Six Simple Twists: The Pleat Pattern Approach to Origami Tessellation Design

130 Pages·2015·15.23 MB·English

by  Benjamin DiLeonardo-Parker

Recreational Math
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“Six Simple Twists is a gentle introduction to the vast, wide world of origami twist tes- e
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sellations that brings the reader through the successive stages of folding, understanding, a
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and, ultimately, designing origami geometric patterns based on the powerful hexagon grid o-
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system. The six twists of the title—interacting configurations of paper pleats—may be r
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simple in isolation, but Parker shows how they can be combined in uncountable variety. r
Parker presents more than a bare plan; he gives tips and techniques, guidelines and hints
using standard origami notation where it is useful, but also introducing new notations that
aid both understanding and folding. … photographs of the author’s own beautiful tessel-
lation creations … illustrate the concepts within the book and inspire the reader to her/his
own inventions.”
—Dr. Robert J. Lang, Origami Artist and Consultant, LangOrigami.com SIX SIMPLE
“… unlike more traditional forms of origami, [origami tessellation] does not have a consistent
system of diagramming or even a vocabulary of its own. In this book, Ben Parker addresses
this problem. The interested novice will find important information to get started folding
S TWISTS
tessellations … The advanced folder will find a thoughtful approach to defining and describing I
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the geometry of origami as revealed in tessellation. Whatever level of experience readers S
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bring to this book, they will find something to challenge them. … Ben brings artistry and an M
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analytic perspective to this subject, and provides much useful information and inspiration.”
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The Pleat Pattern Approach
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—Joel Cooper, Origami Artist

T
W to Origami Tessellation Design
Benjamin DiLeonardo-Parker is an origami pattern artist. He has taught and exhibited at
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origami conventions and art shows throughout the United States and at La Escula-Museo S
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Origami de Zaragosa, an origami museum in Spain. S
Benjamin DiLeonardo-Parker
K23496
www.crcpress.com
AN A K PETERS BOOK

SIX SIMPLE
TWISTS
The Pleat Pattern Approach
to Origami Tessellation Design

SIX SIMPLE
TWISTS
The Pleat Pattern Approach
to Origami Tessellation Design
Benjamin DiLeonardo-Parker
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
AN A K PETERS BOOK
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Version Date: 20150407
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2. How to Use the Six Simple Twists ........45
2.1 Intermediate Gridding .................46
2.2 Tessellation Basics, without
Folding ..........................................50
2.3 Applying Tessellation
Knowledge to the Paper .............51
2.4 Your First Tessellation ..................51
2.5 Your Second and Third
Tessellations ..................................57
2.6 Chirality .........................................64
2.7 Back-Twisting and
Releasing Pleats............................64
2.8 Pleat Flattening ............................67
2.9 Different Types of Pleat
Formats ........................................71
2.10 The Referential Method of
Tessellation Formation .................71
2.11 Flagstone Tessellations and
Contents
Offsetting the Pleats ...................74
2.12 Twist Decorations and
Progressions .................................80
Preface ........................................................VII 2.13 Twist Expansion ............................81
Acknowledgments ......................................IX
About the Author .......................................XI 3. Beyond the Six Simple Twists ...............89
3.1 The Next Step ...............................90
1. The Six Simple Twists ..............................1 3.2 Designing a New Tessellation .....94
1.1 The Basics and Preparation ...........2 3.3 Describing Twists .........................94
3.4 We Don’t Care about
1.2 How Tessellations Differ from
Traditional Folding .........................2 the Middle ..................................98
1.3 Fold Bias and Manipulation ...........3 3.5 Tips for Tricky Twists ....................98
1.4 How to Read the Exercises ............3 3.6 The Database ................................99
1.5 Obtaining a Hexagon .....................3
1.6 How and Why to Fold a Grid .........9 Final Thoughts...........................................113
1.7 Basic Pleat Design ........................13
For Further Study .....................................115
1.8 Diagrams of the Six
Simple Twists .............................21

Preface
What are origami tessellations? Why does there is a design and a method of folding, and
this book exist? the student learns that method, step by step.
Origami has risen in popularity around the Some folders go on to modify the design,
world in the past century. It started as a fun but many memorize and repeat. The latter
and educational pastime for schoolchildren extreme allows for a myriad of designs to be
(which is how most people encounter the craft) created through mathematical progression;
and grew into an exciting study for artists, however, all but the most ambitious folders
mathematicians, engineers, teachers, and any- may become paralyzed by the many options
one else with an inquisitive mind. These peo- or intimidated by the mathematics. Neither is
ple meld their varied backgrounds to explore superior to the other, but in my experience,
how the paper interacts with itself. This is a mixture of the two makes the activity acces-
often done through simple play. Origami sible to more folders.
artists certainly know about paper play. They So we look at those who have successfully
enjoy thinking, “If this is 1×1, let’s see what a designed a tessellation that was either never
2×1 looks like!” “I wonder what it would look folded before or was otherwise independently
like if this angle were 30° larger,” or simply discovered. How did they do it? Did they
“Let’s make it bigger!” There are many sub- stumble upon it through experimentation?
fields of study, and more units and models are Or did they use complex algorithms to figure
being discovered and published daily. it out? The truth is most likely a bit of both.
The study of origami tessellating is one An Internet image search for origami tessel-
branch of this medium. Tessellations are lations will yield varying results, from flagstone
patterns of polygons that tile regularly on a tilings to waterbomb-based tessellations, to
plane. Origami tessellations accomplish this corrugations, and other styles within the field.
using one sheet of paper without cuts, gen- If one were to try to study how all of these are
erally using pleats and twists, or mountain- done using one unifying technique, the task
ous corrugations. Oftentimes the paper is would be enormous and daunting, since they
folded into a grid, especially when a folder is are each distinct styles and the approach is
learning. Despite the rise in popularity of this different for each. I restrict my studies to similar
genre over the past decade, the study of tes- techniques and draw conclusions from those
sellations has been difficult to access for many progressions before figuring out how to spill
folders. I believe that this is due to lack of lit- over the concepts into other styles. I began
erature on the subject, the tedium of folding a studying waterbomb-based tessellations, which
grid (a valuable teaching and experimentation are mountainous single-layer patterns. After a
tool), and the many different types of tessella- time, I decided to switch and study something
tions from which to choose. completely different: what I call pleat patterns.
When I started attending origami conven- After years of design, it is time to write my
tions in 2007, there were few people folding thoughts down and describe how I view what
tessellations, and far fewer classes on the is happening when a tessellation is folded.
subject. The classes were either based around No one questions whether it is possible to fold
a procedure for folding a specific tessella- these patterns, because origami tessellations
tion or on the mathematical concept behind can be seen at art galleries and origami con-
a family of tessellations. These represent two ventions. The questions are, “How is it done?”
extremes. The first is an extension of tradi- and “What are the further possibilities?”
tional origami teaching methods in which This book provides some of those answers.
VII
Preface