Six Simple Twists The Pleat Pattern Approach to Origami Tessellation Design

Part I

1.00 - Why Study Pleat Patterns?
1.01 - Basics and Preparation
1.02 - How Pleat Patterns Differ from Traditional Origami
1.03 - How to Read the Diagrams and Fold Parity
1.04 - Folding Uniform Parallel Creases
1.05 - Grid Axes and How to Fold a Hexagon
1.06 - How to Fold a Triangle Grid
1.07 - Simple Pleat
1.08 - The Six Simple Twists
1.09 - Triangle Twist
1.10 - Triangle Spread
1.11 - Hex Twist
1.12 - Hex Spread
1.13 - Rhombic Twist
1.14 - Arrow Twist
1.15 - Anatomy of a Molecule
1.16 - Pleat Intersection Notation

Part II

2.00 - How to Use the Six Simple Twists
2.01 - 32nd’s Grid
2.02 - Locking and Unlocking Pleats
2.03 - Triangle Twist Tessellation
2.04 - 3.6.3.6 Tessellation
2.05 - Tessellation Basics
2.06 - Applying Tessellation Knowledge Folding
2.07 - Triangle Weave Tessellation
2.08 - 3.6.3.6 Weave Tessellation
2.09 - 6.6.6 Hexagonal Failing Cluster
2.10 - Modifications
2.11 - Backtwisting
2.12 - Twist Handedness and Pleat Symmetry
2.13 - Pleat Flattening
2.14 - Triangle Twist Tessellation with Flattened Pleats
2.15 - Hidden Circles Pattern
2.16 - Rhombic Twist Tessellation
2.17 - Rhombic Twist Variants
2.18 - Twist Sinking
2.19 - Twist Expansion
2.20 - Nub Offset Tessellation
2.21 - Shift Rosette Tessellation
2.22 - Ridge Creation
2.23 - Button Molecule
2.24 - Button Molecule Tessellation
2.25 - Triangle Flagstone Tessellation and Offsetting Pleats
2.26 - 3.6.3.6 Flagstone Tessellation
2.27 - Crooked Split
2.28 - Snowflake Tessellation
2.29 - Tulip Split
2.30  - Tulip Split Tessellation
2.31 - Molecule Size and Different Grid Densities
2.32 - “Front” and “Back” Sides
2.33 - Tendril Tessellation
2.34 - Inverting a Pleat
2.35 - Iso-Area Triangle Twist Tessellation
2.36. - Pleat Pushing
2.37 - Platform Tess
2.38 - Triple Twist Tess

Part III

3.00 - Pleat Patterns as Artwork
3.01 - Gallery
3.02 - Pleat-to-Molecule Analysis
3.03 - Twist Archetype Sets
3.04 - Molecule Database
3.05 - Archetype Composition
3.06 - Actions and Notation
3.07 - Splitting Equation
3.08 - Normal Polygon Models
3.09 - Circle Cutout Model
3.10 - Molecule-to-Pleat Analysis
3.11 - Sectioning Model of Perfect Twist Design
3.12 - Brocard Points

Final thoughts

Pleat Notation Thoughts By Matthew Benet

Glossary

Biography

Born in Pittsburgh and living in Connecticut, Benjamin DiLeonardo-Parker has an active student of origami tessellations since 2007. He has taught and exhibited at origami conventions and art shows internationally, including Chi Mei Museum (Tainan City, Taiwan), La Escuela-Museo Origami de Zaragosa (Zaragosa, Spain), The Science Museum Oklahoma (Oklahoma City, OK), the Museum of Mathematics (New York, NY), the Japan Information and Culture Center (Washington D.C.), The New Britain Museum of American Art (New Britain, CT), and the Cooper Union Gallery (New York, NY).

Outside of art, Ben teaches high school mathematics to students with uncommon learning styles, and incorporates origami into his classes as often as he can. Ben approaches his artwork from a holistic standpoint, preferring to view origami as an entry into the vast network of disciplines to which it is connected. This has led him to extend his knowledge of education, engineering, mathematics, CNC fabrication, paper arts, fashion, alternative photography, and other studies. When not teaching high school math, Ben operates a workshop in Essex, CT out of which he creates artwork and runs classes on origami design,

He views his practice of origami as cyclical and recursive. Origami is connected to such as vast network of disciplines, each with its own siren’s call. Each flavor, each culture, cycles back onto its own basics over and over, swirling and interacting with previous knowledge, each enhancing the others in some way.

"The second edition of Six Simple Twists is a marvelous expansion of the original; like the first edition it provides an introduction to the vast, wide world of origami twist tessellations, with a focus on those based on a hexagonal grid. Through copious and detailed photographs, he shows not just the structure, but how individual folds are actuated, and then how they may be combined in beautiful ways. An expanded selection of photographs of works by worldwide artists provides both inspiration and challenge for the budding tessellator and experienced artist alike."
– Robert Lang, author of Origami Design Secrets: Mathematical Methods for an Ancient Art

"In this wonderfully expanded new edition, Ben Parker not only describes the basic elements composing his intricate pleated paper artworks, he also unlocks their secrets and logic and teaches their molecules and several examples in a clear format, allowing the reader to create new folded patterns, through both practical and technical approaches!"
– Alessandro Beber, author of Origami New Worlds

"This new edition of Six Simple Twists is a tremendous expansion of the first edition. Through the thorough investigation of the foundational elements of origami tessellation, Ben has laid the groundwork for understanding design and application of geometry in paper. The second edition expands and refines the insights of the first with copious illustrations and examples, clear definitions and thought provoking exercises. For anyone fascinated by the geometry of origami, or even curious about place of origami in the field of geometry, this edition is absolutely necessary."
– Joel Cooper, Origami artist

"Ben's second edition is full to the brim with new and exciting insights into the craft and mechanisms of origami tessellations, extending the first edition by a couple hundred pages. He provides extensive explanations and deep insights into the maths behind the scenes and accompanies it with a plethora of projects for you to fold at home."
– Peter Keller, Origami artist

"As soon as we open the pages of this book, Benjamin DiLeonardo-Parker’s enthusiasm for the design and creation of origami tessellations is clear. The author reveals the patterns that can be formed by paper pleats and their combinations and demonstrates how by folding paper in different configurations the beautiful structures known as origami tessellations can be constructed. The author explains to his readers how to fold tessellations and encourages them to apply their learning to explore the subject to generate their own folding techniques and create their own designs, offering case studies to experiment with. The book is generously supplied with photographs and diagrams illustrating each step of the techniques, and a comprehensive glossary is provided. The author’s creativity and attention to detail are a delight. This second edition is a greatly expanded version of its predecessor [1], benefiting from more instruction in basic skills and pleat manipulation methods, enhanced illustration of the processes, a greater range of demonstrated patterns, an improved notation system, and further mathematical analysis of the behaviour of twists.

[. . .] The thorough, detailed, methodical treatment of this subject will surely inspire readers of this book to try out the techniques for themselves, and to embark upon creating many of the tessellations described here, and, with experience gained, even attempting their own tessellation designs. I certainly feel encouraged to reach for some paper and get folding."
– Dorothy Leddy, London Mathematical Society