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Origami Axioms and Mathematical Constructions

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Origami Axioms and Mathematical Constructions:
A History and Exploration
Derek Chen
Maggie L. Walker Governor’s School
ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND
EXPLORATION
2
Abstract
The development and application of a mathematically rigorous theory of origami has been a very
recent phenomenon, considering the historical origins of origami and paperfolding in general.
Many civilizations discovered paperfolding as a recreational activity soon after the invention of
paper thousands of years ago, while in contrast, the upsurge in popularity of the discipline, as
well as the rise in mathematical interest surrounding origami, have all taken place largely since
the mid-20th century. Among the most extensively studied mathematical interpretations of
origami are the seven Huzita-Hatori or Huzita-Justin Axioms, which describe the full set of
single folds possible on a plane of paper. The operations described by these ‘axioms’ are similar
to those allowed with a compass and straightedge – they provide a framework under which
certain numbers are constructible. One particular Huzita-Hatori axiom, the sixth, is equivalent to
finding a common tangent of two distinct parabolas in the plane, which may have up to three
solutions. As a result, single-fold origami constructions are more powerful than compass and
straightedge constructions, allowing for a larger set of “origami-constructible” numbers than just
the Euclidean or Thalian numbers. Thus, many problems that stymied the Ancient Greeks, such
as doubling the volume of a given cube, solving a general cubic equation, and trisecting an
arbitrary angle, are possible with origami. Even more powerful operations are possible if
multiple simultaneous folds are allowed, paving the way to solving polynomials of any degree
and constructing regular polygons of increasing size. However, even the single-fold axioms
present an intriguing take on paperfolding that offers concrete solutions to famous mathematical
problems.
ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND
EXPLORATION
Table of Contents
I. Introduction
4
II. Definitions
5
III. The Huzita-Hatori Axioms
6
IV. Geometric and Algebraic Consequences
8
3
IV.i The Construction of √2
9
IV.ii The Trisection of an Angle
12
IV.iii The Real Solutions of a Cubic
13
V. Two-Fold Axioms and More
16
Bibliography
18
3
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I. Introduction
The origins of paperfolding (“origami”) trace their roots to the invention of paper, with
various traditions beginning contemporaneously in many regions of Eastern Asia (Hatori, n.d.).
Origami, the Japanese offshoot of this phenomenon, began following the arrival of paper in the
archipelago, introduced by Buddhist monks during the 6th century C.E. Paperfolding appeared in
Europe during or prior to the 15th century, though it may have been discovered much earlier by
the Moors. In the early 20th century, origami was popularized as a tool for child development and
education, and colorful kami was used to teach origami in the Bauhaus design school in
Germany.
Origami underwent a renaissance in the middle of the 20th century thanks to the
contributions of Akira Yoshizawa, an origami designer and author. In 1954, Yoshizawa
introduced a notational system to indicate origami folds which evolved into the internationally
recognized Yoshizawa-Randlett system still used today (Fox, 2005). Further publications by
other authors popularized the tradition immensely, and the foundation of origami societies like
OrigamiUSA and an increasing online presence have resulted in a substantive rise in awareness
of the craft worldwide.
Along with this revolution came a realization of the mathematical contexts of
paperfolding. For much of its existence, origami has been based on linear folds – only recently
have innovations in wet-folding and techniques employing curved folds begun to depart from the
rigidly defined constructions of ‘pure’ origami. Like the compass and straightedge, linear
origami presents a method of constructing numbers, geometries, and algebraic analogues (i.e.
finding real roots of polynomials). However, the seven Huzita-Hatori axioms that describe
paperfolding are more powerful than the compass and straightedge (Hull, 2011). Namely,
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paperfolding under the Huzita-Hatori axioms allows for the trisection of angles, doubling the
cube, and solving general cubic equations (Alperin & Lang, 2006).
II. Definitions
Point
A point (𝑥, 𝑦) is an ordered pair of Cartesian coordinates in two dimensions.
Line
A line defined by the ordered pair (𝑋, 𝑌) is the set of points that satisfy 𝑋𝑥 + 𝑌𝑦 + 1 = 0. This
definition of a line gives a unique representation of every describable line, except for lines that
pass through the origin. This problem is resolved by translating any given system so no such
lines exist.
