DOCSLIB.ORG

The Mathematics of Juggling

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

The Mathematics of Juggling Stephen Hardy March 25, 2016 Stephen Hardy: The Mathematics of Juggling 1 Introduction Goals of the Talk I Learn a little about juggling. I Learn a little mathematics. I Have some fun. Stephen Hardy: The Mathematics of Juggling 2 Introduction Reference Most of the material for this talk was gleaned from Burkard Polster's great little book The Mathematics of Juggling: Stephen Hardy: The Mathematics of Juggling 3 What is Juggling? What is Juggling? Juggling takes many forms... Stephen Hardy: The Mathematics of Juggling 4 What is Juggling? Examples Jonathan Alexander http://adoreministries.com/, Kevin Rivoli http://www.syracuse.com/ Stephen Hardy: The Mathematics of Juggling 5 What is Juggling? Diablo Tony Frebourg - http://historicaljugglingprops.com/ Stephen Hardy: The Mathematics of Juggling 6 What is Juggling? Flair Bartending Ami Shroff - https://www.facebook.com/amibehramshroff Stephen Hardy: The Mathematics of Juggling 7 What is Juggling? Contact Juggling Michael Moschen - http://www.michaelmoschen.com/ Stephen Hardy: The Mathematics of Juggling 8 What is Juggling? Bounce Juggling Dan Menendez - http://pianojuggler.com/ Stephen Hardy: The Mathematics of Juggling 9 What is Juggling? Juggling is... I Juggling is manipulating more objects than hands you are using. Bob Whitcomb, http://historicaljugglingprops.com/ I We are interested in toss juggling. Stephen Hardy: The Mathematics of Juggling 10 What is Juggling? Toss Juggling Rings Anthony Gatto - http://www.anthonygatto.com/ Stephen Hardy: The Mathematics of Juggling 11 What is Juggling? Toss Juggling Clubs Jason Garfield - http://jasongarfield.com/ Stephen Hardy: The Mathematics of Juggling 12 What is Juggling? Toss Juggling Knives Edward Gosling - http://chivaree.co.uk/ Stephen Hardy: The Mathematics of Juggling 13 What is Juggling? Toss Juggling (Passing) Cirque Dreams Jungle Fantasy - https://en.wikipedia.org/wiki/Juggling_ring Stephen Hardy: The Mathematics of Juggling 14 What is Juggling? Toss Juggling We will focus on a single person toss juggling with two hands. Jason Garfield - http://jasongarfield.com/ Stephen Hardy: The Mathematics of Juggling 15 Simplifications Distractions We are only interested in the various patterns of throws and catches. We will ignore: I Showmanship (costumes, unicycles, balancing...) I Props (kerchiefs, balls, rings, clubs, knives, bowling balls, torches, chainsaws...) I Variations (inside, outside, piston two-in-one-hand...) I Flourishes (pirouettes, throwing under the leg or behind the back...) Stephen Hardy: The Mathematics of Juggling 16 Simplifications Simplifying Assumptions I The balls are juggled at a constant rate, throws alternating between right and left hands. I Patterns are periodic (otherwise it is not a pattern). I At most one ball gets caught and thrown on every beat. If a ball is caught, it is the one which is thrown on that beat. (We ignore the time between catch and throw). These are simple, asynchronous juggling patterns. At the end of the talk we will mention some generalizations. Stephen Hardy: The Mathematics of Juggling 17 Representing Juggling Patterns Throws I We use numbers to denote throws. I A throw of height n lands n beats later. I If n is odd it changes hands. I If n is even it lands in the same hand it was thrown from. I A juggling pattern is denoted by a string of numbers. I The constant string n is the standard way to juggle n balls. Stephen Hardy: The Mathematics of Juggling 18 Representing Juggling Patterns Juggling Patterns The constant string n is the standard way to juggle n balls. I For odd n this is called a cascade. I For even n this is called a fountain { and you actually just juggle n=2 balls in each hand independently. I The circular pattern people think of when they think of juggling is called a shower. Since this pattern is asymmetric and requires throwing higher, is actually a bad way to start learning. Stephen Hardy: The Mathematics of Juggling 19 Representing Juggling Patterns Throws (Continued) The throws 0, 1 and 2 are a bit unusual. I You juggle zero balls by doing nothing (sometimes performers use this opportunity to clap or pirouette or catch a prop thrown by someone else). I You juggle one ball by fast horizontal throws back and forth. I You juggle two balls by just holding them (or very small vertical throws, but more often this is an opportunity for flourishes, or biting an apple). Stephen Hardy: The Mathematics of Juggling 20 Representing Juggling Patterns Throws (Continued) I Humans can only throw so high accurately, even if ceilings and wind aren't