.
Understanding the Magic: Teaching Cryptography with Just the Right Amount of ...Joshua Holden
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.
Modular Arithmetic and Trap Door CiphersJoshua Holden
Like other branches of mathematics, number theory has seen many surprising developments in recent years. One of the most surprising is the fact that number theory, long considered the most "useless" of any field of mathematics, has become vital to the development of modern codes and ciphers. As an example, the RSA cryptosystem, eveloped in the 1970's by Rivest, Shamir, and Adleman, uses some ideas that are very easy to understand. Yet, these ideas underlie large portions of both modern number theory and modern cryptography. We will explore these ideas, and show how they make RSA the first practical "trap door" cipher. This means that anyone can encode a message but only the recipient can decode it!
The Pohlig-Hellman Exponentiation Cipher as a Bridge Between Classical and Mo...Joshua Holden
The Pohlig-Hellman exponentiation cipher is a symmetric-key cipher that uses some of the same mathematical operations as the better-known RSA and Diffie-Hellman public-key cryptosystems. First published in 1978, the Pohlig-Hellman cipher was never of practical importance due to its slow speed compared to ciphers such as DES and AES. The theoretical importance of the Pohlig-Hellman cipher comes from the fact that it relies on the Discrete Logarithm Problem for its resistance against known plain text attacks, as does RSA and several other modern cryptosystems. For this reason, the Pohlig-Hellman systemcan play a very important role pedagogically, since it also shares many features in common with classical ciphers such as shift ciphers and Hill ciphers. Thus, it allows the instructor to introduce the important concepts of the discrete logarithm and known plain text attacks separately from the more conceptually difficult idea of public-key cryptography.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Interplay Between Art and Math: Lessons from a STEM-based Art and Math co...Joshua Holden
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.
Between the Two Cultures: Teaching Math and Art to Engineers (and Scientists ...Joshua Holden
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.
Teaching the Group Theory of Permutation CiphersJoshua Holden
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.
Granny’s Not So Square, After All: Hyperbolic Tilings with Truly Hyperbolic C...Joshua Holden
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.
A Good Hash Function is Hard to Find, and Vice VersaJoshua Holden
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.
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.
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 and algorithms for maze traversalsJoshua Holden
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.
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.
Understanding the Magic: Teaching Cryptography with Just the Right Amount of ...Joshua Holden
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.
Modular Arithmetic and Trap Door CiphersJoshua Holden
Like other branches of mathematics, number theory has seen many surprising developments in recent years. One of the most surprising is the fact that number theory, long considered the most "useless" of any field of mathematics, has become vital to the development of modern codes and ciphers. As an example, the RSA cryptosystem, eveloped in the 1970's by Rivest, Shamir, and Adleman, uses some ideas that are very easy to understand. Yet, these ideas underlie large portions of both modern number theory and modern cryptography. We will explore these ideas, and show how they make RSA the first practical "trap door" cipher. This means that anyone can encode a message but only the recipient can decode it!
The Pohlig-Hellman Exponentiation Cipher as a Bridge Between Classical and Mo...Joshua Holden
The Pohlig-Hellman exponentiation cipher is a symmetric-key cipher that uses some of the same mathematical operations as the better-known RSA and Diffie-Hellman public-key cryptosystems. First published in 1978, the Pohlig-Hellman cipher was never of practical importance due to its slow speed compared to ciphers such as DES and AES. The theoretical importance of the Pohlig-Hellman cipher comes from the fact that it relies on the Discrete Logarithm Problem for its resistance against known plain text attacks, as does RSA and several other modern cryptosystems. For this reason, the Pohlig-Hellman systemcan play a very important role pedagogically, since it also shares many features in common with classical ciphers such as shift ciphers and Hill ciphers. Thus, it allows the instructor to introduce the important concepts of the discrete logarithm and known plain text attacks separately from the more conceptually difficult idea of public-key cryptography.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Braids, Cables, and Cells II: Representing Art and Craft with Mathematics and Computer Science
1. Braids, Cables, and Cells
Representing Art and Craft with Mathematics and Computer
Science
Joshua Holden
Rose-Hulman Institute of Technology
http://www.rose-hulman.edu/~holden
1 / 43
2. “Knotwork” has been used in visual arts for many
centuries.
Left: Detail from Roman mosaic at Woodchester, c. 325 CE
Right: Detail from the “Book of Kells”, c. 800 CE
2 / 43
3. “Knotwork” has been used in visual arts for many
centuries.
