Robert.J.Lang.-.Origami.and.Geometric.Constructions.(article).pdf

Lang, Origami and Geometric Constructions

Origami and Geometric Constructions

By Robert J. Lang

Introduction....................................................................................................................................2

Preliminaries and Definitions .......................................................................................................2

Binary Divisions .............................................................................................................................4

Binary Folding Algorithm.........................................................................................................5

Binary Approximations .............................................................................................................9

Rational Fractions........................................................................................................................11

Crossing Diagonals...................................................................................................................12

Fujimoto’s Construction..........................................................................................................15

Noma’s Method ........................................................................................................................18

Haga’s Construction ................................................................................................................20

Irrational Proportions .................................................................................................................22

Continued Fractions ................................................................................................................22

Quadratic Surds .......................................................................................................................26

Angle Divisions .........................................................................................................................31

Axiomatic Origami.......................................................................................................................37

Preliminaries.............................................................................................................................39

Folding.......................................................................................................................................41

Alignments ................................................................................................................................41

Bringing a point to a point

P ↔ P

.....................................................................................42

Bringing a point onto a line (

P ↔ L

)..................................................................................42

Bringing one line to another line (

L ↔ L

) .........................................................................42

Alignments by folding..............................................................................................................43

Multiple Alignments ................................................................................................................44

Constructability........................................................................................................................44

Axiom 6 and Cubic Curves .....................................................................................................45

Approximation by Computer..................................................................................................49

References.....................................................................................................................................53

Lang, Origami and Geometric Constructions

Introduction

Compass-and-straightedge geometric constructions are familiar to most students from high-

school geometry. Nowadays, they are viewed by most as a quaint curiosity of no more than

academic interest. To the ancient Greeks and Egyptians, however, geometric constructions were

useful tools, and for some, everyday tools, used for construction and surveying, among other

activities.

The classical rules of compass-and-straightedge allow a single compass to strike arcs and

transfer distances, and a single unmarked straightedge to draw straight lines; the two may not be

used in combination (for example, holding the compass against the straightedge to effectively

mark the latter). However, there are many variations on the general theme of geometric

constructions that include use of marked rules and tools other than compasses for the

construction of geometric figures.

One of the more interesting variations is the use of a folded sheet of paper for geometric

construction. Like compass-and-straightedge constructions, folded-paper constructions are both

academically interesting and practically useful—particularly within origami , the art of folding

uncut sheets of paper into interesting and beautiful shapes. Modern origami design has shown

that it is possible to fold shapes of unbelievable complexity, realism, and beauty from a single

uncut square. Origami figures posses an aesthetic beauty that appeals to both the mathematician

and the layman. Part of their appeal is the simplicity of the concept: from the simplest of

beginnings springs an object of depth, subtlety, and complexity that often can be constructed by

a precisely defined sequence of folding steps. However, many origami designs—even quite

simple ones—require that one create the initial folds at particular locations on the square:

dividing it into thirds or twelfths, for example. While one could always measure and mark these

points, there is an aesthetic appeal to creating these key points, known as reference points, purely

by folding.

Thus, within origami, there is a practical interest in devising folding sequences for particular

proportions that overlaps with the mathematical field of geometric constructions. Within this

article, I will present a variety of techniques for origami geometric constructions. The field is

rich and varied, with surprising connections to other branches of mathematics. I will show

origami constructions based on binary divisions, and then show how these can be extended

construction of proportions that are arbitrary rational fractions. Certain irrational proportions are

also constructible with origami; I will present several particularly interesting examples. I’ll then

turn to the topic of approximate folding sequences, which, though perhaps not as mathematically

interesting, are of considerable practical utility. Along the way, I’ll present the axiomatic theory

of origami constructions, which not only stipulates what classes of proportions are foldable, but

also provides the basis for finding extremely efficient approximate folding sequences by

computer solution—a technique that has found application in a number of published origami

books of designs.

Preliminaries and Definitions

Origami, like geometric constructions, has many variations. In the most common version, one

starts with an unmarked square sheet of paper. Only folding is allowed: no cutting. The goal of

origami construction is to precisely locate one or more points on the paper, often around the

edges of the sheet, but also possibly in the interior. These points, known as reference points , are

then used to define the remaining folds that shape the final object. The process of folding the

model creates new reference points along the way, which are generated as intersections of

creases or points where a crease hits a folded edge. In an ideal origami folding sequence —a step-

Lang, Origami and Geometric Constructions

by-step series of origami instructions—each fold action is precisely defined by aligning

combinations of features of the paper, where those features might be points, edges, crease lines,

or intersections of same.