an issue. I When you start even 5 level throws are hard. I can manage about a 7. I The world records are around 13. I Thus it makes sense we can cap our throws to a finite level. Stephen Hardy: The Mathematics of Juggling 21 Examples Two-Ball Patterns I 31 { the 2-ball shower. A bad habit. I 330 { training for 3 balls. I 40 { two in one hand - training for 4 balls. I 501 { the interesting and fairly challenging 2-ball pattern! Stephen Hardy: The Mathematics of Juggling 22 Examples Three-Ball Patterns I 3 { lots of variations, inside/outside/tennis etc. I 42 { two in one hand. A good opportunity for showmanship. I 423 { a good first trick, time to bite the apple. I 522 { how you juggle when you learn. I 55500 { the flash - add a clap or pirouette to impress. I 51 { the shower - be sure to practice both ways. Stephen Hardy: The Mathematics of Juggling 23 Examples More Three-Ball Patterns I 504 { kind of weird. I 441 { the asynchronous box. I 4414413 { my favorite, fairly easy to do but impressive looking. I 50505 { the snake - good training for 5. I 531 { very pretty when done properly. Stephen Hardy: The Mathematics of Juggling 24 Examples Four-Ball Patterns I 4 { lots of variations: inside/outside/pistons, etc. I 552 { kind of ugly, not great practice for 5. I 55550 { better practice for 5. I 53 { the four-ball half-shower. I 71 { the four-ball shower. I 5551 { fun. I 534, 7531, 7131, 633, 741... Stephen Hardy: The Mathematics of Juggling 25 Examples Five-Ball Patterns I 5 I 73 { the five-ball half-shower. I 91 { the five-ball shower. I 64 { three in one hand, two in the other. I 645, 771, 726, 66661, 663, 744, 77731... Stephen Hardy: The Mathematics of Juggling 26 Questions Questions I What strings of natural numbers are valid juggling patterns? I How can we generate all valid juggling patterns? I How many balls are needed to juggle a pattern? Stephen Hardy: The Mathematics of Juggling 27 Questions Observations I The period of a pattern is the minimal length before it repeats: 33333 = 3 has period 1, 441441 = 441 has period 3. I Cyclic permutations give the same pattern up to chirality (51 vs. 15 changes direction of the shower). I Not all strings of natural numbers give valid juggling patterns. We can never have n(n − 1) for example: Stephen Hardy: The Mathematics of Juggling 28 A Theorem Notation Let's tackle the question: given a valid juggling pattern, how many balls are needed to juggle it? th I We let t(i) be the height of of the throw on the i beat. That ball lands on the i + t(i)th beat. I We assume this is a periodic function with some period p: t(i + p) = t(i). I We want to calculate balls(t), the number of balls needed to juggle this pattern. I We define σ(i) = i + t(i) − balls(t). I This will be a permutation of Z exactly when t(i) is a valid juggling pattern. I Conversely, given such a permutation we can recover t(i). Stephen Hardy: The Mathematics of Juggling 29 A Theorem An Example Stephen Hardy: The Mathematics of Juggling 30 A Theorem More Notation I height(t) = maxi t(i) is the maximum throw height that occurs (again we assume this is finite). I balls(t), the number of balls needed to juggle the pattern t, corresponds to the number of infinite orbits in our diagram (singleton orbits correspond to zero throws, and there are no other possibilities since balls cannot vanish or appear). Stephen Hardy: The Mathematics of Juggling 31 A Theorem Theorem (Average Theorem) I The number of balls needed to juggle a valid pattern is the average height of the throws. I If t is a valid juggling pattern with finite height, then P t(i) balls(t) = lim i2I jI j!1 jI j Where I = fa; a + 1;:::; bg is an integer interval and jI j = b − a + 1 is the number of elements in that interval. Stephen Hardy: The Mathematics of Juggling 32 A Theorem The Proof Let I be an integer interval with jI j > height(t). Then each ball lands at least once in the interval I . Let O be an infinite orbit. Stephen Hardy: The Mathematics of Juggling 33 A Theorem Proof Continued From the diagram, we can see that X jI j − height(t) ≤ t(i) ≤ jI j + height(t) i2I \O Summing over all the orbits (singleton orbits do not contribute because then t(i) = 0), we get X balls(t)[ jI j − height(t)] ≤ t(i) ≤ balls(t)[ jI j + height(t)] i2I Dividing by jI j and taking the limit as jI j ! 1 we get the desired P i2I t(i) result: balls(t) = lim jI j!1 jI j Stephen Hardy: The Mathematics of Juggling 34 A Theorem Applications I Corollary: If the average of a string of natural numbers is not an integer, then that string is not a valid juggling pattern.
Recommended publications
  • How to Juggle the Proof for Every Theorem an Exploration of the Mathematics Behind Juggling

