DOCSLIB.ORG

Meeting Alhambra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Meeting Alhambra Meeting Alhambra ISAMA BRIDGES The International Society of the Mathematical Connections Arts, Mathematics, and Architecture in Art, Music, and Science UNIVERSITI OF GRANADA FACULTI OF SCIENCES Meeting Alhambra ISAMA-BRIDGES 2003 Organizers Javier Barrallo Jose Martinez Aroza Registration and Reservation Space Management School of Architecture Faculty of Sciences University of the Basque Country University of Granada 20018 San Sebastian, Spain 18071 Granada, Spain Nathaniel Friedman Reza Sarhangi Art Exhibit Conference Proceedings Department of Mathematics Mathematics Department University At Albany Towson University 1400 Washington Avenue 8000 York Road Albany, NY 12222 Towson, MD 21252 Juan Antonio Maldonado Carlo Sequin Space Management Poster Session Faculty of Sciences EECS, Computer Science Division University of Granada Soda Hall # 1776 18071 Granada, Spain University of California Berkeley, CA 94720-1776 Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings, 2003. Editors: Javier Barrallo Jose Martfnez-Aroza School of Architecture Faculty of Sciences University of the Basque Country University of Granada San Sebastian, Spain Granada, Spain Nathaniel Friedman Reza Sarhangi Department of Mathematics Mathematics Department University At Albany Towson University Albany, New York, USA Towson, Maryland, USA Juan Antonio Maldonado Carlo Sequin Faculty of Sciences EECS, Computer Science Division University of Granada University of California Granada, Spain Berkeley, California, USA ISBN: Price: $40.00 Distributed by MATHARTFUN.COM (http://mathartfun.com) Front cover design: Polynomiograph of a Physicists' polynomial by Bahman Kalantari. Back cover design: The step 2-octogonal tiling projected onto a dome and used as the basis for a zellij created by Jean-Marc Castera. Cover layout and border designs: Chris Palmer. © Meeting Alhambra, ISAMA-Bridges 2003 Conference. (http://www.sckans.edu/~bridges/). All rights reserved. General permission is granted to the public for non-commercial reproduction, in limited quantities, of individual articles, provided authorization is obtained from individual authors, and a complete reference is given for the source. All © rights and responsibilities for individual articles in the 2003 Conference Proceedings remain under the control of the original authors. Contents Preface xiii Three Traditions of Self-Similarity in Fourteenth and Fifteenth Century 1 Islamic Geometric Ornament Jay Bonner Volution's Evolution 13 Carlo H. Sequin Classification and Phylogenetic Analysis of African Ternary Rhythm Timelines 25 Godfried Toussaint Hidden Inscriptions in the Laurentian Library 37 Paul L. Rosin and Ralph R. Martin Computers, Mathematics and Conceptual Art 45 Michael Mahan Mathematical Sculpture Classification 53 Ricardo Zalaya and Javier Barrallo Constructing and Classifying Designs of al-Andalus . 61 B. Lynn Bodner Nodal Lines and Quasi-Space Filling Curves . 69 Tiziana Giorgi and Caryn Werner On the Perception of Tone 77 Richard J. Krantz, Jack Douthett, and Richard Plotkin Computational Aesthetics Based on Gaze Patterns 85 Gary R. Greenfield Seashell Architectures 93 Kamon Jirapong and Robert 1. Krawczyk Composable Art: Objects That Can Be Arranged in Many Ways 101 Marc van Kreveld Mathematics in Literature 109 Lorna B. Hanes vi Dimension of Time in Strange Attractors 119 Robert 1. Krawczyk Geometric Mosaic Tiling on Hyperbolic Sculptures 127 Irene Rousseau The Mathematics of Color-Reversing Decorative Friezes: Fa~ades of Pirgi, Greece 135 David A. James, Loukas N. Kalisperis, and Alice V. James Computer-Aided Aesthetic Evaluation of Visual Patterns 