DOCSLIB.ORG

The Mathematics of Juggling

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Mathematics of Juggling The Mathematics of Juggling Yuki Takahashi University of California, Irvine November 12, 2015 Co-sponcered by UCI Illuminations and Juggle Buddies Yuki Takahashi (UC Irvine) November 12, 2015 1 / 51 Introduction: What is Juggling? Figure : 2013, winter, 5-ball cascade Juggling is the manipulation of objects (balls, clubs, rings, hats, cigar boxes, diabolos, devil sticks, yoyos, etc). Yuki Takahashi (UC Irvine) November 12, 2015 2 / 51 Club Juggling Figure : Korynn Aguilar (Chair), April 2015, UC Irvine, One World Concert. Yuki Takahashi (UC Irvine) November 12, 2015 3 / 51 Passing Figure : October 2015, University Hills, Fall Fiesta. Yuki Takahashi (UC Irvine) November 12, 2015 4 / 51 (Dragon) Ball Juggling Figure : Vegeta, juggling 7 (dragon) balls Yuki Takahashi (UC Irvine) November 12, 2015 5 / 51 Papers about Siteswaps A. Engsr¨om,L. Leskel¨a,H. Varpanen, Geometric juggling with q-analogues, Discrete Math. 338 (2015), 1067{1074. A. Ayyer, B, Arvind, S. Corteel, F. Nunzi, Multivariate juggling probabilities, Electron. J. Probab. 20 (2015). A. Knutson, T. Lam, D. Speyer, Positroid varieties: Juggling and geometry, Compos. Math. 149 (2013) 1710{1752. C. Elsner, D. Klyve, E. Tou, A zeta function for juggling sequences, J. Comb. Number Theory 4 (2012) 53{65. S. Butler, R. Graham, Enumerating (multiplex) juggling sequences, Ann. Comb. 13 (2010) 413{424. F. Chung, R. Graham, Primitive juggling sequences, Amer. Math. Monthly 115 (2008) 185{194. Yuki Takahashi (UC Irvine) November 12, 2015 6 / 51 Juggling Patterns Definition (Simple Juggling Patterns) In this talk, we assume the balls are juggled in a constant beat, that is, the throws occur at discrete equally spaced moments in time, patterns are periodic, and at most one ball gets caught and thrown at every beat. Figure : Juggling diagram of the ?-ball cascade. Yuki Takahashi (UC Irvine) November 12, 2015 7 / 51 Example (1) Figure : Juggling diagram of the 4-ball fountain. This pattern is denoted by 4. Yuki Takahashi (UC Irvine) November 12, 2015 8 / 51 Example (2) Figure : Juggling diagram of the 3-ball shower. This pattern is denoted by ...? Yuki Takahashi (UC Irvine) November 12, 2015 9 / 51 Example (3) Figure : 4-ball shower. This pattern is denoted by 71. Yuki Takahashi (UC Irvine) November 12, 2015 10 / 51 Example (4) This pattern is denoted by ...? The number of ball juggled is ...? Yuki Takahashi (UC Irvine) November 12, 2015 11 / 51 Terminologies We call a finite sequence of non-negative integers arising from a juggling pattern a juggling sequence. (Example: 3, 4, 51, 441, 7, 51515151, are all juggling sequences) The length of a finite sequence of integers is called its period. (Example: 441441 has period 6) A juggling sequence is minimal if it has minimal period among all the juggling sequences representing the same juggling pattern. (Example: 144 is minimal, but 441441 is not) Yuki Takahashi (UC Irvine) November 12, 2015 12 / 51 Example (4) 546 is NOT a juggling sequence. Figure : What is wrong...? In general, if s = n(n − 1) ··· , then s is not a juggling sequence. Yuki Takahashi (UC Irvine) November 12, 2015 13 / 51 The Average Theorem Theorem (The Average Theorem) The number of balls necessary to juggle a juggling sequence is its average. If the average is not an integer, then the sequence is not a juggling sequence. Example The number of balls necessary to juggle 441 is 3. 