    How to Juggle the Proof for Every Theorem an Exploration of the Mathematics Behind Juggling

    How to juggle the proof for every theorem An exploration of the mathematics behind juggling Mees Jager 5965802 (0101) 3 0 4 1 2 (1100) (1010) 1 4 4 3 (1001) (0011) 2 0 0 (0110) Supervised by Gil Cavalcanti Contents 1 Abstract 2 2 Preface 4 3 Preliminaries 5 3.1 Conventions and notation . .5 3.2 A mathematical description of juggling . .5 4 Practical problems with mathematical answers 9 4.1 When is a sequence jugglable? . .9 4.2 How many balls? . 13 5 Answers only generate more questions 21 5.1 Changing juggling sequences . 21 5.2 Constructing all sequences with the Permutation Test . 23 5.3 The converse to the average theorem . 25 6 Mathematical problems with mathematical answers 35 6.1 Scramblable and magic sequences . 35 6.2 Orbits . 39 6.3 How many patterns? . 43 6.3.1 Preliminaries and a strategy . 43 6.3.2 Computing N(b; p).................... 47 6.3.3 Filtering out redundancies . 52 7 State diagrams 54 7.1 What are they? . 54 7.2 Grounded or Excited? . 58 7.3 Transitions . 59 7.3.1 The superior approach . 59 7.3.2 There is a preference . 62 7.3.3 Finding transitions using the flattening algorithm . 64 7.3.4 Transitions of minimal length . 69 7.4 Counting states, arrows and patterns . 75 7.5 Prime patterns . 81 1 8 Sometimes we do not find the answers 86 8.1 The converse average theorem . 86 8.2 Magic sequence construction . 87 8.3 finding transitions with flattening algorithm .
  • Inspiring Mathematical Creativity Through Juggling

    Inspiring Mathematical Creativity Through Juggling

    Journal of Humanistic Mathematics Volume 10 | Issue 2 July 2020 Inspiring Mathematical Creativity through Juggling Ceire Monahan Montclair State University Mika Munakata Montclair State University Ashwin Vaidya Montclair State University Sean Gandini Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Monahan, C. Munakata, M. Vaidya, A. and Gandini, S. "Inspiring Mathematical Creativity through Juggling," Journal of Humanistic Mathematics, Volume 10 Issue 2 (July 2020), pages 291-314. DOI: 10.5642/ jhummath.202002.14 . Available at: https://scholarship.claremont.edu/jhm/vol10/iss2/14 ©2020 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Inspiring Mathematical Creativity Through Juggling Ceire Monahan Department of Mathematical Sciences, Montclair State University, New Jersey, USA
  • Happy Birthday!

    Happy Birthday!