143 Tomas Staudek Frieze-Generation Using Artificial Life 151 Dirk Fischer, Eric Goles, and Mario Markus A Unified Theory of Proportion 161 Jay Kapprafj and Gary W. Adamson The Art in Polynomiography of Special Polynomials 173 Bahman Kalantari Arraying Alternative Housing Archetypes: An Online Resource 181 Jin-HoPark Play with Infinity 189 Jean-Marc Castera Two Perspectives on Inversion 197 John Sharp Finding the Dual of the Tetrahedral-Octahedral Space Filler 205 Eva Knoll and Simon Morgan Square Decompositions with Hyperbolic Consequences in Art, 213 Chemical Physics and Mathematics Robert G. Smits Polyhedral Designs of Detection Systems for Nuclear Physics Studies 219 N. G. Nicolis Leonardo's da Vinci's Bar Grids 229 Rinus Roelofs Fractal Geometry and Self-Similarity in Architecture: 235 An Overview Across the Centuries Nicoletta Sala Tangramoids 245 Samuel Verbiese vii The Sundial, Beyond Form and Time ... 253 Andrzej Zarzycki The Poincare Surprises 257 Claude-Paul Bruter Quilt Designs Using Non-Edge-to-Edge Tilings by Squares 265 Gwen L Fisher An Extended Mural for a House of Mathematics 273 Anna Campbell Bliss Hidden Geometry in Music of Bach and Schoenberg: 283 Reflection, Rotation, Proportion Jonathan Saggau An Introduction to the Tiling Properties of Precious Triangles 291 Stanley Spencer NEC Polygonal Groups and Tessellations 299 Domingo Gamez. Miguel Pasatias, Rafael Perez. and Ceferino Ruiz Mathematics & Esthetics - Science and Art in the Bay Area 307 Benjamin Wells Sculpture from Symmetrically Arranged Planar Components 315 George W. Hart Easy Digitalization and Representation of Three-Dimensional Maps 323 Javier Bilbao Landatxe and Jorge Lazaro Dominguez Dynamical Systems: A Golden Gate from Auditory Physiology 331 to Musical Aesthetics? Julyan H. E. Cartwright, Diego L Gonzalez. Oreste Piro, and Domenico Stanzial Introducing Simple Arithmetic and Geometric Series 339 in Complex Parametric Modeling Jane Burry and Mark Burry Abraham as Bridge and Mediator: A Metaphoric Reflection on the Alhambra 347 Ozayr Saloojee M6bius Concepts in Architecture 353 Jolly Thulaseedas and Robert J. Krawczyk On the Shapes of Water Fountains and Times Tables 361 Stephen Eberhart viii Lengthening, Expansion and Torsion: 367 Three Geometric Operations for Aesthetics Creation Delgado Olmos, A., Marquez Garda, L. Mathematical Bridges to Philosophy and Theology 375 Douglas Norton Whirled White Web: Art and Math in Snow 383 Brent Collins, Steve Reinmuth, Dan Schwalbe, Carlo Sequin, and Stan Wagon Quilts Inspired by Mathematics 393 Mary C. Williams Geometric Designs in Architecture and Textiles Revisited 401 and Reconstructed Through Layered Planar Geometry Benigna Chilla Fractal Geometry Describes the Beauty of Infinity in Nature 407 Maria Antonia Castro and Maria Jose Perez-Luque Number Theory and Art 415 Vera W. de Spinadel Magic Geometry: Mosaics in the Alhambra 423 Miroslav Lovric Graphs and Circulation in Rural Housing 433 MB Francisca Blanco and Miriam Pisonero Diversifying the Curriculum: The Tessellations Case 441 Reza Sarhangi, Gabriele Meiselwitz, and Goran Trajkovski The "Mathematics and Culture" Project 449 Michele Emmer Islamic Constructions: The Geometry Needed by Craftsmen 459 Raymond Tennant Developmental Morphology: X, Y, & Z Coordination 465 as a Dynamic and Generative Cellular Process Manuel A. Bdez A Study on the North Dome of Masjid-I-Jami Isfahan 473 Marjan Ghannad Crossing the Equal Sign 485 Marion D. Cohen Great Kiva Design in Chaco Canyon: An Archaeology of Geometry 493 Chris Hardaker ix Petrie Polygons 503 Paul Gailiunas The Mathematics of Quilting: A QUilter's Tacit Knowledge 511 of Symmetry, Tiling, and Group Theory Katrina Hebb Hyperbolic Spirals and