Folded Point
The folded image 𝐹𝐿𝐹 (𝑃) of a point 𝑃 = (𝑥, 𝑦) across a line 𝐿𝐹 is the reflection of point 𝑃 across
𝐿𝐹 . The coordinates of 𝐹𝐿𝐹 (𝑃) can be expressed as (
𝑥(𝑌 2 −𝑋 2 )−2𝑋(1+𝑦𝑌) 𝑦(𝑋 2 −𝑌 2 )−2𝑌(1+𝑥𝑋)
,
𝑋 2 +𝑌 2
𝑋 2 +𝑌 2
).
Folded Line
The folded image 𝐹𝐿𝐹 (𝐿) of a line 𝐿 = (𝑋, 𝑌) across fold line 𝐿𝐹 = (𝑋𝐹 , 𝑌𝐹 ) is the reflection of 𝐿
𝑥(𝑋𝐹 2 −𝑌𝐹 2 )+2𝑋𝐹 𝑌𝑌𝐹
across 𝐿𝐹 . The coordinates of 𝐹𝐿𝐹 (𝐿) are given by (𝑋
𝐹
2 −2𝑋𝑋
𝐹 −2𝑌𝑌𝐹 +𝑌𝐹
2
𝑦(𝑋𝐹 2 −𝑌𝐹 2 )−2𝑋𝑋𝐹 𝑌𝐹
,𝑋
𝐹
2 −2𝑋𝑋 −2𝑌𝑌 +𝑌 2
𝐹
𝐹
𝐹
).
Note that folding is the inverse operation of itself – that is, the folded image of the folded image
of a point 𝑃 or line 𝐿 under an arbitrary fold line 𝐿𝐹 is itself (𝐹𝐿𝐹 (𝐹𝐿𝐹 (𝑃) = 𝑃 and 𝐹𝐿𝐹 (𝐹𝐿𝐹 (𝐿) =
𝐿) (Alperin & Lang, 2006).
Point Alignment
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In general, an alignment, or incidence, between two objects 𝐴 and 𝐵 is notated as 𝐴 ↔ 𝐵. Two
points, 𝑃1 = (𝑥1 , 𝑦1 ) and 𝑃2 = (𝑥2 , 𝑦2 ) are said to be aligned, i.e., 𝑃1 ↔ 𝑃2 , iff 𝑥1 = 𝑥2 and 𝑦1 =
𝑦2 .
Line Alignment
Two lines 𝐿1 = (𝑋1 , 𝑌1 ) and 𝐿2 = (𝑋2 , 𝑌2 ) are aligned (𝐿1 ↔ 𝐿2 ) iff 𝑋1 = 𝑋2 and 𝑌1 = 𝑌2 .
Point-Line Alignment
A point P=(x,y) and a line L=(X,Y) are aligned (P↔L) iff xX+yY+1=0. That is, P and L are
aligned if and only if P is a point whose coordinates satisfy the equation defining line L (P is on
line L).
III. The Huzita-Hatori Axioms
The seven elementary single-fold operations in origami were discovered first by Justin,
crediting Peter Messer in the publication (Alperin & Lang, 2006). In 1989, Huzita again
introduced six of the seven known axioms, defining possible folds given an initial set of points
(in context, intersections of creases or where a crease meets the edge of the paper) and lines
(creases, paper edges, etc.). These six axioms have been shown to encompass all Euclidean
constructions possible with a straightedge and compass – Martin further showed that Huzita’s
sixth axiom alone allowed all constructions possible by the first six operations (Alperin R. C.,
2000). On the other hand, Auckly and Cleveland showed that the set of constructible numbers
from the first 4 axioms is smaller than that of compass and straightedge construction (Alperin &
Lang, 2006). In 2001, Hatori rediscovered the existing seventh axiom (first introduced by Justin
two decades prior), completing the set of single-fold operations. It does well to note that the first
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five “axioms” are all constructible using solely the sixth, but following nomenclatural precedent
the set of seven operations are all named the Huzita-Hatori or Huzita-Justin Axioms, as follows:
1. Given two points P1 and P2, there exists a unique fold that connects them.
Axiom 1 is the operation creating a line that passes through two points. This operation has
exactly one solution.
2. Given two points P1 and P2, there exists a unique fold that aligns P1 with P2.
The second axiom describes the operation that folds the perpendicular bisector of P1P2.
Again, there is always exactly one solution to this axiom.
3. Given two lines L1 and L2, there is a fold that aligns L1 with L2.
Axiom 3 is equivalent to finding an angle bisector of the angle between two lines. Note that
when L1 and L2 are parallel, this operation has only one solution, and it has two solutions
given skew lines.