    THE THURSDAY, APRIL 1, 2021 Quote of the Day “That’s what I love about dance. It makes you happy, fully happy.” Although quite popular since the ~ Debbie Reynolds 19th century, the day is not a public holiday in any country (no kidding). Happy Birthday! 1998 – Burger King published a full-page advertisement in USA Debbie Reynolds (1932–2016) was Today introducing the “Left-Handed a mega-talented American actress, Whopper.” All the condiments singer, and dancer. The acclaimed were rotated 180 degrees for the entertainer was first noticed at a benefit of left-handed customers. beauty pageant in 1948. Reynolds Thousands of customers requested was soon making movies and the burger. earned a nomination for a Golden Globe Award for Most Promising 2005 – A zoo in Tokyo announced Newcomer. She became a major force that it had discovered a remarkable in Hollywood musicals, including new species: a giant penguin called Singin’ In the Rain, Bundle of Joy, the Tonosama (Lord) penguin. With and The Unsinkable Molly Brown. much fanfare, the bird was revealed In 1969, The Debbie Reynolds Show to the public. As the cameras rolled, debuted on TV. The the other penguins lifted their beaks iconic star continued and gazed up at the purported Lord, to perform in film, but then walked away disinterested theater, and TV well when he took off his penguin mask into her 80s. Her and revealed himself to be the daughter was actress zoo director. Carrie Fisher. ©ActivityConnection.com – The Daily Chronicles (CAN) HURSDAY PRIL T , A 1, 2021 Today is April Fools’ Day, also known as April fish day in some parts of Europe.
  • Class of 1965 50Th Reunion

    Class of 1965 50Th Reunion

    CLASS OF 1965 50TH REUNION BENNINGTON COLLEGE Class of 1965 Abby Goldstein Arato* June Caudle Davenport Anna Coffey Harrington Catherine Posselt Bachrach Margo Baumgarten Davis Sandol Sturges Harsch Cynthia Rodriguez Badendyck Michele DeAngelis Joann Hirschorn Harte Isabella Holden Bates Liuda Dovydenas Sophia Healy Helen Eggleston Bellas Marilyn Kirshner Draper Marcia Heiman Deborah Kasin Benz Polly Burr Drinkwater Hope Norris Hendrickson Roberta Elzey Berke Bonnie Dyer-Bennet Suzanne Robertson Henroid Jill (Elizabeth) Underwood Diane Globus Edington Carol Hickler Bertrand* Wendy Erdman-Surlea Judith Henning Hoopes* Stephen Bick Timothy Caroline Tupling Evans Carla Otten Hosford Roberta Robbins Bickford Rima Gitlin Faber Inez Ingle Deborah Rubin Bluestein Joy Bacon Friedman Carole Irby Ruth Jacobs Boody Lisa (Elizabeth) Gallatin Nina Levin Jalladeau Elizabeth Boulware* Ehrenkranz Stephanie Stouffer Kahn Renee Engel Bowen* Alice Ruby Germond Lorna (Miriam) Katz-Lawson Linda Bratton Judith Hyde Gessel Jan Tupper Kearney Mary Okie Brown Lynne Coleman Gevirtz Mary Kelley Patsy Burns* Barbara Glasser Cynthia Keyworth Charles Caffall* Martha Hollins Gold* Wendy Slote Kleinbaum Donna Maxfield Chimera Joan Golden-Alexis Anne Boyd Kraig Moss Cohen Sheila Diamond Goodwin Edith Anderson Kraysler Jane McCormick Cowgill Susan Hadary Marjorie La Rowe Susan Crile Bay (Elizabeth) Hallowell Barbara Kent Lawrence Tina Croll Lynne Tishman Handler Stephanie LeVanda Lipsky 50TH REUNION CLASS OF 1965 1 Eliza Wood Livingston Deborah Rankin* Derwin Stevens* Isabella Holden Bates Caryn Levy Magid Tonia Noell Roberts Annette Adams Stuart 2 Masconomo Street Nancy Marshall Rosalind Robinson Joyce Sunila Manchester, MA 01944 978-526-1443 Carol Lee Metzger Lois Banulis Rogers Maria Taranto [email protected] Melissa Saltman Meyer* Ruth Grunzweig Roth Susan Tarlov I had heard about Bennington all my life, as my mother was in the third Dorothy Minshall Miller Gail Mayer Rubino Meredith Leavitt Teare* graduating class.