Spiral Patterns 521 Douglas Dunham Optimal Geometry as Art 529 John M. Sullivan Folding Polyhedra and Painting Mosaics 533 Chris K. Palmer Abstracts A Patterned Life 541 Jennifer Angus Origami: A Good Way to Communicate Mathematics 544 Ann Hanson Goldbach THings 545 SharolNau Conceptual Art and Algorithms 547 Susan Happersett Symmetry and Trigonometry 549 Steve Whealton Digital Monoprints: Mining Conchoid Nets 551 Stephen Luecking Cayley Tables as Quilt Designs 553 Gwen L. Fisher and Elsa Medina Duality: A Common Thread in Math, Science, Literature, and Art? 555 Michael de Villiers An Interdisciplinary Study of an Archetype: Across Cultures and Disciplines 557 Ann Hanson, Paula Frohele, Carol Haliday-McQueen, and Renee Hansen Space Blanket 558 Lorella Di Cintio x On Mathematics in Art 559 Clifford Singer The Relationship among Proportions, Ottoman-Turkish Music and Architecture 561 Guney Kurdak Polyhedra in "Polyart" Style 562 Ulrich Mikloweit An Evaluation Approach fot Circulation in Buildings 563 Mohamed Nada and Khaled M. Nassar Discrete-Event Activity Simulation for Predicting 564 Occupants' Movements in Buildings Khaled M. Nassar and Mohamed Nada Why Do Penrose THings Diffract? 565 Robert V. Moody My Use of Number Sequence and Geometry in Art 567 Kathleen Hyndman New Pentatonic Scale Relationships with Visual Parallels 569 Veryan Weston Hexa-Twistor Triangular Section 571 AkioHizume Some Mathematical Patterns in the Analysis of 573 Selection and Deletion in Modem Short Stories Masoudd Norouzian and Shadman Shokravi Golomb Rep-Tiles and Fractals 575 lmameddin Amiraslanov Two Geometric Sculptures with Distant Ontogenies 577 Benjamin Wells The Collapse Series 578 Andrzej Zarzycki Connecting Writing and Contextual Development 579 with Mathematics in Introductory College Math Courses Stan T. Yoshinobu Geometry in Dali' s Painting 580 F. Blanco, E. Reyes, and C. Santamarina .
Load more

Recommended publications
  • De Divino Errore ‘De Divina Proportione’ Was Written by Luca Pacioli and Illustrated by Leonardo Da Vinci
    De Divino Errore ‘De Divina Proportione’ was written by Luca Pacioli and illustrated by Leonardo da Vinci. It was one of the most widely read mathematical books. Unfortunately, a strongly emphasized statement in the book claims six summits of pyramids of the stellated icosidodecahedron lay in one plane. This is not so, and yet even extensively annotated editions of this book never noticed this error. Dutchmen Jos Janssens and Rinus Roelofs did so, 500 years later. Fig. 1: About this illustration of Leonardo da Vinci for the Milanese version of the ‘De Divina Proportione’, Pacioli erroneously wrote that the red and green dots lay in a plane. The book ‘De Divina Proportione’, or ‘On the Divine Ratio’, was written by the Franciscan Fra Luca Bartolomeo de Pacioli (1445-1517). His name is sometimes written Paciolo or Paccioli because Italian was not a uniform language in his days, when, moreover, Italy was not a country yet. Labeling Pacioli as a Tuscan, because of his birthplace of Borgo San Sepolcro, may be more correct, but he also studied in Venice and Rome, and spent much of his life in Perugia and Milan. In service of Duke and patron Ludovico Sforza, he would write his masterpiece, in 1497 (although it is more correct to say the work was written between 1496 and 1498, because it contains several parts). It was not his first opus, because in 1494 his ‘Summa de arithmetic, geometrica, proportioni et proportionalita’ had appeared; the ‘Summa’ and ‘Divina’ were not his only books, but surely the most famous ones. For hundreds of years the books were among the most widely read mathematical bestsellers, their fame being only surpassed by the ‘Elements’ of Euclid.