5+6+2 562 is NOT a juggling sequence (note that 3 = 4:3333::: is not an integer). Yuki Takahashi (UC Irvine) November 12, 2015 14 / 51 Swapping i j i j Figure : a juggling sequence s is transformed into another juggling sequence si;j . This operation is called swapping. Yuki Takahashi (UC Irvine) November 12, 2015 15 / 51 Example (1) Let s = 441. Then s0;2 = :::? Figure : s to s0;2. Yuki Takahashi (UC Irvine) November 12, 2015 16 / 51 Example (1) Figure : 441 to 342. Yuki Takahashi (UC Irvine) November 12, 2015 17 / 51 Example (2) Let s = 7531. Then s1;3 = :::? Figure : s to s1;3. Yuki Takahashi (UC Irvine) November 12, 2015 18 / 51 Example (3) Let s = 6313. Then s1;2 = 6223. Figure : s to s1;2. Yuki Takahashi (UC Irvine) November 12, 2015 19 / 51 Swapping Definition (Swapping) p−1 Let s = fak gk=0 be a sequence of nonnegative integers. Let i and j be integers such that 0 ≤ i < j ≤ p − 1 and j − i ≤ ai . Let si;j be 8 a + j − i if k = i <> j si;j (k) = ai − (j − i) if k = j > :ak o.w. i j i j Yuki Takahashi (UC Irvine) November 12, 2015 20 / 51 Summary of Swapping For a juggling sequence s, we denote the number of balls necessary to juggle it by ball(s). Then The sequence s is a juggling sequence if and only if si;j is a juggling sequence. The average of s is the same as the average of si;j . If s is the juggling sequence, then ball(s) = ball(si;j ). Yuki Takahashi (UC Irvine) November 12, 2015 21 / 51 Leeeeet's Practice!!! =] Example (Swapping) Let s = 642. Then s0;1 = 552, and s0;2 = :::? (What does this result tell you about s? Is it a juggling sequence?) Let s = 532. Then s0;1 = :::? Let s = 123456789. Then s3;5 = :::? Reminder: p−1 Let s = fak gk=0 be a sequence of nonnegative integers. Then 8 a + j − i if k = i <> j si;j (k) = ai − (j − i) if k = j > :ak o.w. Yuki Takahashi (UC Irvine) November 12, 2015 22 / 51 Getting Sleeply? It's Show Time! :) Yuki Takahashi (UC Irvine) November 12, 2015 23 / 51 Cyclic Shifts Definition (Cyclic Shifts) p−1 Let s = fsk gk=0 be a sequence of nonnegative integers. Let ^s = a1a2a3 ··· ap−1a0: We call the operation of transforming s into ^s the cyclic shift of s. Example: If s = 12345, then ^s = 23451. Yuki Takahashi (UC Irvine) November 12, 2015 24 / 51 Example Let s = 441. Then ^s = 414. Figure : 441 to 414. Yuki Takahashi (UC Irvine) November 12, 2015 25 / 51 Summery of Cyclic Shifts Apparantly, we have the following: The sequence s is a juggling sequence if and only if ^s is a juggling sequence. The average of s is the same as the average of ^s. If s is a juggling sequence, then ball(s) = ball(^s). Yuki Takahashi (UC Irvine) November 12, 2015 26 / 51 Summery!!! Swapping and cyclic shifts both preserve 1 "jugglibility", 2 