WONDROUS ONE SHEET

ORIGAMI

MEENAKSHI MUKERJI AN A K PETERS BOOK

Wondrous One Sheet Origami

Wondrous One Sheet Origami

Meenakshi Mukerji

Cover photos: Front - Fractal Sakura Back - Elm Leaf (top) and Abstract Flowers (bottom) Overleaf: Kusumita Butterfly

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-0-367-20813-4 (Hardback) 978-0-367-20810-3 (Paperback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To All Who Inspire Me

Contents Foreword ix Preface xi Acknowledgments xiii Photo Credits 1 � Origami Basics

xv 1

Origami Paper and Tools.................................................................................................................... 1 Origami Symbols.................................................................................................................................. 4 Common Bases and Terms................................................................................................................. 6 Pentagon from a Square...................................................................................................................... 8 Hexagon from a Square....................................................................................................................... 9 2 �  Simple Designs

10

Simple Diamond................................................................................................................................ 11 Simple Hearts..................................................................................................................................... 12 Ace of Diamonds................................................................................................................................ 13 Traditional and Five Petal Lilies....................................................................................................... 15 Tuberose.............................................................................................................................................. 18 Oleander.............................................................................................................................................. 20 Simple Peacock................................................................................................................................... 23 Simple Swan........................................................................................................................................ 24 Flower Strips....................................................................................................................................... 25 Hexagonal Flower.............................................................................................................................. 27 3 �  Four-Sink Base Flowers

29

Four-Sink Base................................................................................................................................... 30 Mum.................................................................................................................................................... 33 Hollyhock............................................................................................................................................ 35 Marigold.............................................................................................................................................. 36 Poinsettia with Leaves....................................................................................................................... 39 4 �  Kusumita and Derivatives

40

Modified Four-Sink Base.................................................................................................................. 42 Kusumita............................................................................................................................................. 45 Kusumita Tiles.................................................................................................................................... 47 Flowers................................................................................................................................................ 52 Contentsvii

Leaves.................................................................................................................................................. 54 Butterflies............................................................................................................................................ 56 Card Suits............................................................................................................................................ 62 5 � Leaves

66

Maple Leaves....................................................................................................................................... 67 Tropical Leaf....................................................................................................................................... 69 Elm Leaf.............................................................................................................................................. 71 Aspen Leaf.......................................................................................................................................... 74 Birch Leaf............................................................................................................................................ 76 6 �  Octagonal Collapse Flowers

77

Gazania................................................................................................................................................ 78 Gaillardia............................................................................................................................................. 81 Rudbeckia............................................................................................................................................ 84 Abstract Flowers................................................................................................................................. 88 Sunflower............................................................................................................................................ 92 Transposing the Flowers to Other Polygons................................................................................... 95 Folding a Tridecagon......................................................................................................................... 96 Thirteen Petalled Gaillardia.............................................................................................................. 98 7 �  Recursive Designs

101

Fractal Sakura................................................................................................................................... 102 Octospiral.......................................................................................................................................... 110 Whirl.................................................................................................................................................. 116 Hydrangea with Leaves................................................................................................................... 119 12-Pointed EZ Star........................................................................................................................... 121 Star Tower......................................................................................................................................... 124 Floral Perpetua................................................................................................................................. 128 8 � Decorative Boxes

131

Twisted Box....................................................................................................................................... 132 Leafy Box........................................................................................................................................... 135 Four-Leaf Tato Box.......................................................................................................................... 136 Hydrangea Box................................................................................................................................. 138 Author 140 Contributors 142 References 144 Index 146

viiiContents

Foreword Meenakshi Mukerji is one of today’s masters of modular origami, geometric designs comprised of multiple folded pieces of paper assembled without glue. In her newest book, she brings her ingenuity and creativity to designs made from a single piece of paper. Among the most appealing aspects of this book is the way she subtly manipulates a purely geometric design to form a flower (including a beautiful 12-petaled Hollyhock), a leaf, a butterfly, or familiar symbols such as the four suits from a deck of cards. Although many of her designs are complex and three-dimensional, requiring an abundance of both patience and dexterity to fold, she provides crystal-clear diagrams with lucid text and doesn’t stint on providing perspective drawings (which are challenging

and time-consuming to produce) when needed. In addition to presenting her own stunning designs, Mukerji offers up designs and biographies of three “guest folders” whose work is as stunning as her own. Of these designs, my favorite is the mind-boggling Floral Perpetua, which uses a recursive (self-repeating) folding process to mimic the concentric layers of petals in a blooming flower. It’s worth the price of admission all on its own. Like this spectacular model, Mukerji’s book yields riches in perpetuity! Peter Engel, author of Origami Odyssey: A Journey to the Edge of Paperfolding [Eng11], 10-Fold Origami [Eng16], and Folding the Universe: Origami from Angelfish to Zen [Eng94]