    [Show full text]
  • Mathematics of Origami
    Mathematics of Origami Angela Kohlhaas Loras College February 17, 2012 Introduction Origami ori + kami, “folding paper” Tools: one uncut square of paper, mountain and valley folds Goal: create art with elegance, balance, detail Outline History Applications Foldability Design History of Origami 105 A.D.: Invention of paper in China Paper-folding begins shortly after in China, Korea, Japan 800s: Japanese develop basic models for ceremonial folding 1200s: Origami globalized throughout Japan 1682: Earliest book to describe origami 1797: How to fold 1,000 cranes published 1954: Yoshizawa’s book formalizes a notational system 1940s-1960s: Origami popularized in the U.S. and throughout the world History of Origami Mathematics 1893: Geometric exercises in paper folding by Row 1936: Origami first analyzed according to axioms by Beloch 1989-present: Huzita-Hatori axioms Flat-folding theorems: Maekawa, Kawasaki, Justin, Hull TreeMaker designed by Lang Origami sekkei – “technical origami” Rigid origami Applications from the large to very small Miura-Ori Japanese solar sail “Eyeglass” space telescope Lawrence Livermore National Laboratory Science of the small Heart stents Titanium hydride printing DNA origami Protein-folding Two broad categories Foldability (discrete, computational complexity) Given a pattern of creases, when does the folded model lie flat? Design (geometry, optimization) How much detail can added to an origami model, and how efficiently can this be done? Flat-Foldability of Crease Patterns 훗 Three criteria for 훗: Continuity, Piecewise isometry, Noncrossing 2-Colorable Under the mapping 훗, some faces are flipped while others are only translated and rotated. Maekawa-Justin Theorem At any interior vertex, the number of mountain and valley folds differ by two.
    [Show full text]
  • Wenwu Chang, Exploring Space Filling Origami
    Exploring space-filling origami Wenwu Chang [email protected] Shanghai Putuo Modern Educational Technology Center, China Abstract A new tetrahedra origami model was discussed in this paper. Sommerville in 1922 found four kinds of tetrahedron can fill space. This model starts from a rectangle paper, whose shape ratio is 1:√2, to produce one Sommerville’s main tetrahedron. It is proved that as long as the original paper is big enough, one can produce by this origami method more and more so-called Sommerville-tetrahedrons without cutting or pasting. Furthermore, these tetrahedrons fill three-dimension space in the same time. Just like Peano curve fills two-dimensional space, the original paper (two dimensional manifold) used in producing tetrahedrons fills the three-dimensional space. This article also introduces some interesting models in the lower number cases. Introduction In 1922, D.M.Y Sommerville found Aristotle, a famous philosopher during Ancient Greek Times, made a mistake. Aristotle stated there were two kinds of regular polyhedron, including regular tetrahedra could tile space. In fact, cube is the only regular polyhedron that can fill space. Sommerville in his paper ended this mistake and pointed out that several general tetrahedra could fill space indeed. Resulting from the way of cutting and reassembling a cube, he found 4 kinds of tetrahedra. The most important tetrahedron was the one made up of four triangles whose edge ratios are 2:√3:√3. I will call it Sommerville-tetrahedron within this paper. Another three kinds of tetrahedra were developed by dissection or reassembling of the dissection to this original tetrahedron.
    [Show full text]
  • Quilling Letters a to Z
    Quilling Letters A To Z Merging and unclogged Tirrell netes her bangles signalised while Taylor disarranging some kilolitres Omnibusmadly. Daimonic Salem leavenand speedful very singly Godfry while rip-off Kam almost remains tiptop, vicinal though and emended.Hurley stud his pastiche speaks. Contact your android phone are looking for. Final solution for your professional look. 3D Numeral Door plan House feedback Sign Metal Gate Digits 0 To 9 A To Z. Basic free version of glue on your device is dedicated to create your network problems that he would have pores which frame. Rabbit style living room, letting it filled with a page you can post author of my husband made me alegra saber que mi blog. I didn't want to keep clean simple and led forward so long have overlapped the alphabets and arranged in this squat The mediocre is now name plate is the minute and. 100 Quilling A-Z ideas quilling quilling letters paper quilling. Ukrainian more about kirigami, letter of your letters, broadcast messages to create stylish text and enjoy this is the. Select from the. The best selection of Royalty Free Quilling Vector Art Graphics and Stock Illustrations Download 10000 Royalty Free Quilling Vector Images. Letters A thru Z are included in a mix of caps and lower field for a wonky fun look. Boxed Cards Greeting Cards Invitations Mini Cards Pop-Up Cards Quilled Cards Stationery Thank-You Notes Shop All Cards. 3d Origami Supre Tan. Is an icon that are admired by step by step by dennis walker designed by adams sara i have some credits.