Left: by A. Reed Mihaloew, Right: by Christian Mercat
3 / 43
4. In knitting, raised knot-like designs are known as
“cables”.
Left: Design by Meredith Morioka, knitted by Lana Holden
Right: Design by Julie Levy, knitted by Lana Holden
4 / 43
5. “Cables” can also be done in crochet.
Both: Designed and crocheted by Lisa Naskrent
5 / 43
6. A somewhat similar effect is given by “traveling
eyelets” in knitted lace.
From Barbara Walker’s Charted Knitting Designs
6 / 43
7. And of course, there are many actual weaving
patterns.
Left: 2/2 twill weave, woven by Sarah, a.k.a. Aranel
Right: “Noonday Sun” pattern, woven by Peggy Brennan
(Cherokee Nation)
7 / 43
8. Vandermonde was interested in the mathematical
study of knots and braids.
From “Remarques sur les problèmes de situation”, 1771
8 / 43
9. So was Gauss.
From Page 283 of Gauss’s Handbuch 7, c. 1825?
9 / 43
10. Today, braids are studied from the perspectives of
topology and group theory.
Two equal braids (Wikipedia)
Two braids which are the same except for “pulling the
strands” are considered equal.
All strands are required to move from bottom to top.
10 / 43
11. You can make braids into a group by “multiplying”
them.
× =
Multiplying braids (Wikipedia)
You multiply two braids by stacking them and then
simplifying.
11 / 43
12. A Cellular Automaton is a mathematical construct
modeling a system evolving in time.
Finite number of cells in a regular grid
Finite number of states that a cell can be in
Each cell has a well-defined finite neighborhood
Time moves in discrete steps
State of each cell at time t is determined by the states of
its neighbors at time t − 1
Each cell uses the same rule
The “von Neumann neighborhood”
(Wikibooks)
12 / 43
13. “The Game of Life” is an example you might know.
Invented by John Conway
Grid is two-dimensional
Two states, “live” and “dead”
Neighborhood is the eight cells which are directly
horizontally, vertically, or diagonally adjacent
Any live cell with two or three live neighbors stays live.
Any other live cell dies.
Any dead cell with exactly three live neighbors becomes a
live cell.
Any other dead cell stays dead.
(Figures by Paul Callahan, from www.math.com)
13 / 43
21. Another well-known class of automata are the
“Elementary” Cellular Automata.
Popularized by Stephen Wolfram (A New Kind of Science)
Grid is one-dimensional
Two states, “white” and “black”
Neighborhood includes self and one cell on each side
“Rule 30” (Mathworld)
15 / 43
22. Example: “Rule 90”
Second dimension is used for “time”
´
Produces the Sierpinski triangle fractal
An “additive” rule
(Mathworld)
16 / 43
23. Cellular automata can exhibit aperiodic behavior.
Conjecture (Wolfram, 1984)
The sequence of colors produced by the cell at the center of
Rule 30 is aperiodic.
This sequence is used by the pseudorandom number
generator in the program Mathematica.
The center and right portions of Rule 30 appear to have
some of the characteristics of “chaotic” systems.
Theorem (Jen, 1986 and 1990)
(a) At most one cell of Rule 30 produces a periodic sequence
of colors.
(b) The sequence of color pairs produced by any two adjacent
cells of Rule 30 is aperiodic.
17 / 43
25. Cellular automata are also “computationally universal”.
Theorem (Cook, 1994+)
Rule 110 can be used to simulate any Turing machine.
This is important because of the widely accepted:
Church-Turing Thesis
Anything that can be computed by an algorithm can be
computed by some Turing machine.
And for complexity geeks:
Theorem (Neary and Woods, 2006)
Rule 110 can be used to simulate any polynomial time Turing
machine in polynomial time. (I.e., it is “P-complete”.)
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26. Rule 110 on a Single Cell Input
(Mathworld)
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27. How is this possible?
1. Use Rule 110 to simulate a “cyclic tag system”.
A cyclic tag system has:
A data string
A cyclic list of “production rules”
To perform a computation:
If the first data symbol is 1, add the production rule to the
end of the data string. If the first data symbol is 0 do
nothing.
Delete the first data symbol.
Move to the next production rule.
Repeat until the data string is empty.
2. Show that any Turing machine can be simulated by a cyclic
tag system.