Forewordix

Preface After nearly 15 years of designing modular origami and writing four books on the subject, I am excited to present my very first book on single sheet designs. I have folded and enjoyed single sheet designs in the past, but it is only lately that I have delved into it deeper. Such amazing models as Shuzo Fujimoto’s Hydrangea [Fuj10] and Roman Diaz’s [Dia06] Fractal Flower [Dia12] renewed my interest in this original form of the ancient art. More recently, John Montroll’s book Galaxy of Origami Stars [Mon12] completely enraptured me with the single sheet genre. The challenges of one sheet designs are intriguing and refreshing after having designed modular origami for so long. Since my name is somewhat associated with modular designs, I worked the words “one sheet” into the title so no presumptions would be made by folders. Not only are the designs presented here made from a single sheet, they also follow the basic rules of origami, i.e., they are from single squares with no cuts or glue. There are a few designs that begin with other polygons, but they still follow the same rules once the polygon, the starting medium, has been obtained by cutting. As different as these new designs might be from my previous ones, my love for symmetry and geometry still remain noticeably unscathed. Notably, most of the designs in this book incorporate color change (both sides of the paper visible in the final model), so duo paper, i.e., paper with two different colors on each side, is recommended for the best outcome. The book starts with some origami basics that are customarily included in any origami book. If you are already familiar with these basics, please feel free to skip ahead. The general structure of the book follows a natural progression starting with simple designs and gradually going into more complex ones, although there may be a few minor exceptions based on context. I always like to include designs of new and upcoming artists, so I have presented creations of guest folders as well. It is my privilege to introduce them to origami enthusiasts around the world, if one is not already familiar with them. A large number of artists that I have included in my books have gone on to write their own, giving me a huge sense of fulfillment.

The diagrams in this book follow standard origami symbols pioneered by origami master Akira Yoshizawa [Yos84]. The styles of guest contributors have been preserved instead of changing them simply for the sake of consistency across the book. The origami diagramming language is very powerful with barely any need for written instructions with some exceptions when a little explanation may be required. To keep the diagrams simple and uncluttered, I have refrained from showing layers except when necessary. Some crease lines have been omitted at times for the sake of clarity, without compromising the outcomes. Wondrous One Sheet Origami covers a wide range of folding levels from simple to high intermediate, with more emphasis on the latter. The book should appeal to audiences 12 years of age and above or to children folding at higher than age level. Most of the designs are flat and hence suitable for mounting on greetings cards or for framing as gifts. My understanding from social media is that the designs here are quite sought after. So, I am confident that you will enjoy the book because I am proceeding with useful feedback from origami enthusiasts across the globe. Sit back, relax, and enjoy while you explore the huge potential of a single sheet of paper! Cupertino, California December 2013

Second Edition Addendum It is my pleasure to release a new edition of this book, six years later, and to publish with CRC Press. I am excited to present 40% new material, including some leaves that are my absolute favorites. You will also find new simple designs, recursive designs, flowers, and boxes. With compact formatting, you will get about 200 pages worth of goodies squeezed into just a 150-page book without loss of clarity and artfulness, and while keeping it economical. Once again, I hope you enjoy folding from this new edition of the book which has been a labor of love. Cupertino, California May 2019

Prefacexi

Acknowledgments While I have designed, folded and diagrammed most of the origami in this book, and done much of the book layout and photography, it would not have been complete without the invaluable help of the people named here. I would like to express my deepest gratitude to them all. First of all, I would like to thank my guest contributors, Evan Zodl (NJ), Dáša Ševerová (Slovakia), Christiane Bettens (Switzerland), and David Donahue (CA), for being generous with their beautiful designs and allowing me to include them in this book. They have been extremely cooperative with the challenges of importing and exporting diagrams between formats. They already have a huge fan base, but I hope that with this book I am able to introduce them to an even wider audience. Thanks to Roman Diaz (Uruguay), Endre Somos (Hungary), Jorge Jaramillo (Colombia), and Hitoshi Fujimoto (son of Shuzo Fujimoto, Japan). They were immensely kind to let me include in this book designs that are based on theirs. Further details are presented in their respective chapters. I would like to thank Kedar Amladi (CA) for patiently taking photographs using his innovative homemade diffused light apparatus, and for processing the photos as well. He was then a sophomore at Carnegie Mellon University, and I was lucky that he was available during a break. Thanks also to Rosalinda Sanchez (AZ) for enhancing the cover photograph. Thanks to Sara