4. Given a point P and a line L not aligned with P, there exists a unique fold that passes
through P and is perpendicular to L.
Axiom 4 describes the construction of a perpendicular passing through a point. There exists
one solution for this operation.
5. Given two points P1 and P2 and a line L, there exists a fold that aligns P1 with L and that
passes through P2.
The fifth axiom may have 0, 1, or 2 distinct solutions. The operation described finds the
point(s) of intersection of a circle with center P2 passing through P1 and the line L.
6. Given two points P1 and P2, and two lines L1 and L2, there exists a fold that aligns P1 with
L1 and aligns P2 with L2.
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The sixth operation finds a line simultaneously tangent to two parabolas, with foci at P1 and
P2, and directrices defined by L1 and L2. An expanded discussion on this interpretation of
HHA 6 may be found in Section IV.iii. Since two parabolas in the plane have at most three
distinct lines of tangency, this axiom has 0, 1, 2, or 3 solutions, allowing for the solving of
cubic equations.
7. Given a point P and two lines L1 and L2, there exists a fold that aligns P with L1 and is
perpendicular to L2.
The final axiom has no concise geometric equivalent other than the operation described. It
may have 0 solutions (i.e. if L1 and L2 are parallel and P is not already aligned with L1) or 1
solution. It has been shown that, though this is not equivalent to any of the previous six
operations, HHA 7 does not expand the set of origami-constructible numbers (Alperin &
Lang, 2006).
IV. Geometric and Algebraic Consequences
As discussed, the seven Huzita-Hatori Axioms (HHAs) permit the construction of all
numbers and figures possible using a compass and straightedge (Alperin R. C., 2000). However,
it has been shown that several constructions not possible with the compass and straightedge are
possible with origami – for example, constructing the cube root of 2 and trisecting an arbitrary
angle (Lee, 2017). There exist limitations to the HHAs, among them the quintisection of an angle
and the construction of the endecagon, as well as the solving of quintic equations (Alperin &
Lang, 2006). All of these constructions are possible under origami, though they require
operations outside the HHAs, specifically the simultaneous creation of two folds.
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It has been proven that the seven HHAs cover all possible single-fold operations in the
plane through exhaustive enumeration of all possible alignments of points and lines (Alperin R.
C., 2000). For two-fold operations, there exist 489 distinct operations, which lie firmly outside
the scope of this paper. Further, the definition of three-fold operations allows for the solving of
quintic operations – as a general rule, an nth-degree general equation may be solved with
operations of n-2 folds (Alperin & Lang, 2006).
Noting the limitations of the HHAs, they nonetheless represent a more powerful set of
operations than those defined for the straightedge and compass. A discussion of several
applications of the HHAs follows: the doubling of the cube, the trisection of an angle, and the
solving of cubic equations.
3
IV.i The Construction of √2
The construction of the cube root of 2 is an ancient problem – that of doubling the
volume of a cube, which was tackled by the Ancient Greeks with compass and straightedge
operations, which are not powerful enough to complete the construction (Hull, 2011). However,
this problem is possible under one-fold planar origami, as described here.
To begin, consider the Beloch Square. Given two distinct points P1 and P2, and two lines,
L1 and L2, the Beloch Square is a square formed by points WXYZ such that adjacent corners X
and Y are aligned with L1 and L2, respectively, and the two parallel lines defined by WX and YZ
are aligned with P1 and P2, respectively. Such a square can be constructed with origami. First,
two lines L1’ and L2’ are constructed, such that L1’ is parallel to L1 and L1 is equidistant to P1
and L1’, and L2’ is parallel to L2 and L2 is equidistant to P2 and L2’. Then, the sixth Huzita-Hatori
operation is performed, aligning P1 with L1’ and P2 with L2’. The fold line created, LF, intersects
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L1 and L2 at points X and Y, respectively. Further, since L1 is halfway between P1 and L1’, X is
collinear with P1 and its folded image, and thus similarly, Y is collinear with P2 and its image.
Further, LF is necessarily perpendicular to the line segments connecting P1 and P2 with their
folded images, and as a result, we know P1X and P2Y are parallel and both perpendicular to XY.
Thus, we let XY be one side of the Beloch Square, with adjacent sides to XY aligned with P1X
and P2Y.