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28. Cyclic Tag System Example
Production rules Data string
010 11001
000 1001010
1111 001010000
010 01010000
000 1010000
1111 010000000
010 10000000
.
. ..
. .
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29. To simulate a cyclic tag system with Rule 110, you
need:
a representation of the data string (stationary)
a representation of the production rules (left-moving)
“clock pulses” (right-moving)
(Wikipedia)
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31. CAs have been used in fiber arts before.
Left: Designed and crocheted by Jake Wildstrom
Right: Knitted by Pamela Upright, after Debbie New
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32. Each of our cells will store 4 bits of information in 8
states.
upright slanted
no strands
left only
right only
both
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33. The neighborhood will be a “brick wall” neighborhood.
(Time moves from bottom to top, like a knitting pattern.)
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34. The CA rule system can actually be thought of as four
simpler CAs. The first two just control whether strands
are present or not.
no left left
no right
right
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35. The CA rule system can actually be thought of as four
simpler CAs. The first two just control whether strands
are present or not.
no left left
no right
right
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36. The third CA controls whether the strands are upright
or diagonal, specified by a numbered rule.
“Turning Rule 39”
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37. And the fourth CA controls which strand is on top if the
strands cross, also specified by a numbered rule.
“Crossing Rule 39”
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38. There are several possible choices for what to do at
the edges of the grid.
Make the grid infinite?
Have a special kind of state for edge cells?
Make the grid cylindrical? (“Periodic boundary conditions”)
Reflect cells at the edges? (Where to put the axis?)
I have so far only implemented the cylindrical case.
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39. The rules can produce fractal patterns, . . .
Rules 68 and 0 33 / 43
41. . . . traditional braids, . . .
’
Left: Wikipedia, Right: Rules 333 and 39
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42. . . . slightly less traditional braids, . . .
Left: backstrapweaving.wordpress.com
Right: Rules 333 and 99
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43. . . . and other sorts of “cable” patterns.
Left: Rules 47 and 0, Right: Rules 201 and 39
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44. If all strands are present and only one rule is active,
previously known results on “elementary” CA’s apply.
Rules 68 and 0 give the same result as Wolfram’s Rule 90
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45. Since the width is finite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width m, no
repeat can be longer than m2m − 2m rows.
Proof.
After 2m rows, all of the strands
have returned to their original
positions. The only question is
which strand of each crossing
is on top.
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46. Since the width is finite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width m, no
repeat can be longer than m2m − 2m rows.
Proof.
If there are m crossings then
there are 2m possible
arrangements of the crossings
but only 2 different ways the
row can be shifted. So the
maximum repeat is the lcm of a
number ≤ 2m and a
number ≤ 2.
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47. Since the width is finite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width m, no
repeat can be longer than m2m − 2m rows.
Proof.
If there are m − 1 crossings,
then there are 2m−1 possible
arrangements but 2m different
shifts, so the maximum repeat
is the lcm of a number ≤ 2m−1
and a number ≤ 2m.
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48. Since the width is finite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width
m ≥ 2k , the maximum repeat is at least lcm(2k +1 , 2m) rows
long.
Proof.
Consider the starting row with
one single strand and m − 1
crossings. Crossing Rule 100
(which is additive) acts on this
with a repeat (modulo cyclic
shift) of 2k +1 if m > 2k . The
cyclic shift gives the 2m.
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49. Since the width is finite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width
m ≥ 2k , the maximum repeat is at least lcm(2k +1 , 2m) rows
long.
Remark
For m ≤ 5, this is sharp. For
m = 23, m crossings and
Crossing Rule 257 (which is
also additive) does better.
For large m, neither the upper
bound above nor this lower
bound seems especially likely
to be sharp.
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50. If only one or two strands are present then the
maximum length of a repeat can be determined.
Proposition
Assume only one or two strands are
present. For a given width m ≤ 5, the
maximum repeat is (2m)(2m + 1) rows long.
Proof.
This is achieved by Turning Rule 97 and two
strands.
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51. There is much future work to be done.
If all strands are present and both rules are active, then we
have two “elementary” CA’s where one can “overwrite” the
other.
The length of a maximum repeat in other cases is open.
What is the computational complexity of predicting things
that the CA might do?
More work can be done with different edge conditions.
Which braids can be represented? (In the sense of braid
groups)
Which rules are “reversible”?
Two-dimensional grids with time as the third dimension
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