Adams (Germany) [Ada07] for patiently making video instructions for some of the designs in this book. I am fortunate to have a brilliant team of testers who patiently tested and pointed out errors easy to overlook. They came up with wonderful suggestions for improvement as well. I would like to thank my tester-folders: Rui Roda (Portugal), JC Nolan (CA), Rebecca Harris (CT), Scott Cramer (NH), and Charul Patil (India). Rui has been the power tester, test-folding a large part of the book. JC came up with meaningful suggestions to improve my Fractal Sakura diagrams, and helped develop a crease pattern for it. Scott tested many of the leaves for me. Special thanks to Mark Kennedy (PA), Hilli Zenz (Germany), and JC Nolan for the beautiful handmade paper that they generously gave me. I have used the paper to fold some of the designs for very nice effects. As usual, Vinita Singhal (CA) has been my taken-for-granted technical adviser, and Jean Jaiswal (OR) my resourceful proofreader. My heartfelt thanks go to both of them. Thanks to my copy editor, Charlotte Byrnes (MA), for her patience and her keen eye for the minutest of errors that one may easily overlook. Thanks to Callum Fraser and the rest of the team at CRC Press, and Arun Kumar of Nova Techset for making the book possible. And last but not least, I would like to thank everyone who inspired me including my family, friends, visitors of my website, and fans of my origami.

Acknowledgmentsxiii

An arrangement of Abstract Flowers (p. 88) folded with hand colored harmony paper. Leaf diagrams not included.

xiv

Photo Credits I would like to credit and thank the following profoundly for folding, photography, or both:

””Evan Zodl: Folding and photos of 12-Pointed EZ Star and Star Tower on p. 120 (first two rows).

””Teresa Montero-Cañibe: Flower arrangement, folding and photo on p. xiv.

””Dáša Ševerová: Folding and photos of Floral Perpetua on p. 120 and Twisted Boxes on p. 131 (top).

””Billie Griebler: Wreath arrangement, folding and photo on p. xvi. ””Kedar Amladi: Photos of Butterfly on title page; Mum, Hollyhock, and Marigold on p.  29; Hollyhock on p. 35; Kusumita Flowers and Leaves on p. 40; Butterflies on p. 41; and Fractal Sakura Finish 2 on p. 108. ””Rui Roda: Folding and photos of Simple Swan on p.  22 (bottom), Butterflies on p.  56, 13-petalled Gaillardia on p. 95 (bottom right).

””Christian Bettens: Folding and photos of Leafy Boxes on p.  131 (bottom) and Four-Leaf Tato Boxes on p. 137. ””Sara Adams: Photo of Hydrangea Box on p. 139. ””Indra Singhal: Photo of author on p. 140. ””Hilli Zenz: Photo of Dáša Ševerová on p. 142. ””All other folding and photos are by the author.

””JC Nolan: Folding of Fractal Sakura Base on p. 108 (top row).

Photo Creditsxv

WONDROUS ONE SHEET

ORIGAMI MEENAKSHI MUKERJI Wondrous One Sheet Origami is a how-to book full of beautiful origami designs covering a wide range of folding levels from simple to high intermediate, with more emphasis on the latter. The book is meant for audiences 12 years of age and above, and children folding at higher than age level. Most of the designs are flat and suitable for mounting on cards or framing as gifts. Features • Richly illustrated full-color book with clear, crisp diagrams following international standard, and an abundance of photographs of finished models • Select designs hand-picked by the author based on social media responses • Most of the designs incorporate color-change, a technique showing both sides of paper for enhanced beauty “Meenakshi Mukerji’s work is both intricate and lovely. She’s greatly respected in the origami world, one of the well-known world leaders in modular origami. Her books offer a nice exposition of the mathematical elements, but you’re not being hit over the head with math lessons. You learn things without even realizing that you have.” – Dr. Robert J. Lang “Meenakshi Mukerji is one of today’s masters of modular origami, designs comprised of multiple pieces of paper. She also brings her ingenuity and creativity to designs made from a single piece of paper. Among the most appealing aspects of her single sheet work is the way she subtly manipulates a purely geometric form to fold a flower, a leaf, a butterfly, or card suits.” – Peter Engel

Meenakshi Mukerji is the winner of 2005 Florence Temko Award presented by OrigamiUSA. In 2007, her first book Marvelous Modular Origami was published followed in quick succession by six more books listed inside. She has been a featured artist and special guest at various origami conventions both in the USA and abroad. Born and raised in Kolkata, India, Meenakshi earned her BS in electrical engineering from the Indian Institute of Technology, Kharagpur, and a MS in computer science from Portland State University, Oregon, and has worked in the software industry.

RECREATIONAL MATHEMATICS

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