Fig. 1: The Beloch Square and its Construction
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The construction of Beloch’s Square can be applied to the construction of the cube root
of 2. In this case, we define L1 to be the y-axis and L2 to be the x-axis. Further, we let P1 = (-1,0)
and P2 = (0,-2). Consequently, L1’ is the line x = 1, and L2’ is the line y = 2. Using HHA 6, a fold
can be made that aligns P1 with L1’ and P2 with L2’, which will intersect the x- and y-axes at two
points, X and Y, respectively. Calling the origin O, notice that the right triangle OP1Y is similar
to the two triangles OXY and OP2X, since XY is perpendicular to P1FLF(P1) and P2FLF(P2). Thus,
|OY|
|OX|
we know that |OP | = |OY| =
1
|OX|
|OP2 |
|OX|
. We know |OP1| = 1 and |OP2| = 2, and therefore substituting
2
3
yields |OY| = |OY| = |OX|. Solving this for |OY| yields |OY|3 = 2, and thus |OY| = √2 and Y =
3
3
(0, √2). Note that with an arbitrary P = (0, −k), Y will have coordinates (0, √k), allowing the
construction of the cube root of any real (Hull, 2011).
3
Fig. 2: Using Beloch’s Square to Construct √2
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IV.ii The Trisection of an Angle
Based on the work of Hull, Alperin, and Lang, Hisashi Abe developed a method of
trisecting any given angle θ using single-fold operations in 1980 (Lee, 2017). As with the
doubling of the cube, this was a problem attempted by the Greeks, but left unsolved, as its
generalization requires the construction of solutions to a cubic equation, which cannot be done
with a compass and straightedge.
Given an arbitrary angle θ represented by two lines L1 and L2 intersecting at point P in
the plane, position L2 horizontally and fold a perpendicular to it passing through P. This new
line, L3, can be thought of as our y-axis, and L2 as our x-axis. Let Q be an arbitrary point on L3.
Fold a perpendicular, LP, to L3 through Q, and then make the fold that aligns LP with L2 – since
LP and L2 are parallel, this will construct a line L4 that is also parallel to the two lines and
halfway between them. Next, HHA 6 is used to align Q onto L1 and P onto L4. The fold, LF, that
accomplishes this is unique, and the line connecting P and its folded image from this operation
θ
will form an angle of 3 with L2.
To prove this, we will consider P’ and Q’ to be the folded images of P and Q across LF.
Let R be the intersection of PQ’ and P’Q, S be the intersection of L3 and L4, and T be the
intersection of PP’ and LF. We know that R is aligned with LF since P’ and Q’ are the reflections
of P and Q across LF, and we also know that |PS| = |QS|. Further, since P’ lies on L4, we know
P’QP is an isosceles triangle. Also, |PT| = |P’T| since P’ is the reflection of P across LF, and since
R lies on a perpendicular equidistant from P and P’, RPP’ is also an isosceles triangle. letting
m∠RPP′ = 𝛼, 𝑚∠𝑃𝑃′𝑆 = 𝛽, & 𝑚∠𝑅𝑃′𝑆 = 𝛾, we then know that 𝛽 = 𝛾 and 𝛼 = 𝛽 + 𝛾. Since L2
and L4 are parallel, the angle formed by PP’ and L2 also has measure 𝛽. Therefore, 𝜃 = 𝛼 + 𝛽 =
1
(𝛽 + 𝛾) + 𝛽 = 3𝛽, and so 𝛽 = 3 𝜃.
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Fig. 3: Trisection of an Arbitrary Angle
While the process described assumes an acute angle θ, the method can be extended to
obtuse angles by dividing said angle into a right angle and an acute angle, for example by folding
a perpendicular to one of the initial rays defining θ (Lee, 2017). Following the trisection process
on the two new angles will construct the trisection of θ between the two respective segments,
P1P1’ of the acute angle, and P2P2’ of the right angle.
IV.iii The Real Solutions of a Cubic
The sixth Huzita-Hatori Axiom may also be called the Beloch Fold, as it was
independently discovered in 1936 by Margharita P. Beloch, who was the first to realize and
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prove that origami constructions can find the real solutions of general cubic equations (Hull,
2011).
First, consider an arbitrary fold line LF that aligns a given point P and line L, given P and
L are not already aligned. This fold aligns some point P’ on L with P. A perpendicular of L
aligned with P’ will necessarily intersect with LF at some point X. Note that, since P’ is the
folded image of P across line LF, the point X on LF must be equidistant from P and P’. Since the
line segment between P’ and X is perpendicular to L, consequently X is equidistant from P and
L, and thus is a point on the parabola defined by focus P and directrix L. Thus, any arbitrary fold
that aligns P and L will be a tangent line of this parabola (Lucero, 2017).