the average, and 3 the number of balls (if it is a juggling sequence). Yuki Takahashi (UC Irvine) November 12, 2015 27 / 51 Flattening Algorithm The flattening algorithm takes as input an arbitrary sequence s. 1. If s is a constant sequence, stop and output this sequence. Otherwise, 2. use cyclic shifts to arrange the elements of s such that one of maximum height, say e, comes to rest at beat 0 and one not of maximum height, say f , comes to rest at beat 1. If e and f differ by 1, stop and output this new sequence. Otherwise, 3. perform a swapping of beats 0 and 1, and return to step 1. Yuki Takahashi (UC Irvine) November 12, 2015 28 / 51 Examples! =) Denote the map s 7! s0;1 by Sw, and cyclic shifts s 7! ^s by Cy. Example (flattening algorithm) Cy Cy Cy 2 1) 264 −! 642 −!Sw 552 −! 525 −!Sw 345 −! 534 −!Sw 444: (This implies ...what?) Cy 2 Cy 2 2) 514 −!Sw 244 −! 424 −!Sw 334 −! 433: (Similarly, this implies...?) This proves the Average Theorem! (why??) Yuki Takahashi (UC Irvine) November 12, 2015 29 / 51 Test Vector Definition (Test Vector) p−1 Let s = fak gk=0 be a nonnegative sequence. Then we call the vector (0 + a0; 1 + a1; ··· ; (p − 1) + ap−1) mod p the test vector of s. Example Let s = 6424. Then (0 + 6; 1 + 4; 2 + 2; 3 + 4) = (6; 5; 4; 7), so the test vector is (2; 1; 0; 3). Let s = 543. Then (0 + 5; 1 + 4; 2 + 3) = (5; 5; 5), so the test vector is (2; 2; 2). Yuki Takahashi (UC Irvine) November 12, 2015 30 / 51 Permutation Test Theorem (The Permutation Test) p−1 Let s = fak gk=0 be a nonnegative sequence, and let φs be the test vector of s. Then, s is a juggling sequence if and only if φs is a permutation. Example Let s = 6424. Then the test vector is (2; 1; 0; 3), so 6424 is a juggling sequence. Let s = 444. Then the test vector is (1; 2; 0), so 444 is a juggling sequence. Let s = 433. Then the test vector is (1; 1; 2), so 433 is NOT a juggling sequence. Yuki Takahashi (UC Irvine) November 12, 2015 31 / 51 Idea of the Proof The proof is based on the flattening algorithm. The key fact is that: "permutationability" is preserved under swapping and cyclic shift!!! Example Let s = 7531. Then s1;2 = 7441. The test vector of 7531 is (3; 2; 1; 0), and the test vector of 7441 is (3; 1; 2; 0). Let s = 6534. Then s2;3 = 6552. The test vector of 6534 is (2; 2; 1; 3), and the test vector of 6552 is (2; 1; 3; 1). Yuki Takahashi (UC Irvine) November 12, 2015 32 / 51 Proof of the Permutation test WTS: s is a juggling sequence if and only if the test vector of s is a permutation. Example (Flattening Algorithm and Test Vector) Cy 1) 642 (0; 2; 1) −!Sw 552 (2; 0; 1) −! 525 (2; 0; 1) Cy 2 −!Sw 345 (0; 2; 1) −! 534 (2; 1; 0) −!Sw 444 (1; 2; 0): (This implies ...what?) Cy 2 2) 514 (2; 2; 0) −!Sw 244 (2; 2; 0) −! 424 (1; 0; 0) Cy 2 −!Sw 334 (0; 1; 0) −! 433 (1; 1; 2): (Similarly, this implies...?) This proves the Permutation Test! :) (why??) Yuki Takahashi (UC Irvine) November 12, 2015 33 / 51 Corollary Corollary (Vertical Shifts) p−1 Let s = fak gk=0 be a sequence of nonnegative integers.
Load more

Recommended publications
  • Midwest Flow Fest Workshop Descriptions!