  • CNO Awarded at IHS Tribal Urban Awards Ceremony

    State-of-the-art Chahta Oklahoma press at Texoma Foundation teams play Print Services works to secure in Stickball Choctaw legacy World Series Page 3 Page 9 Page 18 BISKINIK CHANGE SERVICE REQUESTED PRESORT STD P.O. Box 1210 AUTO Durant OK 74702 U.S. POSTAGE PAID CHOCTAW NATION BISKINIKThe Official Publication of the Choctaw Nation of Oklahoma August 2012 Issue CNO awarded at IHS Tribal Urban Awards Ceremony By LISA REED services staff, the Choctaw Nation Choctaw Nation of Oklahoma has several new programs aimed at educating us on improving our life- The ninth annual Oklahoma styles.” City Area Director’s Indian Health Receiving awards were: Service Tribal Urban Awards Cer- • Area Director’s National Impact emony was held July 19 at the Na- – Mickey Peercy, Choctaw Nation’s tional Cowboy & Western Heritage Executive Director of Health. Museum in Oklahoma City. Chief • Area Director’s Area Impact – Gregory E. Pyle assisted in present- Jill Anderson, Clinic Director of the ing awards to the recipients from the Choctaw Health Clinic in McAles- Choctaw Nation of Oklahoma. Thir- ter. teen individuals and one group from • Area Director’s Lifetime the Choctaw Nation’s service area Achievement Award – Kelly Mings, were recognized for their dedica- Chief Financial Officer for Choctaw tion and contributions to improving Nation Health Services. the health and well-being of Native • Exceptional Group Performance Americans. Award Clinical – Chi Hullo Li, The “I would like to commend all who Choctaw Nation’s long-term com- are here today,” said Chief Pyle. prehensive residential treatment pro- “Their hard work and dedication gram for Native American women Choctaw Nation: LISA REED are exemplary.
  • The Neuroscience of Learned Behaviors

    The Neuroscience of Learned Behaviors

    Train Your Brain The Neuroscience of Learned Behaviors by Gregory L. Vogt, Ed.D., Barbara Z. Tharp, M.S., Christopher Burnett, B.A., and Nancy P. Moreno, Ph.D. © 2015 Baylor College of Medicine © 2015 by Baylor College of Medicine. All rights reserved. Field-test version. Printed in the United States of America. ISBN: 978-1-944035-03-7 BioEdSM TEACHER RESOURCES FROM THE CENTER FOR EDUCATIONAL OUTREACH AT BAYLOR COLLEGE OF MEDICINE The mark “BioEd” is a service mark of Baylor College of Medicine. Development of The Learning Brain educational materials was supported by grant number 5R25DA033006 from the National Institutes of Health, NIH Blueprint for Neuroscience Research Science Education Award, National Institute on Drug Abuse (NIDA), administered through the Office of the Director, Science Education Partnership Award program (Principal Investigator, Nancy Moreno, Ph.D.). The activities described in this book are intended for school-age children under direct supervision of adults. The authors, BCM, NIDA and NIH cannot be responsible for any accidents or injuries that may result from conduct of the activities, from not specifically following directions, or from ignoring cautions contained in the text. The opinions, findings and conclusions expressed in this publication are solely those of the authors and do not necessarily reflect the views of BCM, image contributors or the sponsoring agencies. No part of this book may be reproduced by any mechanical, photographic or electronic process, or in the form of an audio recording; nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use without prior written permission of the publisher.
  • IJA Enewsletter Editor Don Lewis (Email: Enews@Juggle.Org) Renew at Http

    IJA Enewsletter Editor Don Lewis (Email: [email protected]) Renew at Http

    THE INTERNATIONAL JUGGLERS! ASSOCIATION August 2011 IJA eNewsletter editor Don Lewis (email: [email protected]) Renew at http:www.juggle.org/renew IJA eNewsletter IJA Stage Championships Results, July 21 - 22, 2011 Individuals Contents: 1st: Tony Pezzo 2011 Championships Results 2nd: Kitamura Shintarou IJA Busking Competition 3rd: Tomohiro Kobayashi Joggling Correction Video Download Help Teams Club Tricks Handouts 1st: Showy Motion - Stefan Brancel and Ben Hestness Magazine Process 2nd: Smirk - Reid Belstock and Warren Hammond Will Murray in Afghanistan 3rd: The Jugheads - Rory Bade, Michael Barreto, Alex Behr, Daniel Burke, Sean YEP Report Carney, Tom Gaasedelen, Joe Gould, Danny Gratzer, Conor Hussey, Reid Johnson, Griffin Kelley, Jonny Langholz, Jack Levy, Chris Lovdal, Mara Moettus, Chris Olson, Montreal Circus Festival Evan Peter, Scott Schultz, Joey Spicola, and Brenden Ying Stagecraft Corner Regional Festivals Juniors Best Catches IJA Games Winners 1st: David Ferman 2nd: Jack Denger FLIC Auditions 3rd: Patrick Fraser Busking Competition 1st: Cate Flaherty 2nd: Kevin Axtell 3rd: Gypsy Geoff See all the results online at: http://www.juggle.org/history/champs/champs2011.php Juggling Festivals: Davidson, NC S.Gloucestershire, UK Kansas City, MO Portland, OR Philadelphia, PA Asheville, NC St. Louis, MO Baden, PA Waidhofen, Austria North Goa, India Bali, Indonesia photo: Martin Frost WWW.JUGGLE.ORG Page 1 THE INTERNATIONAL JUGGLERS! ASSOCIATION August 2011 IJA Busking Competition, Most of us know there are a lot of great buskers amongst IJA members, but most of us don!t get to see their street acts in the wild. Stage shows and street shows are totally different dynamics, and different again from the zaniness that occurs on the Renegade stage.
  • Envisaging a Comprehensible Global Brain As a Playful Organ