Two examples of creating such alignments are shown in Figures 1 and 2. Figure 1 illustrates

folding a sheet of paper in half along its diagonal. The fold is defined by bringing one corner to

the opposite corner and flattening the paper. When the paper is flattened, a crease is formed that

(if the paper was truly square) connects the other two corners.

1. Fold the bottom right

corner up to the top left.

2. Unfold.

3. “Fold and unfold” is

indicated by a double-

headed arrow.

Figure 1. The sequence for folding a square in half diagonally.

As a shorthand notation, the two steps of folding and unfolding are commonly indicated by a

single double-headed arrow as in the third step of Figure 1.

Figure 2 illustrates another way of folding the paper in half (“bookwise”). This fold can be

defined in 3 distinct, but equivalent ways:

(1)

Fold the bottom left corner up to the top left corner.

(2)

Fold the bottom right corner up to the top right corner.

(3)

Fold the bottom edge up to be aligned with the top edge.

For a square, these three methods are equivalent. However, if you start with slightly skew paper

(a parallelogram rather than a square), you will get slightly different results from the three.

Lang, Origami and Geometric Constructions

1. Fold the bottom edge up

to the top edge.

2. Unfold.

3. The new crease defines

two new points.

Figure 2. The sequence for folding a square in half bookwise.

In both cases, if you unfold the paper back to the original square, you will find you have created

a new crease on the paper. For the sequence of figure 2, you will also have now defined two new

points: the midpoints of the two sides. Each point is precisely defined by the intersection of the

crease with a raw edge of the paper.

These two sequences also illustrate the rules we will adopt for origami geometric constructions.

The goal of origami geometric constructions is to define one or more points or lines within a

square that have a geometric specification (e.g., lines that bisect or trisect angles) or that have a

quantitative definition (e.g., a point 1/3 of the way along an edge). We assume the following

rules:

(1)

All lines are defined by either the edge of the square or a crease on the paper.

(2)

All points are defined by the intersection of two lines.

(3)

All folds must be uniquely defined by aligning combinations of points and lines.

(4)

A crease is formed by making a single fold, flattening the result, and (optionally)

unfolding.

Rule (4), in particular, is fairly restrictive; it says that folds must be made one at a time . By

contrast, all but the simplest origami figures include steps in which multiple folds occur

simultaneously. Later in this article, I will discuss what happens when we relax this constraint.

Binary Divisions

One of the most common origami constructions that turns up in practical folding is the problem

of dividing one or both sides of the square into N equal divisions, where N is some integer.

Figure 2 illustrated the simplest case—dividing the edge of a square into two parts—and its

solution. Of course, this method is not restricted to a square; it works equally well on any line

segment in a square. Thus, the two halves of the square may be individually divided into two

parts, and so on. By repeatedly dividing the segments in half, it is possible to divide the edge of a

square (or rectangle) into 4ths, 8ths, and so forth, as shown in Figure 3.

Lang, Origami and Geometric Constructions

Division into 4ths.

Division into 8ths.

Division into 16ths.

Figure 3. Division of a square into 4ths, 8ths, and 16ths.

This method allows us to divide a square into proportions of 1/2, 1/4, 1/8,…and in general,

1/2

n of the side of the square. By scaling all numbers to the size of

the square, we can say we have constructed the fraction

1/2

n , where the fraction is given in terms

of the side of the square.

n . (In all the

discussion that follows, we will consider only fractions between 0 and 1.) The most direct

method is to subdivide the edge of the square completely into

m /2

n for any integer

m < 2

n ths, then count up m divisions

creases, and is not very efficient, because

completely subdividing the square results in the creation of many unnecessary creases. There is

an elegant method for constructing any fraction of this type that uses the minimal number of

folds. A rational fraction whose denominator is a perfect power of two is called a binary

fraction ; the folding method is called the binary folding algorithm .

− 1

Binary Folding Algorithm

The binary folding algorithm was described by Brunton [1] and expanded upon by Lang [2]. It

produces an efficient folding sequence to construct any proportion that is a binary fraction and is

based on binary notation. In binary notation, there are only two digits, 1 and 0; all numbers are

written as strings of ones and zeros. Any number can be written in binary notation as a string of

ones and zeros. For example, the numbers 1 through 10 can be written in binary as shown in

Table 1.

for integer n . Each division is

It is also possible to construct a fraction of the form

from the bottom. This method clearly requires

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