    Ping Tom Memorial Park Chicago, IL Saturday, September 9 Area 1 Area 2 Area 3 Area 4 FREE! Intro Contact Mini Hoop Technicality VTG 1:1 with Fans Tutting for Flow Art- 11am Jay Jay Kassandra Morrison Jessica Mardini Dushwam Fancy Feet FREE! Poi Basics Performance 101 No Beat Tosses 12:30pm Perkulator Jessie Wags Matt O’Daniel Zack Lyttle FREE! Intro to Fans Better Body Rolls Clowning Around Down & Dirty- 2pm Jessica Mardini Jacquie Tar-foot Jared the Juggler Groundwork- Jay Jay Acro Staff 101 Buugeng Fundamentals Modern Dance Hoop Flowers Shapes & Hand 3:30pm Admiral J Brown Kimberly Bucki FREE! Fearless Ringleader Paths- Dushwam Inclusive Community Swap Tosses 3 Hoop Manipulation Tosses with Doubles 5pm Jessica Mardini FREE! Zack Lyttle Kassandra Morrison Exuro 6:30pm MidWest Flow Fest Instructor Showcase Sunday, September 10 Area 1 Area 2 Area 3 Area 4 Contact Poi 1- Intro Intro to Circle Juggle Beginner Pole Basics Making Organic 11am Matt O’Daniel FREE! Juan Guardiola Alice Wonder Sequences - Exuro Intermediate Buugeng FREE! HoopDance 101 Contact Poi 2- Full Performance Pro Tips 12:30pm Kimberly Buck Casandra Tanenbaum Contact- Matt O’Daniel Fearless Ringleader FREE! Body Balance Row Pray Fishtails Lazy Hooping Juggling 5 Ball 2pm Jacqui Tar-Foot Admiral J Brown Perkulator Jared the Juggler 3:30pm Body Roll Play FREE! Fundamentals: Admiral’s Way Contact Your Prop, Your Kassandra Morrison Reels- Dushwam Admiral J Brown Dance- Jessie Wags 5pm FREE! Cultivating Continuous Poi Tosses Flow Style & Personality Musicality in Motion Community- Exuro Juan Guardiola Casandra Tanenbaum Jacquie Tar-Foot 6:30pm MidWest Flow Fest Jam! poi dance/aerial staff any/all props juggling/other hoop Sponsored By: Admiral J.
    [Show full text]
  • 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 .
    [Show full text]
  • Physical Activity Resource Guide for Child Care Centres
    Physical Activity Resource Guide for Child Care Centres 2009 C R E A T E D B Y : J O S É E D IGNARD AND D O M I N I C F ORTIN C H I L D R E N S E R V I C E S S EC TION Table of Contents Outline p. 3 Children’s Developmental Characteristics p. 4 - 6 Guideline for Adapting your Activities p. 7 Stretches for Warm ups and Cool downs p. 8 - 15 Warm ups and Cool downs Activities p. 16 - 22 Grouping of Children for Activities p. 23 - 24 Locomotors movements for Games and Activities p. 25 – 26 Games and Activities p. 27 - 71 Other Fun Games and Activities p. 72 -77 Walking Programs p. 78 - 79 Health and Well- Being Information p. 80 - 87 Reference & Resources p. 88 - 89 Josée & Dominic – Summer Physical Activity Programmers 2009 2 Outline of the Activities The activities provided in this guide include warm-ups, moderate to vigorous physical activity for limited spaces and outdoors, cool-downs, stretches and other fun ideas. These activities can be used throughout the year. Activities should be adapted for different age groups and suggestions are provided throughout the guide. Repetition of a physical activity five or six times during the course of a month, for example, will allow children to become familiar with the activity, and reduces the time required for instruction in the activity. As a result, children have more time to be physically active. Leaders can also create variations on the activities, and can also encourage kids to create their own variations.
    [Show full text]
  • THE BENEFITS of LEARNING to JUGGLE for CHILDREN Written by Dave Finnigan
    THE BENEFITS OF LEARNING TO JUGGLE FOR CHILDREN Written by Dave Finnigan Fitness, Motor Skills, Rhythm, Balance, Coordination Benefits . Students can start acquiring pre-juggling skills in pre-school and kindergarten by learning to toss and catch one big, colorful, slow-moving nylon scarf. Once tossing and catching becomes routine, you can use one nylon scarf anywhere that you would normally use a ball or beanbag for most games and many individual challenges. Just think of the scarf as a "ball with training wheels." Scarf juggling and scarf play progresses in a step-by-step manner from one, to two, and on to three. Scarf play and scarf juggling requires big, flowing movements. Children get a great cardio-vascular and pulmonary work-out when they juggle scarves, exercising the big muscles close to the head and close to the heart. They will find scarf juggling to be a great deal of work and lots of fun at the same time. Once they move on to beanbags or other faster moving equipment, they get exercise not only from tossing, but from bending over to pick up errant objects as well. Once a student can juggle continuously, they can "workout" with heavy or bulky objects such as heavyweight beanbags, basketballs, or other items which provide strenuous exercise while they improve focus and dexterity. Because all of this tossing and catching activity can be done to music, you can use different types of music to help children acquire a sense of rhythm and a natural "beat." By working together, students can participate in a group activity requiring concentration and attention to task and reinforcing rhythm.
