METHODS AND INSTRUCTIONS

Here are diagrams that show, chapter by chapter, how to fold and further develop, if not all the models, then at least the basic ones. We sincerely hope that these explanations will help readers understand these works and fire their imagination.

LEGENDS

CHAPTER ONE

CONSTRUCTING WITH THE KNOT

1 THE PENTAGONAL KNOT (illustrations on pp. 24–25)

How to Make a Star

Make a pentagonal knot with a strip of paper, then refold one of the outer strips and insert it inside the knot. Now look at the pentagon you have made against the light. The layers of paper will be superimposed in regular sections, and you will be able to see a perfect star. After you fold the strip inwards, you can wrap it around the other strip and repeat the operation a number of times. You will end up with the bookmark shown in the illustration.

THREADING A STRIP THROUGH THE KNOT

Another way to obtain a regular star by layering paper is by threading a second strip through the knot. If you apply the same “knotted bookmark” method to two strips, you can create a ring. A decagonal ring similar to the one in diagram 7 was included in Kōji and Mitsue Fushimi’s volume, Origami no kikagaku [Geometric Origami], published by Nippon Hyōronsha in 1979. There, however, the ring was created from a single strip of paper in which the strips were alternately superimposed and knotted, while here two strips are used.

6 A RING OF PENTAGONAL STARS (illustrations on p. 24)

Make a knot with one of the two strips and wrap the other around it.

7 A RING OF DECAGONAL STARS (illustrations on p. 24)

Knot the two strips, overlapping one on top of the other.

2 CONSTRUCTING WITH MOLECULES

The Basic Structure of the Molecule

A molecule is made out of a long narrow sheet of paper. It is folded gradually at an angle so that it wraps around itself until the two ends are finally closed by being inserted into each other. Flat or solid shapes can be created by introducing thin strips of paper into the slots of the molecule. It is fun to use the molecule as a guide or as a central element for the creation of new forms.

1 THE SQUARE MOLECULE (illustrations on pp. 26, 28–31)

Planar Figures

Insert narrow strips of paper through the openings of the molecule. It is possible to come up with a variety of compositions by folding and interweaving these strips. Carefully calculate where the ends of the strips should begin and end.

HOW TO JOIN STRIPS

Solid Figures

Before inserting the strip, mark the fold lines. Illustrated here is the process for making a cuboctahedron.

COMPOSITION OF A BASIC CUBOCTAHEDRON

2 THE PENTAGONAL MOLECULE (illustrations on pp. 38–43)

There are two versions of the pentagonal molecule—with or without an opening at the center. The first is better suited for further development. An unbroken version can be achieved through the gradual insertion of strips.

3 THE HEXAGONAL MOLECULE (illustrations on pp. 27, 32–37)

The hexagonal molecule can be constructed with the same procedure used for the pentagonal molecule. Here, however, we prefer to adopta procedure that requires 2 strips of paper, as in the figure on the right. In this system, the impact of the paper’s thickness is reduced.

HOW TO CREATE THE MOLECULE

CHAPTER TWO

TESSELLATIONS

1 FLAT FOLDING A SQUARE (illustrations on pp. 17, 44–83, back endpapers)

The Twist

To make this type of fold, use two fingers to lift the sheet and make a crimp in each of the four directions, then flatten out by twisting the point of intersection. (It is actually more a question of lowering, or opening the intersection between the folds than one of flattening.) You can continue joining as you twist to the right and left.

Exercises with a Square Twist

TWIST TOWARDS THE RIGHT

TWIST TOWARDS THE LEFT

Exercises in Square Tessellation

Composing modules with a twist. Mark the folds needed to achieve the result you desire, then alternate modules that are twisted to the right and to the left. In this way you can come up with numerous different motifs.

Flattened squares should be arranged with maximum precision.

Unlimited Square Tessellation

This model is obtained by dividing it in half, and then in half again along the diagonal of each square.

2 FLAT FOLDING A TRIANGLE (illustrations on pp. 60–61, 64–67, 81)

The Twist

For the triangle, use 60 degree folds. The basic folds will be mountain folds alternating with valley folds. Lift the sheet with two fingers and make a crimp in three directions, then, twisting, flatten the triangle at the point of intersection. Each time, twist towards the right and towards the left. By doing so, you will end up with a motif composed of equilateral triangles.

Exercise with a Triangle Twist

Exercises in Triangular Tessellation

3 THE SPIRAL (illustrations on pp. 17, 44–48, 50–51, 80)

The Twist Superimposed

These are folds that are twisted more than once, that is, structures made from folds on which other, similar folds can be made. Thanks to these overlaps, the sheet wraps around itself, creating a spiral.

Double Twisted Exercises

Exercises with Double Twisted Spirals

CHAPTER THREE

INFINITE FOLDS

Four Types of Infinite Folds

Here we present four varieties of infinite folds known as the dovetail, the arrowhead, the double pleat, and the consecutive pleat. Within certain limits, the angles of the folds can be varied as desired. We therefore offer diagrams for four different angle choices for each variant. Each angle produces a different form. For the dovetail and arrowhead types, you will use grids with trisections and quadrisections as well.

VARIATIONS OF THE DOVETAIL

With a 45-degree angle, the faces will all be equal.

With an angle greater than 45 degrees, the smallest folds will end up inside the larger ones.

VARIATIONS OF THE ARROWHEAD

DOUBLE PLEATS

CONSECUTIVE PLEATS

THREE OR MORE PARTITIONS

Applying the Dovetail and Arrowhead Variations

With the variations of the dovetail and arrowhead, it is possible, to some extent, to vary the angle of the folds and the manner of joining the modules in order to create different motifs.