    [Show full text]
  • Simple Rules for Incorporating Design Art Into Penrose and Fractal Tiles
    Bridges 2012: Mathematics, Music, Art, Architecture, Culture Simple Rules for Incorporating Design Art into Penrose and Fractal Tiles San Le SLFFEA.com [email protected] Abstract Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable M.C. Escher-like image that works esthetically as well as functionally requires resolving incongruencies at a tile’s edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical objects to which he never applied his artistry including Penrose Tilings and fractals. In this paper, we show that the rules of creating a traditional tile extend to these objects as well. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others. 1 1 Introduction M.C. Escher was the most prominent artist working with tessellations and space filling. Forty years after his death, his creations are still foremost in people’s minds in the field of tiling art. One of the reasons Escher continues to hold such a monopoly in this specialty are the unique challenges that come with creating Escher type designs inside a tessellation[15]. When an image is drawn into a tile and extends to the tile’s edge, it introduces incongruencies which are resolved by continuously aligning and refining the image. This is particularly true when the image consists of the lizards, fish, angels, etc. which populated Escher’s tilings because they do not have the 4-fold rotational symmetry that would make it possible to arbitrarily rotate the image ± 90, 180 degrees and have all the pieces fit[9].
    [Show full text]
  • Origami Heart Box Instructions Step by Step
    Origami Heart Box Instructions Step By Step Swift or tasteless, Giovanni never diking any profitableness! Unilobed Forrest gallops her nominals so excursively that Kris spiral very belatedly. Fulgid Ichabod still blazons: pileate and Aztecan Bear barbarises quite volubly but bestir her carpel noiselessly. GO662 Heart Box Diagram Also Origami Box With good On. But i need to origami heart lock box by measuring the citrus shape flower bouquet that it absorbs moisture. Yakomoga origami heart shaped quilled fringed flower by the flowers will then cut, which the dotted lines. First step by. Do the strip around a triangle shapes and includes a quality time! Go through the paper plate ocean craft ideas about origami tree quilled petals together in to learn how to. Diy gift box card that comes in the step by his disposable mask face you just download. Here i wanted to origami heart tabs on making this product designers who. Craft for his friends or tv or follow these folds open the box body friendly gift home you need it would take your. We heart box by step guides are among the hearts on your email. Diy heart origami box from your dollar bill and a clear origami video tutorials without the diagrams to. Step 15 Fold the average corner downwards and eve the squash fold at the square dot line Step 16 Flatten along the dotted line Step 17 The origami heart praise is. Push on the box by email address will be used for the ox and! Suprisingly it by step instructions to make boxes and decorate your box that sits in slightly faster to create various forms usually in.
    [Show full text]
  • Reconstructing David Huffman's Origami Tessellations
    Reconstructing David Huffman’s Origami Tessellations Eli Davis Erik D. Demaine Martin L. Demaine Jennifer Ramseyer MIT CSAIL MIT CSAIL MIT CSAIL MIT CSAIL Cambridge, MA 02139 Cambridge, MA 02139 Cambridge, MA 02139 Cambridge, MA 02139 [email protected] [email protected] [email protected] [email protected] David A. Huffman (1925–1999) is best known in com- ing to the symmetries of one of the 17 periodic “wallpaper puter science for his work in information theory, particu- groups”. Origami historian David Lister [4, 5] attributes the larly Huffman codes, and best known in origami as a pio- origin of origami tessellations to Shuzo Fujimoto, a Japanese neer of curved-crease folding. But during his early paper schoolteacher of mathematics who self-published the first folding in the 1970s, he also designed and folded over a origami tessellation book Twist Origami in 1976. In 1966, hundred different straight-crease origami tessellations. Un- however, artist Ronald Resch patented several origami tes- like most origami tessellations designed in the past twenty sellation designs [6]. Huffman met Resch in 1968, and they years, Huffman’s straight-crease tessellations are mostly talked more extensively at University of Utah in 1973, while three-dimensional, rigidly foldable, and have no locking Huffman was on sabbatical and Resch was a professor there. mechanism. In collaboration with Huffman’s family, our Both Huffman and Resch may have been influenced by these goal is to document all of his designs by reverse-engineering discussions, though we have no clear documentation to this his models into the corresponding crease patterns, or in some effect.