    [Show full text]
  • 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
    [Show full text]
  • Book Review: the Mathematics of Juggling, Volume 51, Number 1
    Book Review The Mathematics of Juggling Reviewed by Allen Knutson The Mathematics of Juggling • (to avoid considering boundary conditions) The Burkard Polster juggler has been juggling since the infinite past Springer-Verlag, New York, 2003 and will continue into the infinite future. 226 pages, $39.95 • The throws come one per second, with right ISBN 0-387-95513-5 throws at even times and left throws at odd times (this can be ensured by reparametrizing Around 1985 three groups of jugglers (in Santa the time coordinate). Cruz, California; in Pasadena, California; and in More notable, though, is the assumption about Cambridge, England) independently created the what one actually wants to record: same notational system for juggling patterns. These • The only information kept about the ball thrown ∈ Z numerical descriptions have been well publicized at time n is when it is next thrown, at time in the juggling world under the name “siteswaps” f (n). (and, even so, continue to be rediscovered by new- In particular, this assumption loses any extra de- comers). tail about the throw like “under the leg”, “with a triple spin” (if the objects juggled are pins rather As we will see shortly upon presentation of the than balls), “while telling joke #1729”, etc. But it basic definitions, a rather simple list of restric- suggests a very nice mathematical theory of cer- tions leads easily to a unique notational system. tain functions f : Z → Z: More surprising is the near-simultaneity of the dis- 1. The function f should be one-to-one, since we covery! The best explanation of this seems to be do not want two balls landing at the same time.
    [Show full text]
  • The Beginner's Guide to Circus and Street Theatre
    The Beginner’s Guide to Circus and Street Theatre www.premierecircus.com Circus Terms Aerial: acts which take place on apparatus which hang from above, such as silks, trapeze, Spanish web, corde lisse, and aerial hoop. Trapeze- An aerial apparatus with a bar, Silks or Tissu- The artist suspended by ropes. Our climbs, wraps, rotates and double static trapeze acts drops within a piece of involve two performers on fabric that is draped from the one trapeze, in which the ceiling, exhibiting pure they perform a wide strength and grace with a range of movements good measure of dramatic including balances, drops, twists and falls. hangs and strength and flexibility manoeuvres on the trapeze bar and in the ropes supporting the trapeze. Spanish web/ Web- An aerialist is suspended high above on Corde Lisse- Literally a single rope, meaning “Smooth Rope”, while spinning Corde Lisse is a single at high speed length of rope hanging from ankle or from above, which the wrist. This aerialist wraps around extreme act is their body to hang, drop dynamic and and slide. mesmerising. The rope is spun by another person, who remains on the ground holding the bottom of the rope. Rigging- A system for hanging aerial equipment. REMEMBER Aerial Hoop- An elegant you will need a strong fixed aerial display where the point (minimum ½ ton safe performer twists weight bearing load per rigging themselves in, on, under point) for aerial artists to rig from and around a steel hoop if they are performing indoors: or ring suspended from the height varies according to the ceiling, usually about apparatus.
    [Show full text]
  • 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.
