Curved-folding origami design

123 Pages·2019·29.92 MB·english

by  Mitani

CURVED-FOLDING
ORIGAMI DESIGN

CURVED-FOLDING
ORIGAMI DESIGN
Jun Mitani
CRC Press
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Names: Mitani, Jun, 1975- author.
Title: Curved-folding origami design / Jun Mitani
Other titles: Curved folding origami design
Description: Boca Raton, Florida : CRC Press, c2019. | Includes
bibliographical references and index.
Identifiers: LCCN 2018060524| ISBN 9780367180256 (pbk. : alk. paper) | ISBN 9780367180270 (hardback : alk. paper) |
ISBN 9780429059179 (ebook : alk. paper)
Subjects: | LCSH: Origami. | Folds (Form) | Paper work |
Classification: LCC TT872.5 .M5644 2019 | DDC 736/.982--dc23
LC record available at https://lccn.loc.gov/2018060524
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Contents
Preface ........................................................VII 5. Folding Cones ........................................57
Prologue: Folding a Curve Means a Lot .....IX 5.1 Making a Cone without Cut .....57
Author ........................................................XIX 5.2 Making Conjoined Cones .............60
5.3 Making a Partial Cone ..................62
Appendix: Folding Back a Cone .........66
1. Fold a Curve ............................................1
1.1 Folding a Simple Curve ..................1
6. Folding Back at Straight Line ...............69
1.2 Folding a Winding Curve ...............4
1.3 Folding a Squiggle .........................6 6.1 Combining Straight and
Curved-Fold Lines .........................69
6.2 Aligning Straight and Curved
2. Aligning Curves .....................................13
Lines Alternately ...........................71
2.1 Pleat-Folding Two Aligned 6.3 Aligning Line-and-Curve Pairs
Curves ............................................13 Rotationally Symmetrically ..........72
2.2 Aligning Many Curves ..................16
2.3 Aligning Curve and Reversed
7. Other Techniques ..................................77
Curve .............................................20
7.1 Providing Projections and
Hollows ..........................................77
3. Aligning Rotated Curves ......................35
7.2 Folding Cylinders ..........................77
3.1 Making Rotationally Symmetric 7.3 Laying Out Alternatingly.............80
Structures ......................................35 7.4 Folding Closed Curves .................83
3.2 Decentering ..................................38 Appendix: Example of Whipped
3.3 Aligning Whirls .............................38 Cream .........................................88
Appendix: Is Paper Really Inelastic? .....43
Afterword ....................................................97
4. Tucking ..................................................45
Index ............................................................99
4.1 Tucking One Line .........................45
4.2 Tucking Multiple Curves...............47
4.3 Repeating Tucking .......................48
V
Contents

Preface
It hink most of you enjoyed origami when computer-based designing is a high hurdle
you were kids. The most basic origami is to and not an easy method. Do we really need
fold a sheet flat along a straight line, corner a computer to make shapes by folding paper
to corner or side to side. If you were to “fold along curves? I am a computer science
this paper as you like,” you would start with researcher, but I gave it a try with computer-
folding it flat. It is widely considered that paper free curved-fold origami. Paper can be folded
should be tightly folded. However, paper along curves relatively freely, but not that freely.
is bendable. The paper can also be folded Most randomly drawn curves do not fold well.
along a curve by adding a fold on its bended, To obtain beautifully designed modelings, we
curved surface. This “curved folding” has not at least need knowledge about how fold lines
drawn much attention. However, the surface are laid out. This book presents shapes that are
of the paper created from curved folding formed by curve-folding paper, together with
produces geometric, elegant shading. I have fold line patterns. Its contents are a record of
created a number of geometric solid objects fun experiments from my own real experience
through computer-calculated designing of with sheets of paper. You will be able to create
shapes obtained from folding curves. Those a variety of shapes by combining the fold
techniques are organized in the book titled line patterns in this book. I hope to share the
3D Origami Art (CRC Press, 2016). Fortunately, beauty of curved surfaces created by paper
that book inspired many people to know with as many people as possible.
the world of curved-fold origami. At the
same time, it made me recognize again that Jun Mitani
VII
Preface

Prologue: Folding a Curve
Means a Lot
P.1 Shapes from Bending and tangent surfaces. These are called “devel-
opable surfaces”. A developable surface is
Paper
a shape consisting of lines. Each shape in
Figure P.2 is also an assembly of lines, though
First, let’s be clear about the difference
they are not visible. So, a ruler tightly fits on
between “bending” and “folding” a sheet of
its surface. Figure P.3 illustrates the consisting
paper. Bending is to change the entire paper
lines of the three types of developable sur-
into a smooth shape without creating edges,
faces. By actually bending a sheet and closely
as in Figure P.1.
observing its surface, you can roughly estimate
This produces a smoothly curved surface.
how the lines are arranged. And, thinking the
However, the bent shape quickly opens back
direction of these lines are very important in
flat if it is not held or taped due to paper’s
creating a curved surface out of a sheet.
ability to return to its original state. The end
There are only three types of developable
result is the shape balanced between this
surfaces. However, you can make shapes with
property of the paper and the fixed point. A
different types of developable surfaces, as
somewhat thick sheet produces a beautifully
seen in Figure P.4, by smoothly connecting a
curved, tense surface with no deflection.
curved surface with another curved one or a
Paper is inelastic and does not deform
curved surface with a plane. The left is esti-
freely like a rubber sheet. There are only three
mated to be a shape that combines a cylinder
known curved surfaces that can be made from
bending a sheet of paper—cone, cylinder,
Figure P.1 Bent paper. Figure P.2 Paper bent into a cone and a cylinder.
IX
Prologue: Folding a Curve Means a Lot