An Origami-Constructible Number is defined to be a real number r where two points of distance apart can be constructed in a finite number of steps. [1] This can be defined by the set such that. In addition, we can say that any element is Origami-Constructible if and only if is also Origami-Constructible. We can further define Origami-Constructible points as the set and is closed under origami constructions, where are points from the Cartesian coordinate system in the plane. [2]
Defining an origami pair , where is a set of points in and is a collection of lines in , is more complicated since we need to ensure that a set of conditions are satisfied. These conditions are:
Origami spans over 1,000 years and is the art of Japanese paper folding. [3] However, Origami was first referenced in China when paper was introduced in 200AD as a cheaper alternative to silk and was known as Zhezhi. It was brought to Japan by Monks in the 6th century and became popular from then on. The word "Origami" is from the Japanese language where "ori" means fold and "kami" (or "gami") means paper. [4] Paper-folding was restricted to formal ceremonies and religious rituals since paper was expensive. [5] Origami was linked with Mathematics when Akira Yoshizawa used it to help teach children geometry. He realised that it would help them understand geometrical problems, such as angles, lines and shapes, in a simpler form. Akira Yoshizawa continued to enjoy origami and developed a technique of wet folding which allowed for much more intricate designs to be created, these advances interested mathematicians who also recognised the link between origami and geometry. [4]
After the link between origami and mathematics was found, Humiaki Huzita, a Japanese-Italian mathematician, founded six single-fold axioms in 1989. A further seventh axiom was discovered by Koshiro Hatori in 2003. Hence these seven axioms were to be known as the Huzita-Hatori axioms. However, it was later found that Jacques Justin had discovered these same seven axioms in 1986 but they were overlooked, so Huzita and Hatori had just rediscovered them. Huzita was born in Japan but emigrated to Italy to study nuclear physics at University of Padua. [1]
Although the Huzita-Hatori axioms are an important aspect of origami geometry, especially in solving two of the Greek problems: Trisecting the Angle and Doubling the Cube, this would not be possible without the discovery of the Beloch fold. Margherita Piazzolla Beloch was able to use her Beloch fold and Beloch square to construct solutions to cubic equations. However, Beloch needed to use Eduard Lill's method of finding real roots of polynomials of any degree before she could consider solving cubic equations. It was in 1936 when Beloch was able to extend Lill's method to show that we can solve cubic equations using purely origami. [6]
A Japanese Astrophysicist, Koryo Miura, developed an origami design where you can fold a flat surface, namely a piece of paper, into a tesselation of parallelograms and each part of the fold remains flat. [7] This method of paper folding was given the name of the Miura Fold, after the founder, and has since been used for modern day developments. In 1995, Japan's Space Flyer Unit used the Miura Fold to deploy its solar panels. This choice of design method was possible since the Miura Fold is easily expanded and collapsed and hence the solar panels can be collapsed for the launch of the Space Unit and expanded once in space. [8]
Robert J Lang, an American Physicist, is the author who showed that the Huzita-Hatori axioms are complete, that is, there are no other folds in origami construction. [9] Lang is also an origami artist and considered one of the pioneers of modern maths-based origami. He was intrigued by the idea of a computer being able to design an origami model and so in 1990 he created a computer program called TreeMaker. Lang continued to develop this program and by 1998 it was at an advanced stage and was capable of constructing full crease patterns for a range of designs. [4]
Euclid, a Greek Mathematician, is widely known for being the "founder of geometry" and wrote 'The Elements', one of the leading books in the history of mathematics. Within this book, Euclid stated five assumptions which he used to solve geometric constructions, these assumptions are now more commonly known as Euclid's Axioms. Although these axioms can be used to solve some of the most complex geometric problems, they were unable to provide a solution to two of the most famous Greek problems: Trisecting the Angle and Doubling the Cube.
Similar to Euclid's axioms devised for planar geometry, there are a set of axioms to describe origami geometry. These axioms were developed by mathematicians Humiaki Huzita and Koshiro Hatori and are called the Huzita-Hatori axioms. [4]
The axioms are as follows:
Properties in origami geometry can now be solved using purely the Huzita-Hatori axioms.
It is possible to add two given lengths and using the Huzita-Hatori axioms. Axiom O3 is used to fold the first line onto the second, which in turn will place the line segment somewhere on that line. Now copying the line segment to a particular point on the line will allow for the other end of this line segment to lie in the preferred direction. Axiom O4 is now used to create a perpendicular fold through one of the endpoints of the line segment. Then we copy the line segment to that part of the line and unfold. This allows our desired point to be free and untouched by the line segment. Using Axiom O2, we can fold the near endpoint of the line segment to the desired point. We can now use Axiom O4 if our line segment is not in the desired direction, simply fold the line segment in the other direction using Axiom O4 through our desired point.
Similarly, to subtract two given lengths and , we want to move the line segment on top of the other in order to find the difference in their lengths. [10]
It is possible to multiply two given lengths and using the Huzita-Hatori axioms. We begin by placing the two lengths so that they form an acute angle, this can be done using the same method described in the addition of two lengths. Now, we want one end of the unit length segment to lie at our angle vertex, and to do this, we need to copy the unit length segment onto the line which contains line segment . Axiom O1 is then used to create a new line from the end of the line segment to the end of the unit line segment. Axiom O4 is now used, twice, to construct a parallel line which goes through the point lying at the end of the line segment . The line segment which contains should now intersect this constructed parallel line. Now, from similar triangles, the length from our original vertex to this intersection is exactly .