    [Show full text]
  • JUGGLING in the U.S.S.R. in November 1975, I Took a One Week Tour to Leningrad and Ring Cascade Using a "Holster” at Each Side
    Volume 28. No. 3 March 1976 INTERNATIONAL JUGGLERS ASSOCIATION from Roger Dol/arhide JUGGLING IN THE U.S.S.R. In November 1975, I took a one week tour to Leningrad and ring cascade using a "holster” at each side. He twice pretended Moscow, Russia. I wasfortunate to see a number of jugglers while to accidentally miss the catch of all 9 down over his head before I was there. doing it perfectly to rousing applause and an encore bow. At the Leningrad Circus a man approximately 45 years old did Also on the show was a lady juggler who did a short routine an act which wasn’t terribly exciting, though his juggling was spinning 15-inch square glass sheets. She finished by spinning pretty good. He did a routine with 5, 4, 3 sticks including one on a stick balanced on her forehead and one on each hand showering the 5. He also balanced a pole with a tray of glassware simultaneously. Then, she did the routine of balancing a tray of on his head and juggled 4 metaf plates. For a finish, he cqnter- glassware on a sword balanced point to point on a dagger in her spun a heavy-looking round wooden table upside-down on a 10- mouth and climbing a swaying ladder ala Rosana and others. foot sectioned pole balanced on his forehead, then knocked the Finally, there were two excellent juggling acts as part of a really pole away and caught the table still spinning on a short pole held great music and variety floor show at the Arabat Restaurant in in his hands.
    [Show full text]
  • 19730007257.Pdf
    fit ENG A SPECIAL BIBLIOGRAJ'HY / V '^x £'* "" •' V: WITH INDEXES $'. - • ''? Supplement 23 "13 Ks oo oc NATIONAL AERONAUTICS AND SPACE ADMINISTRATION PREVIOUS BIBLIOGRAPHIES IN THIS SERIES ftttCUftWIt! NASA SP-703? September 1970 Jan.-Aug. |9?0 NASA SP-7037 (Oh Januarv 1971 Sept.-Dec. 1970 NASA SP-7037 f 02) Februar> 1971 January 1971 NASA SP-7037(U3» March 1*971 K'briiarv 1971 NASA April |V?1 March 1971 NASA SP-7037 f 05) Ma> 1971 April |97| , NASA SP-7037 »06) June 1971 Ma> 1971 NASA SP-7037 «. i?) Jul\ !*»?! June 1971 NASA SP-7037 (08) Auuu\t 197! Julv 19" I NASA SP-7037 (09) September 1971 August 1971 ,NASA SP-7037 (10) October 1971 September 1971 'NASA SP- 7037(1 1) November 1971 October 1971 NASA SP-7037 (12) December 1971 November 197! NASA SP-7037 (13) JUmuur> 1972 December 1971 NASA SP-7037 (14} January 1972 Annual Indexes 197J NASA SP-7037 (15) February 19t2 January 1972 NASA SP-7037 (16) March 1972 February 1972 NASA SP-7037 (17) April 1972 March 1972 NASA SP-7037 (18) May 1972 April 1972 NASA SP-7037 (19) June 1972 May 1972 NASA SP-7037 (20) July 1972 June 1972 . NASA SP-7037 (2 1) August 1972 July 1972 NASA SP-7037 (22) September 1972 Aueust 1972 Thi> bibliography w-as prepared by the NAS\ Scientific jnd Tt-'chnical information Fac»Hl\ for the National Aeronautic* j«d Space -Vdmini^ration b\ Informatics Th>co, Inc. The Administrator of the Nat'Onaf Aeronautics and Space Admimsration Has determined that tfte pubiicatjon of this persodtcai is neeessa«y in the transaction of the public business rsqui'ea by law of this Agency.