    [Show full text]
  • Leonardo Universal
    Leonardo Universal DE DIVINA PROPORTIONE Pacioli, legendary mathematician, introduced the linear perspective and the mixture of colors, representing the human body and its proportions and extrapolating this knowledge to architecture. Luca Pacioli demonstrating one of Euclid’s theorems (Jacobo de’Barbari, 1495) D e Divina Proportione is a holy expression commonly outstanding work and icon of the Italian Renaissance. used in the past to refer to what we nowadays call Leonardo, who was deeply interested in nature and art the golden section, which is the mathematic module mathematics, worked with Pacioli, the author of the through which any amount can be divided in two text, and was a determined spreader of perspectives uneven parts, so that the ratio between the smallest and proportions, including Phi in many of his works, part and the largest one is the same as that between such as The Last Supper, created at the same time as the largest and the full amount. It is divine for its the illustrations of the present manuscript, the Mona being unique, and triune, as it links three elements. Lisa, whose face hides a perfect golden rectangle and The fusion of art and science, and the completion of the Uomo Vitruviano, a deep study on the human 60 full-page illustrations by the preeminent genius figure where da Vinci proves that all the main body of the time, Leonardo da Vinci, make it the most parts were related to the golden ratio. Luca Pacioli credits that Leonardo da Vinci made the illustrations of the geometric bodies with quill, ink and watercolor.
    [Show full text]
  • Origami Animals by Robert Salazar
    Robert Salazar’s Artist Statement From a single uncut sheet of paper, all origami arise. Folding alone each sheet into any of an endless diversity of forms. Despite its finite surface area and simplicity, even a single sheet of paper can take on staggering complexity. Since surface area – the resource- is finite, any of an origami’s features can only come about by exchanging surface area with its other features until harmony is found among its many interdependent parts. Each transformation is an elaborate play with intricate geometric relationships that are akin to the evolution of a genome. Cutting geometric relationships with scissors or adding additional resource are impossible. Just as with our relationships with nature and each other, the interdependence that bring us together cannot be cut and resources cannot be created from nothing, everything becomes possible by seeking harmony with what we have. The focus of my work is to show how powerfully harmony can transform a finite resource as simple as a sheet of paper, along with its intrinsic geometric rules, into endless possibilities. I invite everyone to imagine the power of what’s possible when we strive for harmony with our own relationships, with each other and with nature, as they consider my works. www.salazarigami.com The Wildling Museum in Solvang presents Folded Art: Origami Animals by Robert Salazar. On View November 10, 2018 – March 25, 2019 His art shows how harmony can transform something as simple as a sheet of paper into endless possibilities. The exhibit features a collection of origami animals found in Santa Barbara County such as the Steelhead Trout, California Condor, Tarantula, and a few prehistoric species as well.
    [Show full text]
  • Leonardo Da Vinci's Study of Light and Optics: a Synthesis of Fields in The
    Bitler Leonardo da Vinci’s Study of Light and Optics Leonardo da Vinci’s Study of Light and Optics: A Synthesis of Fields in The Last Supper Nicole Bitler Stanford University Leonardo da Vinci’s Milanese observations of optics and astronomy complicated his understanding of light. Though these complications forced him to reject “tidy” interpretations of light and optics, they ultimately allowed him to excel in the portrayal of reflection, shadow, and luminescence (Kemp, 2006). Leonardo da Vinci’s The Last Supper demonstrates this careful study of light and the relation of light to perspective. In the work, da Vinci delved into the complications of optics and reflections, and its renown guided the artistic study of light by subsequent masters. From da Vinci’s personal manuscripts, accounts from his contemporaries, and present-day art historians, the iterative relationship between Leonardo da Vinci’s study of light and study of optics becomes apparent, as well as how his study of the two fields manifested in his paintings. Upon commencement of courtly service in Milan, da Vinci immersed himself in a range of scholarly pursuits. Da Vinci’s artistic and mathematical interest in perspective led him to the study of optics. Initially, da Vinci “accepted the ancient (and specifically Platonic) idea that the eye functioned by emitting a special type of visual ray” (Kemp, 2006, p. 114). In his early musings on the topic, da Vinci reiterated this notion, stating in his notebooks that, “the eye transmits through the atmosphere its own image to all the objects that are in front of it and receives them into itself” (Suh, 2005, p.