Since HHA 6 aligns two given points, P1 and P2, with two given lines, L1 and L2, it can
be thought of as finding a single line, L’, that is tangent to the two parabolas defined by foci P1
and P2, and directrices L1 and L2, respectively (Lucero, 2017). As mentioned, two parabolas in
the plane can have at most three unique common tangents, suggesting that this operation is
equivalent to solving a general cubic equation.
Beloch’s development of a method to construct real roots relies on an 1867 paper by
Eduard Lill, an Austrian engineer (Hull, 2011). This paper contains a geometric procedure for
finding real roots of any given polynomial. An abridged description of Lill’s method follows.
Given some polynomial 𝑓(𝑥) = 𝑎0 + 𝑎1 𝑥1 + 𝑎2 𝑥 2 +. . . +𝑎𝑛 𝑥 𝑛 , we can geometrically
create a path representing the coefficients of 𝑓(𝑥), and in so doing find the real solutions of the
polynomial. To construct this path, begin at the origin and face in the positive x-direction, and
move a distance of an. Then, turn 90° counterclockwise and move a distance of an-1. This process
repeats until the final segment, of distance a0, is traveled, at which point the path ends at some
point T. For any negative coefficients, movement is opposed to the direction faced. Then,
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beginning at the origin, a line at angle θ from the positive x-axis is extended until it intersects
with the extension of the segment corresponding to an-1, at which point it will turn 90° towards
the segment corresponding to an-2. This process repeats until the new path intersects the line
extending from the a0 segment. If this new path also ends at the point T, then x = -tanθ is a root
of f(x) (Hull, 2011). The proof of this, which relies on the recognition that the second path is
composed of hypotenuses of similar right triangles, whose legs lie exclusively along lines
defined by the first path, has been omitted for brevity, but is left as an exercise to the reader.
Fig. 4: Lill’s Method for Cubic Polynomials (Note that a1 < 0)
Lill’s method can be applied to polynomials of any degree, but Beloch recognized the
power of HHA 6 to execute Lill’s method for cubic equations, and therefore to solve the general
cubic. First, Lill’s method is followed, constructing a path of four segments corresponding to a3,
a2, a1, and a0 of the given cubic equation. Considering the origin as P1, and the ending point T as
P2, and further considering the lines extending from the a2 segment and a1 segment as L1 and L2,
respectively, then the sides of the Beloch Square (see IV.i) constructed from these givens will lie
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along the lines forming the second path, thus constructing the desired angle θ with the a3 segment
extending from the origin (Hull, 2011). Folding a perpendicular to the a3 segment of unit
distance from the origin then results in a construction of |x|, where x is a solution of the given
cubic polynomial. Since a maximum of three unique Beloch Squares can be constructed from a
given set of two points and two lines, all real roots of any cubic may be found with this method.
V. Two-Fold Axioms and More
It does well to establish that the Huzita-Hatori Axioms do not encompass all possible
operations in origami – rather, they describe only those performed with a single fold. Allowing
two, three, or more simultaneous folds to be made drastically increases the number of possible
operations – as noted before, the exhaustive list of two-fold operations contains 489 distinct
combinations (Alperin & Lang, 2006). The full list of these is omitted, though it is notable that
just two such operations, in combination with the seven HHAs, allow for the solving of some
quintic polynomials and the quintisection of an angle.
One limitation of two-fold axioms is that it is unknown whether or not they allow for the
solution of a general quintic, only a select subset of them. However, it can be shown that any
quintic polynomial can be solved by expanding to three-fold axioms. The method devised by
Alperin and Lang to find such solutions again relies on Lill’s Method, and it has been proven that
the application of Lill’s Method using N-Fold axioms allows for the solution of general
polynomials of degree N + 2.
In considering two- and three-fold axioms, it is clear that the theory yields broad
consequences in angle division and polynomial solutions, but even two-fold axioms become
increasingly impractical as a physical method for solving quartic equations, as methods to
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perform such operations simply do not exist in most cases (Alperin R. C., 2000). As a
consequence, the theory developed for single-fold operations has far more meaningful
implications for practical mathematics – that is, for defining and exploring the physical limits of
origami in the real world.
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