Arrowhead Lampshade (Fan) (illustrations on pp. 168–169)

With the variations of the dovetail and arrowhead, it is possible, to some extent, to vary the angle of the folds and the manner of joining the modules in order to create different motifs.

CHAPTER FOUR

REPETITIVE FOLDING

THE ZIGZAG (12 Angles) (illustrations on pp. 1, 114–123)

After making the valley and mountain folds indicated in the chart, begin from one end and gradually proceed to form a cylinder.

NOTE: The basic creases needed to create the patterns are indicated in the chart.

Applications of the Triangular Base

Diagram 1 shows the procedure for folding the triangular base. By repeating the folds in the illustrated grid, you can obtain the pattern known as “pineapple” or “sea cucumber,” which has been used by several origami artists, most notably Shuzo Fujimoto. By developing this scheme further, you can come up with the different motifs presented in this section and even create “infinite” folds. Either side can work as the front, but as we prefer the main side to be the one with the mountain fold, we have illustrated its fold lines here.

1 TRIANGULAR BASE

2 PINEAPPLE OR SEA CUCUMBER PATTERN (REPETITION OF DIAGRAM 10)

3 TRIANGULAR BASE FOR INFINITE FOLDS (MOUNTAIN AFTER MOUNTAIN) (illustrations on pp. 140–141)

4 TRIANGULAR BIFACIAL BASE (RED WAVE) (illustration on p. 153)

A model developed from a triangular base in which the front and back are identical.

The triangular base appears on both sides.

Other means of bisection are possible as well.

5 DOUBLE TRIANGULAR BASE (BRAIN) (illustrations on pp. 148–149)

Multiple Folds (illustrations on pp. 134–135)

Gradually draw together squares of uniform size, aligning their axes, then proceed by flattening the folds. Once the creases are flattened, twist to create the fan and fold the ends inside. The number of squares and creases depends on the time available.

Wave Series (illustrations on pp. 150–151)

Illustrated here are the basic folds. Fold the paper by combining the indicated folds.

CHAPTER FIVE

THE LIGHT BEHIND THE FOLDS

1 NON-FOLDABLE MODELS (illustrations on pp. 154–155)

Helical Columns with Square Base A, B and C

None of these three columns with square bases, obtained by varying the angle of the folds, can be resealed. Their height can vary according to your wishes. The top and bottom openings of the large models in the illustrations were obtained with the easiest of the systems, that is, by refolding the final level inwards. The number of levels in the fold grids and diagrams depicting the final result is not necessarily the same.

The Pyramid (illustrations on pp. 160, 161)

A simple form obtained by cutting. The base here is square but can also be triangular.

Square Section Columns with Fold-out A and B (illustrations on p. 162)

The fold-out portion becomes an unusual decoration for the column and can be made in a variety of ways.

The Hexagonal, Heptagonal, Octagonal, and Dodecagonal Dome (illustrations on p. 157)

Each of these domes uses the model known as the “box cover.” Different colors were used for the valley and mountain folds, but actually any of these lines can become a valley or mountain fold, with no distinction. In this model we chose to refold the base strip inwards in order to end up with a double layer of paper; but if you use thicker paper you may find that folding it a few centimeters inwards is sufficient. Folding at an angle of 45 degrees will make all these domes look very similar. Once you have made the folds, glue one edge to the opposite one to form a cylinder, then close it at the top.

Intertwining Zigzag Strips (illustrations on pp. 159, 160)

Here is how to make a solid by folding zigzag strips of paper and then intertwining them as you would to make a basket. The basic module consists of a triangle, either concave or convex, created from three strips. The structure will be perfectly sturdy even without glue.

How to Build a Solid Icosahedral Structure

Use 6 strips of 12 units apiece, as shown in the diagram. Instead of cutting the strip of paper, then folding it, it is more practical to make the folds first, then cut the segment with the portions that you need. Looking at the icosahedron from the perspective from which it is depictedin diagram 1, you may distinguish a kind of lateral band consisting of a decagonal ring. Slowly interlock 10 units of the strip around this ring. The remaining 2 units will overlap.

How to intertwine the strips into a zigzag.

Composition based on 3 strips.

The triangle can be concave or convex.

2 FOLDABLE MODELS

Pagoda (illustrations on p. 162)

In this model, the base of the lampshade is not resealable. Here is the crease pattern of the foldable part. After making the folds shown in the diagram, form the quadrangular pyramid and gradually fold the layers, starting from the base.

Taiko Bridge (illustrations on p. 160)

After gluing together the two shorter sides to create a tube with a square section, draw one side closer, then glue it.

Fold both ends inward.

Kabura - Decagono (illustrations on pp. 166–167)

After making the folds indicated in the grid, glue the ends to form a cylinder. Starting at both ends, flatten the folds, level after level.

The number of levels is dictated by preference.

Foldable Lantern (illustrations on p. 162)

After making the folds indicated in the grid, close the cylinder and glue. Then flatten the folds, proceeding gradually from both ends. Choose whatever size and number of levels you desire. Flatten inwards from time to time, in keeping with the opening.

The pentagonal lantern as it is closed and opened.

Watering Can—Dodecagon (Muku Muku)(illustrations on p. 163)

Slowly twist each level in the opposite direction to the previous level. As it grows taller, the top will lean slightly to one side, a peculiar feature of this model.

Drop—Decagon(illustration on p. 4)

The two modules can be inserted into each other once the indicated folds have been made. After gluing the model into a decagonal cone, gradually flatten each level starting from the base. To create the shape of a spindle, set the bases of two mirror models side by side and insert the tabs into the incisions indicated by the bold lines.

Choose however many levels you like.

The fold grid includes the edges meant to be glued.