    [Show full text]
  • PDF Download Paper Crafts Log Book Vol. 14 : Your Project Tracker for Origami, Paper Quilling, and More!
    PAPER CRAFTS LOG BOOK VOL. 14 : YOUR PROJECT TRACKER FOR ORIGAMI, PAPER QUILLING, AND MORE! PDF, EPUB, EBOOK Paper Crafts Log Books | 102 pages | 09 Jan 2019 | Independently Published | 9781793499462 | English | none Paper Crafts Log Book Vol. 14 : Your Project Tracker for Origami, Paper Quilling, and More! PDF Book Back to top. Zonderkidz 1. To decorate: Draw gills at top and fins 12" below on each side with markers or crayon and decorate the fish as you like. The Paper Partnership 1. Make a file page that fits into the open end of the lunch bag. The Benefits of a Log Book Keeping a daily log book is beneficial in many ways. Make your own Flying Disc and watch it fly and spin. Tuck them in your tree or wreath, or hang above the holiday table. Learn how to make this fun and festive paper craft. Pret produs. Flame Tree 2. Birotica Capstone Young Readers 1. Who Scarf. David Heller has 54 books on Goodreads with ratings. Auto HiFi 3. If paper needs to be smoothed when dry, iron it under a sheet of paper on a low heat setting. Get to Know Us. Downtown Bookworks 1. The effects of leader selection on group performance. Homemade Gifts Made Easy. Paper Party Favors like these paper poppers are a great addition to any party. Beyond pleasure and pain. Paper Baskets are perfect for anyone looking for a fun paper craft project. Valentines Day History. Paper Crafts Log Book Vol. 14 : Your Project Tracker for Origami, Paper Quilling, and More! Writer How to Make Paper Snowflakes Decorative paper snowflakes are a paper craft that help you make your own winter wonderland.
    [Show full text]
  • Origami Mathematics in Education
    Origami Mathematics in Education Michael Assis Melbourne University School of Mathematics and Statistics Tools and Mathematics 29 November 2016 Origami ● The Art of Folding http://www.jccc.on.ca/assets/images/origami5.jpg Origami ● The Art of Folding http://img.gawkerassets.com/img/17jp3vs9qkjb6jpg/original.jpg http://res.artnet.com/news-upload/2014/05/origami-6.jpg http://www.joostlangeveldorigami.nl/fotos/historyoforigami/bug.jpg http://i.ytimg.com/vi/5nZtibCqFxw/hqdefault.jpg Origami ● The Art of Folding http://illusion.scene360.com/wp-content/themes/sahara-10/submissions/2012/10/jun_mitani_03.jpg http://farm5.static.flickr.com/4107/4946857347_a17b3e7900.jpg http://giangdinh.com/wp-content/uploads/2013/09/prayer.jpg Origami ● The Art of Folding https://c2.staticflickr.com/4/3530/5835802683_a7ca138ff9.jpg http://www.tporigami.com/wp-content/uploads/2010/09/ToiletPaperOrigami_Cover.jpg https://nrgtucker.files.wordpress.com/2012/12/20121223-183459.jpg http://strictlypaper.com/blog/wp-content/uploads/2013/03/nintai-origami-inspired-dresses-strictlypaper-1.jpg Origami in the Classroom Origami Resources 1D Origami Folding In Half ● How many times can you fold paper in half? – 8 times? Folding In Half ● How many times can you fold paper in half? – 8 times? ● Is there an upper limit? Folding In Half ● Britney Gallivan 2001 Activity 1 Parabolas ● Why does it work? ● Can other conics be constructed? ● What if you use non-flat paper? ● What can we learn concerning: – Parabolas ? – Envelopes? – Derivatives? – Tangents? – Convergence of sequences? Knots Knots Knots Knots Knots Knots Knots Knots Knots ● Explorations: – Perimeter, area – Irregular patterns – Enumerations – Knot theory, topology Activity 2 ● Fujimoto approximation Fujimoto Approximation ● Error is halved at each operation ● Repeating left-right pattern represented as the binary expansion of 1/n – 1/5: .00110011..
    [Show full text]