    [Show full text]
  • TERRAIN INTELLIGENCE Ifope~}M' Oy QUARTERMASTER SCHOOL Lubflar
    MHI tI"Copy ? ·_aI i-,I C'opy 31 77 /; DEPARTMENT OF 'THE ARMY FIELD MANUAL TERRAIN INTELLIGENCE IfOPE~}m' Oy QUARTERMASTER SCHOOL LUBflAR. U.S. ARMY QUA. TZ. MSIR SCL, FORT LEE, VA. 22YOlS ,O.ISTx L.at. : , .S, HEAD QU ARTER S, DEPARTMENT OF T HE ARMY OCTOBER 1967 *FM 30-10 FIELD MANUAL HEADQUARTERS DEPARTMENT OF THE ARMY No. 30-10 I WASHINGTON, D.C., 24 October 1967 TERRAIN INTELLIGENCE CHAPTER 1. INTRODUCTION ..........................-- 1-3 2. CONCEPTS AND RESPONSIBILITIES Section I. Nature of terrain intelligence _____.______ .__. 4-7 II. Responsibilities -. ------------------- 8-11 CHAPTER 3. PRODUCTION OF TERRAIN INTELLIGENCE Section I. Intelligence cycle __ …...__…____-----__-- 12-16 II. Sources and agencies ….................... .--- 17-23 CHAPTER 4. WEATHER AND CLIMATE Section I. Weather ..................................... 24-39 II. Climate -.................................. 40-47 III. Operations in extreme climates _................. 48-50 CHAPTER 5. NATURAL TERRAIN FEATURES Section I. Significance .................................. 51, 52 II. Landforms __________ _------------- ------- 53-61 Drainage............................. 62-69 IV. Nearshore oceanography ................. 70-75 V. Surface materials ............................ 76-81 VI. Vegetation ................................... 82-91 CHAPTER 6. MANMADE TERRAIN FEATURES Section I. Significance -.............................. 92, 93 II. Lines of communication ….................... 94-103 III. Petroleum and natural gas ____. _._______.___ 104-108 IV. Mines, quarries, and pits ….................... 109-112 V. Airfields ----------------------------- 113-115 VI. Water terminals _______…-------- ----------- 116-118 VII. Hydraulic structures -........................ 119-121 VIII. Urban areas and buildings ................... 122-128 IX. Nonurban areas ___ _--------------------- 129-131 CHAPTER 7. MILITARY ASPECTS OF THE TERRAIN Section I. Military use of terrain _....._________._____._ 132-139 Ii. Special operations ......................... 140, 141 III.
    [Show full text]
  • Revolutionary Flyer V2n2 Fall84for a PDF with Full
    A FLYING DISC & FOOTBAG SPORTS DIGEST ~~~~~~~~~ ULTIMATE THE SINKHOLE CLASSIC DISC GOLF WORLD + STATE CHAMPIONSHIPS FOOTBAG NATIONAL CHAMPIONSHIPS «UPCOMING EVENTS» The Revolutionary Flyer FROM THE EDITORS With the passing of summer , autumn is falling upon us leaving time to reflect on the commitment we ' ve made to you , the readers . Making this flyer a subscription publication was essential for its growth. We are pleased with the response we have received over the summer , for it has helped to establish a foundation o~ which to build. Therefore we can pro cede with our goal of making this the best flyer of its type to be circulated throughout Florida. This issue covers Ultimate ' s exciting Sinkhole City Classic , disc golf from the State to the World Championshi ps . W. F. A. Footbag Nationals , and other little interesting tidbits that make for enjoyable reading. So go ahead , read , enjoy , and (if you haven ' t already) subscribe. It will be well worth the investment. Fantastic Flights! Bret & Dori Executive Editor EDITORIAL OFFICE : Bret Segrest The Revolutionary Flyer Assistant Editor 351 Montgomery Road Dori Segrest Alt. Spgs ., Fl. 32714 (05)862- 4852 Contributors Ed Aviles Jr. Lynn Finch Terry Hamlin Next Issue Nick Hart January 1st. Ace Mason The W.F.A. .................. r :' :' I ~ I . BRAND W s¢- S~ The ~ NEW DISC! I Danciny Footbag WOISCRAFT »The PHANTOM W ~ PROOUCTS ~ In,""""", the N'l:W'P.ST H 10 R QUAUTY P'CXYJ'BAO - B~R BY D!'.810N &nd ~NJOY!:D ~ Discraft Products has come out with by bolh recre&Uonal player and profes.slr, M.! altke a new golf and distance disc , the EASIEST 1'0 KICK I ITS PU$I1C BUDS MAK1!! IT PHANTOM.
    [Show full text]