    Envisaging a Comprehensible Global Brain As a Playful Organ

    Alternative view of segmented documents via Kairos 2 December 2019 | Draft Envisaging a Comprehensible Global Brain -- as a Playful Organ Patterns connecting the dots between hemispheres, epicycles and quavers - / - Introduction Patterning and framing a global brain? Systemic feedback cycles of global brain interrelationships in 2D Various representations of cyclic dynamics with implications for a global brain Implication of 3D representation of a global brain Dynamic patterns of play engendered by Homo ludens and Homo undulans? Requisite helical cognitive engagement within a global brain Brainwaves and feedback loops in a global brain? Pathology of the global brain? Global brain as an organ: playable, playful or neither? References Introduction Collectively distributed cognition: This exploration is partly inspired by the work of Stafford Beer, as a management cybernetician, into the design of an appropriate collective brain (Brain of the Firm: the managerial cybernetics of organization, 1981), especially given the requisite complexity for a "viable system" (Giuliana Galli Carminati, The Planetary Brain: From the Web to the Grid and Beyond, 2011). Use of the metaphor followed from the influential study by an earlier cybernetician, W. Ross Ashby (Design for a Brain: the origin of adaptive behavior, 1952). The design preoccupation was the primary feature of a presentation by Shann Turnbull (Design Criteria for a Global Brain, 2001) as presented to the First Global Brain Workshop (Brussels, 2001). The argument follows from the controversial assertion recently made by President Macron of France with respect to the "brain death" of NATO and the potential implications for any "global brain" (Are the UN and the International Community both Brain Dead -- given criteria recognizing that NATO is brain dead? 2019).
  • Creating a Board Game to Influence Student Retention and Success

    Creating a Board Game to Influence Student Retention and Success

    ENROLLED: CREATING A BOARD GAME TO INFLUENCE STUDENT RETENTION AND SUCCESS by Andrew H. Peterson This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Education Ferris State University November 2018 © 2019 Andrew H. Peterson All Rights Reserved ENROLLED: CREATING A BOARD GAME TO INFLUENCE STUDENT RETENTION AND SUCCESS by Andrew H. Peterson Has been approved November 2018 APPROVED: Jonathan Truitt, PhD Committee Chair David Simkins, PhD Committee Member Paul Wright, PhD Committee Member Dissertation Committee ACCEPTED: Roberta C. Teahen, PhD, Director Community College Leadership Program ABSTRACT This dissertation documents the justification, creation, and refinement of a traditional board game to be used in first-year experience (freshman seminar) courses. The board game enRolled allows students to play college in a guided classroom environment. The game creates a low-stakes environment for students to be able to fail at college without risking time and money. Further, it is designed to give students a relatable topic to debate. By having a common experience, students can distance themselves from what their expectations of college are and deliberate on the tangible game they have played. The discussions that stem from this play and the modification of the game elements are designed to help the students take ownership over the experience. Having a game experience to discuss allows a student to objectively analyze their player’s path through the academic setting and debate realistic and perceived unrealistic challenges. Through these discussions, it is possible for students to engage with content while having little to no firsthand knowledge. It is the hope of this author that the creation of enRolled leads to a more engaging curriculum in first-year experience courses.
  • Braids and Juggling Patterns Matthew Am Cauley Harvey Mudd College