Similarly, to divide two given lengths and , we use Axiom O1 to join the end of the line segment to the end of the line segment . Again, we construct a parallel line which goes through the end of the unit line segment. The line segment which contains should now intersect this constructed parallel line and this intersection point is the end point of a line segment which has length . [10]
It is possible to find the square root of a length using the Huzita-Hatori axioms. We start by considering the parabola as this will help us to find the square root of our length. We copy our length onto the -axis and using Axiom O4, twice, we can construct the horizontal line . Next we can use Axiom O5 to make a fold through point which takes the endpoint to our horizontal line . There are two possibilities where our image point can lie, on the horizontal line , since it will lie on the parabola and the distance between the image point and our point will be (the same as the distance between the image point and the point ). Hence, the distance from the two possible image points to the -axis () is exactly . [10]
Robert J. Lang elevated the art form of origami by using his extensive knowledge, as an American Mathematical Society member [15], to practical applications in modern technology. Examples include a laser cutter which is used to score paper to assist in complicated folds [16] and developing designs of the folding patterns that would be used by a manufacturer of a German airbag [17]. Moreover, Lang also worked with a team of fellow scientists to solve the problem of creating a space telescope that would have a lens measuring 100 m, which became known as the Eyeglass telescope, such that it could fold compactly and be placed into a small rocket without causing damage to it or compromising its quality and purpose [18].
In addition, Lang created a computer program, TreeMaker, which uses descriptions of numbers, lengths and the connectedness of the flaps to design origami figures. Lang stated that "TreeMaker could solve crease patterns that I couldn't construct by pencil and paper". In order to create your origami base, you first need to create a stick figure of your base where each stick will be represented by a flap. You then place relevant constraints on these flaps in order for TreeMaker to compute the crease pattern. [19]
If ruler and compass constructible, the set of constructible numbers, and origami constructible, the set of origami-constructible numbers, then we have that [20], where is the set of rational numbers, is the set of algebraic numbers and is the set of real numbers. This chain of containment is depicted in the diagram on the right.
Given a unit length, a real number is constructible if and only if a line segment of length can be constructed from the unit length in a finite number of steps using only a ruler (i.e. a straight unmarked edge) and pair of compasses. It is clear that 0 and 1 are constructible and it can be shown that a real number is constructible if and only if is constructible. [21]
Origami-Constructible Numbers are a powerful field of mathematics and are somewhat more powerful than Constructible Numbers. All numbers that are constructible using a ruler and compass are also constructible using origami. [22] In addition, steps and processes which are allowed in ruler and compass constructions can also be performed by origami. The Huzita-Hatori Axioms describe origami geometry but Axioms 1-5 can also be performed using only ruler and compass constructions. [20]
In 1837 it was proved by Pierre Laurent Wantzel, a French Mathematician, that Doubling the Cube and Trisecting the Angle was impossible using only straightedge and compass construction. However, when Margherita Piazzolla Beloch showed that Origami can be used to solve cubic equations (and hence quartic equations) it became known that it was also possible to solve two of the classical Greek problems that the straightedge and compass constructions could not. With the help of Beloch's findings and Axiom 6 of the Huzita-Hatori axioms, it is possible to trisect angles, double cubes, and otherwise solve cube roots using origami. [23]
This simply means finding the real root to the cubic equation , and hence constructing the cube root of 2, .
Steps taken to solve the problem of Doubling the Cube using Origami:
Steps taken to solve the problem of Angle Trisection when the angle is acute using Tsune Abe's method:
In 1936 Margharita Piazolla Beloch, an Italian Mathematician, found that Origami constructions can be used to solve general cubic equations. Hence, it can be concluded that Origami constructions are more powerful than straightedge and compass constructions. It must be remembered that Beloch's discovery would not have been possible without Eduard Lill, an Austrian Engineer, and his discovery of Lill's Method. [26]
The Beloch Origami Fold is an axiom or folding move considered to set Origami apart from straightedge and compass constructions.
This method is used to find a real root, if one exists, of a given polynomial which has real coefficients. Lill's method is graphical whereby based on the coefficients of , you create a path in the plane. [26]
Given two points and and two lines and we can, whenever possible, make a single fold that places onto and onto simultaneously. [26]
Given two points and and two lines and in the plane, construct a square with two adjacent corners and lying on and , respectively, and the sides and , or their extensions, passing through and , respectively. [26]
Squaring the circle is the problem of constructing a square of equal area to the unit circle, which has radius of length 1 and therefore area equal to . Then in order for the area of the square to be , each side must have length . Hence one needs to prove that is it impossible to construct a line segment of length in order to prove that squaring the circle is impossible. [27]
The Greeks did know that the problem was impossible, but they were unable to prove so. In 1761, it was proved by John Heinrich Lambert that is irrational, and it was proposed as conjectures that and were both transcendental numbers in his 1768 paper. Ferdinand von Lindemann, a German mathematician, published his first complete proof in 1882 for the transcendence of , but in 1880, Lindemann proved that is transcendental, using Euler's identity (), and the fact that is transcendental. This meant that could not be the root of any non-zero polynomial which has rational coefficients [28] and that is not constructible, so it is impossible to construct a line segment of length or , and hence squaring the circle is impossible using straightedge and compass construction.
Since is still a transcendental number in the set of origami-constructible numbers, then the problem of squaring the circle is also impossible using origami. [29]
Even though one is unable to construct a line segment of length exactly equal to , the following are two examples of very close approximations:
Besides using straightedge and compass construction, there are other methods that does allow the problem of squaring the circle to be possible, and we will discuss one of these called "rolling a circle". If a unit circle is rolled halfway along a horizontal, straight line, beginning at point and ending up at point , then because is equal to half of the circumference of the unit circle which is then . Next, allow segment to be extended to the point so that we have , and construct a semicircle with diameter . Then let be the point of intersection of the semicircle with a line drawn through that is perpendicular to . We see that together with and implies that . Hence if one constructs a square with as one of the sides, then the area of this square is exactly equal to the area of a unit circle. [30]
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