    Braids and Juggling Patterns Matthew Am Cauley Harvey Mudd College

    Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2003 Braids and Juggling Patterns Matthew aM cauley Harvey Mudd College Recommended Citation Macauley, Matthew, "Braids and Juggling Patterns" (2003). HMC Senior Theses. 151. https://scholarship.claremont.edu/hmc_theses/151 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Braids and Juggling Patterns by Matthew Macauley Michael Orrison, Advisor Advisor: Second Reader: (Jim Hoste) May 2003 Department of Mathematics Abstract Braids and Juggling Patterns by Matthew Macauley May 2003 There are several ways to describe juggling patterns mathematically using com- binatorics and algebra. In my thesis I use these ideas to build a new system using braid groups. A new kind of graph arises that helps describe all braids that can be juggled. Table of Contents List of Figures iii Chapter 1: Introduction 1 Chapter 2: Siteswap Notation 4 Chapter 3: Symmetric Groups 8 3.1 Siteswap Permutations . 8 3.2 Interesting Questions . 9 Chapter 4: Stack Notation 10 Chapter 5: Profile Braids 13 5.1 Polya Theory . 14 5.2 Interesting Questions . 17 Chapter 6: Braids and Juggling 19 6.1 The Braid Group . 19 6.2 Braids of Juggling Patterns . 22 6.3 Counting Jugglable Braids . 24 6.4 Determining Unbraids . 27 6.4.1 Setting the crossing numbers to zero. 32 6.4.2 The complete system of equations .
  • Skill Toy Kit User Guide

    Skill Toy Kit User Guide

    Skill Toy Kit User Guide Introduction to Skill Toys 2 Coaching Tips Flower Sticks 3 Kendama 5 Spinning Plate 7 Diabolo 9 Flop Ball 11 Yo-Yo 12 Juggling 14 Creating a Routine 17 DIY Instructions Kendama 20 Flower Sticks 22 Juggling Balls 24 Activity Ideas Trick of the Week/Month 25 Everyday Objects 25 Video Contests 25 60 Second Challenges 26 Skill Toy Bingo 27 Other Resources Reading List & On-line Resources 28 Kit Inventory Form 29 Introduction to Skill Toys You and your team are about to embark on a fun adventure into the world of skill toys. Each toy included in this kit has its own story, design, and body of tricks. Most of them have a lower barrier to entry than juggling, and the following lessons of good posture, mindfulness, problem solving, and goal setting apply to all. Before getting started with any of the skill toys, it is important to remember to relax. Stand with legs shoulder width apart, bend your knees, and take deep breaths. Many of them require you to absorb energy. We have found that the more stressed a player is, the more likely he/she will be to struggle. Not surprisingly, adults tend to have this problem more than kids. If you have players getting frustrated, remind them to relax. Playing should be fun! Also, remind yourself and other players that even the drops or mess-ups can give us valuable information. If they observe what is happening when they attempt to use the skill toys, they can use this information to problem solve.
  • Circus for Children and Families Inspiration

    Circus for Children and Families Inspiration

    ACTION FOR CHILDREN’S ARTS Circus For Children And Family Audiences Inspiration Day SO WHAT’S SO INSPIRATIONAL ABOUT CIRCUS FOR CHILDREN AND FAMILIES? Dear Reader, CONTENTS The image of the traditional travelling circus coming to town is one So what’s so inspirational about circus? that has resonated with many since childhood; it is also an image that many contemporary circus practitioners distance themselves from in their artistic explorations. Circus has been a developing Did you know – random research & facts about circus art form in the U.K, and the quality of the work being produced goes from strength to strength. Circus has also attracted the attention of theatre practitioners and choreographers that seek Circus: A beginner’s guide to incorporate it within their work – with some innovative offerings from contemporary UK companies having emerged with recent years. Clearly there is a growing interest in the art form, Take the Quiz: What circus performer would I be? but what is on offer for young people and their families? On 25th October 2011, a group of circus practitioners, artists, Up Close and Personal – we talk hard facts with circus artists producers, arts managers and educationalists gathered to think the creation of circus for young people and their families. Action Making work for young people – Circus Practitioners Speak Out for Children’s Arts has an excellent track record for delivering ‘Inspiration Days’ on various art-forms – it was wonderful that circus was at the heart of this lively debate. Fancy a bit of circus? Find out where and how Please enjoy the read – it’s a quick peek into some of the debates and discoveries of this particular Inspiration Day.